3554 lines
204 KiB
Text
3554 lines
204 KiB
Text
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Calculations to find the quadratic form of the octahedral packing"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$P$ is the matrix of the quadratic form corresponding to $h_1^2 + h_2^2 - \\tilde{b}b = 1$."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, \\pi - \\frac{2 \\, R}{r {\\left(\\frac{R^{2}}{r^{2}} + 1\\right)}} + 2 \\, \\arctan\\left(\\frac{R}{r}\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2 \\, \\pi - \\frac{2 \\, R}{r {\\left(\\frac{R^{2}}{r^{2}} + 1\\right)}} + 2 \\, \\arctan\\left(\\frac{R}{r}\\right)$$"
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],
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"text/plain": [
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"-2*pi - 2*R/(r*(R^2/r^2 + 1)) + 2*arctan(R/r)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"var('r, R')\n",
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"d = diff(2 * atan(R / r) * r - 2 * pi * r, r)\n",
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"show(d)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"scrolled": true
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},
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"outputs": [],
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"source": [
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"P = matrix([\n",
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" [0, -1/2, 0, 0],\n",
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" [-1/2, 0, 0, 0],\n",
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" [0, 0, 1, 0],\n",
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" [0, 0, 0, 1],\n",
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"])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}2$$"
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],
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"text/plain": [
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"2"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,0,\\,0,\\,0\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,0,\\,0,\\,0\\right)$$"
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],
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"text/plain": [
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"(1, 0, 0, 0)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,1,\\,0,\\,0\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,1,\\,0,\\,0\\right)$$"
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],
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"text/plain": [
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"(0, 1, 0, 0)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,0,\\,1,\\,0\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,0,\\,1,\\,0\\right)$$"
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],
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"text/plain": [
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"(0, 0, 1, 0)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,0,\\,0,\\,1\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,\\,0,\\,0,\\,1\\right)$$"
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],
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"text/plain": [
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"(0, 0, 0, 1)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"mat = matrix([\n",
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" [2, -2, -2, -2],\n",
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" [-2, 2, -2, -2],\n",
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" [-2, -2, 2, -2],\n",
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" [-2, -2, -2, 2]\n",
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"])\n",
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"\n",
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"a1 = vector([1, 0, 0, 0])\n",
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"a2 = vector([0, 1, 0, 0])\n",
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"a3 = vector([0, 0, 1, 0])\n",
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"a4 = vector([0, 0, 0, 1])\n",
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"\n",
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"av1 = 2 / (a1 * mat * a1) * a1\n",
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"av2 = 2 / (a2 * mat * a2) * a2\n",
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"av3 = 2 / (a3 * mat * a3) * a3\n",
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"av4 = 2 / (a4 * mat * a4) * a4\n",
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"\n",
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"show(a1 * mat * a1)\n",
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"\n",
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"show(av1)\n",
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"show(av2)\n",
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"show(av3)\n",
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"show(av4)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 0 -2 0 0]\n",
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"[-2 0 0 0]\n",
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"[ 0 0 1 0]\n",
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"P.inverse()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"For an Apollonian packing, you can make a matrix, $W$, where the rows are the coordinates of each of the circles in a quadruple. In an octahedral packing, this is impossible, as there are six circles in a unit. You can try creating a $6\\times 4$ matrix, but, later, we end up needing to invert $WPW^T$, the product of a $6\\times 4$, $4\\times 4$, and $4\\times 6$ matrix, which is singular. Fortunately, the sextuples come in three pairs of circles. Each pair consists of circles that aren't tangent to each other. It turns out that the average of the coordinates of the circles in a pair is the same across the sextuple. So, we can make a matrix where the first three rows are the coordinates of circles from different pairs, i.e. three mutually tangent circles, and the fourth is the average of the coordinates in a pair. From this, you can recover the coordinates for all the circles in a sextuple.\n",
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"\n",
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"Here $W$ is such a matrix computed for the $(0, 0, 1, 1, 2, 2)$ root sextuple."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"W = matrix([\n",
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" [2, 0, 0, 1],\n",
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" [2, 0, 0, -1],\n",
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" [-1, 1, 0, 0],\n",
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" [6, 2, 2*sqrt(2), 0]\n",
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"])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here we have $$\n",
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" WPW^T = \\left(\\begin{matrix}\n",
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" 1 & -1 & -1 & -1\\\\\n",
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" -1 & 1 & -1 & -1\\\\\n",
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" -1 & -1 & 1 & -1\\\\\n",
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" -1 & -1 & -1 & -1\n",
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" \\end{matrix}\\right) = M\n",
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"$$\n",
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"So, inverting both sides, we get $$\n",
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" (WPW^T)^{-1} = M^{-1}\n",
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"$$ $$\n",
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" (W^T)^{-1}P^{-1}W^{-1} = M^{-1}\n",
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"$$ $$\n",
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" P^{-1} = W^TM^{-1}W\n",
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".$$\n",
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"\n",
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"Like in the case with Apollonian packings, this is true for any sextuple $W$. So, we can substitute an arbitrary sextuple for $W$ and it must be equal to $P^{-1}$, letting us derive some useful quadratic forms."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 1 -1 -1 -2]\n",
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"[-1 1 -1 -2]\n",
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"[-1 -1 1 -2]\n",
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"[-2 -2 -2 -4]"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"M = W * P * W.transpose()\n",
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"M"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$M^{-1}$ is the matrix of the quadratic form, although it is nice to normalize it to have ones along the diagonal, which is fine since it is equal to zero.\n",
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"$$\n",
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"\\left(\\begin{matrix}\n",
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" 1 & 0 & 0 & -1/2\\\\\n",
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" 0 & 1 & 0 & -1/2\\\\\n",
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" 0 & 0 & 1 & -1/2\\\\\n",
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" -1/2 & -1/2 & -1/2 & 1/4\n",
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"\\end{matrix}\\right)\n",
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"$$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 1/2 0 0 -1/4]\n",
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"[ 0 1/2 0 -1/4]\n",
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"[ 0 0 1/2 -1/4]\n",
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"[-1/4 -1/4 -1/4 1/8]"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"M.inverse()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"bt1 = var('bt1')\n",
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"b1 = var('b1')\n",
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"h11 = var('h11')\n",
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"h12 = var('h12')\n",
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"bt2 = var('bt2')\n",
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"b2 = var('b2')\n",
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"h21 = var('h21')\n",
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"h22 = var('h22')\n",
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"bt3 = var('bt3')\n",
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"b3 = var('b3')\n",
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"h31 = var('h31')\n",
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"h32 = var('h32')\n",
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"b5_avg = var('b5_avg')\n",
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"b_avg = var('b_avg')\n",
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"h1_avg = var('h1_avg')\n",
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"h2_avg = var('h2_avg')\n",
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"\n",
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"\n",
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"W2 = matrix([\n",
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" [bt1, b1, h11, h12],\n",
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" [bt2, b2, h21, h22],\n",
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" [bt3, b3, h31, h32],\n",
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" [b5_avg, b_avg, h1_avg, h2_avg],\n",
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"])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"D = W2.transpose() * M.inverse() * W2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 0 -2 0 0]\n",
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"[-2 0 0 0]\n",
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"[ 0 0 1 0]\n",
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"P.inverse()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"b1^2 + b2^2 + b3^2 - b1*b_avg - b2*b_avg - b3*b_avg + 1/4*b_avg^2"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"2 * factor(simplify(D[1][1]))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"So, we end up deriving the quadratic form $$\n",
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" b_1^2 + b_2^2 + b_3^2 + b_{\\text{sum}}^2/4 - b_{\\text{sum}}(b_1 + b_2 + b_3) = 0\n",
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".$$\n",
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"\n",
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"This means that, given three mutually tangent circles with curvatures $b_1,b_2,b_3$, there are two solutions for $b_{\\text{avg}}$, allowing us to derive two new sets of three mutually tangent circles with curvatures $b_1' = 2b_{\\text{avg}} - b_1$ etc."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Calculations to find the generators for the \"Apollonian\" Group for the octahedral packing\n",
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"\n",
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"We can try the Weyl group, since that is one way to derive the generators for the Apollonian group and see if it works. My worry is that it won't since the coordinate system is so different."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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"def weyl_generators(matrix, alphas):\n",
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" retval = []\n",
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" for alpha in alphas:\n",
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" scale_factor = (alpha.transpose() * matrix * alpha)[0][0]\n",
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" retval.append(identity_matrix(len(alphas)) - 2 * alpha * alpha.transpose() * matrix / scale_factor)\n",
|
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" return retval"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [],
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"source": [
|
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"def standard_basis(dim):\n",
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" return [ matrix(dim, 1, [0] * i + [1] + [0] * (dim - i - 1)) for i in range(dim) ]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 2 0 0 -1]\n",
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"[ 0 2 0 -1]\n",
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"[ 0 0 2 -1]\n",
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"[ -1 -1 -1 1/2]"
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]
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},
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"4 * M.inverse()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[\n",
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"[-1 0 0 1] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [ 0 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [ 0 0 -1 1] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 4 4 4 -1]\n",
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"]"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
|
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"S_i = weyl_generators(4 * M.inverse(), standard_basis(4))\n",
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"S_i"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"metadata": {},
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"outputs": [],
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"source": [
|
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"S1 = S_i[0]\n",
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"S2 = S_i[1]\n",
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"S3 = S_i[2]\n",
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"S4 = S_i[3]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
|
|
"We can test this out on the packing on page 3 of the Guettler and Mallows. The root sextuple is (-1, 2, 2, 4, 4 7), so the coordinates would be (-1, 2, 4, 6). After applying $s_1$, it should swap out $s_1$ for its pair, resulting in (7, 2, 4, 6). Likewise for $s_2$ and $s_3$. Then, for $s_4$, it should give the average between the triple (-1, 2, 4) and the other tangent triple, i.e. (-1, 2, 4, 14)."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 17,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(7, 2, 4, 6)"
|
|
]
|
|
},
|
|
"execution_count": 17,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"root = vector([-1, 2, 4, 6])\n",
|
|
"S1 * root"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 18,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(-1, 4, 4, 6)"
|
|
]
|
|
},
|
|
"execution_count": 18,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"S2 * root"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 19,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(-1, 2, 2, 6)"
|
|
]
|
|
},
|
|
"execution_count": 19,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"S3 * root"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 20,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"(-1, 2, 4, 14)"
|
|
]
|
|
},
|
|
"execution_count": 20,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"S4 * root"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Calculations to find the curvatures of the dual circles of the octahedral packing\n",
|
|
"\n",
|
|
"Here the basic idea was to work out the intersection points of the circles and let the computer algebraically find the radii of the circles. The end up coming out, rather anticlimactically, to $(0, 0, \\sqrt 2, \\sqrt 2, \\sqrt 2, \\sqrt 2, 2\\sqrt 2, 2\\sqrt 2)$. It's rather satisfying and a bit surprising that the dual of the root sextuple of the octahedral packing is the root of the cubic packing, since the dual polyhedron of the octahedron is the cube."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 21,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def circle_from_points(pta, ptb, ptc):\n",
|
|
" a = var('a')\n",
|
|
" b = var('b')\n",
|
|
" r = var('r')\n",
|
|
" x = var('x')\n",
|
|
" y = var('y')\n",
|
|
" \n",
|
|
" circle_func = (x - a)^2 + (y - b)^2 == r^2\n",
|
|
" \n",
|
|
" eq1 = circle_func.subs(x == pta[0]).subs(y == pta[1])\n",
|
|
" eq2 = circle_func.subs(x == ptb[0]).subs(y == ptb[1])\n",
|
|
" eq3 = circle_func.subs(x == ptc[0]).subs(y == ptc[1])\n",
|
|
" \n",
|
|
" res = solve([eq1, eq2, eq3], a, b, r)[1]\n",
|
|
" \n",
|
|
" return (res[0].rhs(), res[1].rhs(), res[2].rhs())"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 22,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def abbc_coords(b, h1, h2):\n",
|
|
" return [(h1^2 + h2^2 - 1) / b, b, h1, h2]"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 63,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(1/2*sqrt(2), 1)\n",
|
|
"sqrt(2)\n",
|
|
"(1/4*sqrt(2), 0)\n",
|
|
"2*sqrt(2)\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"image/png": 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\n",
|
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"text/plain": [
|
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"Graphics object consisting of 14 graphics primitives"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"c1 = line([(-3, 1), (3, 1)])\n",
|
|
"c2 = line([(-3, -1), (3, -1)])\n",
|
|
"c3 = circle((-sqrt(2), 0), 1)\n",
|
|
"c4 = circle((sqrt(2), 0), 1)\n",
|
|
"c5 = circle((0, 1/2), 1/2)\n",
|
|
"c6 = circle((0, -1/2), 1/2)\n",
|
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"\n",
|
|
"b1 = line([(-sqrt(2), -3), (-sqrt(2), 3)], rgbcolor=(1, 0, 0))\n",
|
|
"b2 = line([(sqrt(2), -3), (sqrt(2), 3)], rgbcolor=(1, 0, 0))\n",
|
|
"\n",
|
|
"x, y, r = circle_from_points((sqrt(2) / 3, 1 / 3), (0, 1), (sqrt(2), 1))\n",
|
|
"print('({}, {})'.format(x, y))\n",
|
|
"print(1 / r)\n",
|
|
"\n",
|
|
"btest = []\n",
|
|
"coordsnew = abbc_coords(1/r, x/r, y/r)\n",
|
|
"bnew = coordsnew[0]\n",
|
|
"btest.append(circle((coordsnew[2]/bnew, coordsnew[3]/bnew), 1/bnew))\n",
|
|
"coordsnew = abbc_coords(1/r, -x/r, y/r)\n",
|
|
"bnew = coordsnew[0]\n",
|
|
"btest.append(circle((coordsnew[2]/bnew, coordsnew[3]/bnew), 1/bnew))\n",
|
|
"coordsnew = abbc_coords(1/r, x/r, -y/r)\n",
|
|
"bnew = coordsnew[0]\n",
|
|
"btest.append(circle((coordsnew[2]/bnew, coordsnew[3]/bnew), 1/bnew))\n",
|
|
"coordsnew = abbc_coords(1/r, -x/r, -y/r)\n",
|
|
"bnew = coordsnew[0]\n",
|
|
"btest.append(circle((coordsnew[2]/bnew, coordsnew[3]/bnew), 1/bnew))\n",
|
|
"\n",
|
|
"#show(sum(btest))\n",
|
|
"\n",
|
|
"b3 = circle(( x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b4 = circle((-x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b5 = circle(( x, -y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b6 = circle((-x, -y), r, rgbcolor=(1, 0, 0))\n",
|
|
"\n",
|
|
"x, y, r = circle_from_points((sqrt(2) / 3, 1 / 3), (0, 0), (sqrt(2) / 3, -1 / 3))\n",
|
|
"print('({}, {})'.format(x, y))\n",
|
|
"print(1 / r)\n",
|
|
"\n",
|
|
"b7 = circle(( x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b8 = circle((-x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"\n",
|
|
"show(c1 + c2 + c3 + c4 + c5 + c6 + b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# The quadratic form for the cubical packing\n",
|
|
"\n",
|
|
"We first made a matrix, $W_c$, whose rows are the abbc coordinates of the circles in the root octuple. Then we found the row echelon form of that matrix, resulting in a system of linear relations the coordinates must satisfy. Then, from there, we could derive the rest of the coordinates from the first four (we chose the first four to be the \"cubicle\" from the Stange), allowing us to derive the quadratic form."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"The idea here is that by multiplying $W_c$ by an arbitrary vector in $\\mathbf{R}^8$ and setting that equal to $\\vec{0}$, we can find a basis for the null space of $W_c$, which will end up giving us a bunch of linear relations the curvatures must satisfy. The thing to notice is that $W_c\\vec{v} = 0$ for some $\\vec{v}\\in\\mathbf{R}^8$ gives the system of equations with coefficient matrix $W^T$. So finding the row echelon form of $W^T$ will give us the coefficients in the simplified system of linear relations, with each row equal to 0."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 24,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[ 1 0 0 0 1/2 1/2 1/2 -1/2]\n",
|
|
"[ 0 1 0 0 1/2 1/2 -1/2 1/2]\n",
|
|
"[ 0 0 1 0 1/2 -1/2 1/2 1/2]\n",
|
|
"[ 0 0 0 1 -1/2 1/2 1/2 1/2]"
|
|
]
|
|
},
|
|
"execution_count": 24,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"Wc = matrix([\n",
|
|
" [4, 0, 0, 1],\n",
|
|
" [0, 2, 0, 1],\n",
|
|
" [2, 1, -sqrt(2), -1],\n",
|
|
" [2, 1, sqrt(2), -1],\n",
|
|
" [2, 1, -sqrt(2), 1],\n",
|
|
" [2, 1, sqrt(2), 1],\n",
|
|
" [4, 0, 0, -1],\n",
|
|
" [0, 2, 0, -1],\n",
|
|
"])\n",
|
|
"\n",
|
|
"Wc.transpose().rref()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"From here we can conclude that $$\n",
|
|
"\\begin{align*}\n",
|
|
" 2b_1 &= -b_5 - b_6 - b_7 + b_8\\\\\n",
|
|
" 2b_2 &= -b_5 - b_6 + b_7 - b_8\\\\\n",
|
|
" 2b_3 &= -b_5 + b_6 - b_7 - b_8\\\\\n",
|
|
" 2b_4 &= b_5 - b_6 - b_7 - b_8\n",
|
|
"\\end{align*}\n",
|
|
"$$\n",
|
|
"This differs slightly from Stange's system of equations because we put the circles in a slightly different order."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"In any case, this gives us the tools to derive the full octuple from just four coordinates, letting us use those four coordinates to represent the entire octuple, and finally making $WPW^T$ nonsingular, allowing us to find the quadratic form the same way we did for the Descartes quadratic form and the octahedral quadratic form."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 25,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"W = matrix([\n",
|
|
" [4, 0, 0, 1],\n",
|
|
" [0, 2, 0, 1],\n",
|
|
" [2, 1, -sqrt(2), -1],\n",
|
|
" [2, 1, sqrt(2), -1],\n",
|
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"])"
|
|
]
|
|
},
|
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{
|
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"cell_type": "code",
|
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"execution_count": 26,
|
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"metadata": {},
|
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"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
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"[ 1 -3 -3 -3]\n",
|
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"[-3 1 -3 -3]\n",
|
|
"[-3 -3 1 -3]\n",
|
|
"[-3 -3 -3 1]"
|
|
]
|
|
},
|
|
"execution_count": 26,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"m = W * P * W.transpose()\n",
|
|
"m"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 27,
|
|
"metadata": {},
|
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"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[ 2 -6/5 -6/5 -6/5]\n",
|
|
"[-6/5 2 -6/5 -6/5]\n",
|
|
"[-6/5 -6/5 2 -6/5]\n",
|
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"[-6/5 -6/5 -6/5 2]"
|
|
]
|
|
},
|
|
"execution_count": 27,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"32 * m.inverse() * 2/5"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 28,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[bt1 b1 h11 h21]\n",
|
|
"[bt2 b2 h12 h22]\n",
|
|
"[bt3 b3 h13 h23]\n",
|
|
"[bt4 b4 h14 h24]"
|
|
]
|
|
},
|
|
"execution_count": 28,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"W2 = matrix([\n",
|
|
" [\n",
|
|
" var('bt' + str(i)),\n",
|
|
" var('b' + str(i)),\n",
|
|
" var('h1' + str(i)),\n",
|
|
" var('h2' + str(i)),\n",
|
|
" ] for i in range(1, 5)\n",
|
|
"])\n",
|
|
"W2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 29,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"D = W2.transpose() * m.inverse() * W2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 30,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"5*b1^2 - 6*b1*b2 + 5*b2^2 - 6*b1*b3 - 6*b2*b3 + 5*b3^2 - 6*b1*b4 - 6*b2*b4 - 6*b3*b4 + 5*b4^2"
|
|
]
|
|
},
|
|
"execution_count": 30,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"32 * simplify(expand(D[1][1]))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"This spits out the lovely quadratic form $$\n",
|
|
" 8(b_1^2 + b_2^2 + b_3^2 + b_4^2) = 3(b_1 + b_2 + b_3 + b_4)^2\n",
|
|
".$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Here's the generators for the Weyl group. They might or might not be generators for the packing itself, however."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 31,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[\n",
|
|
"[ -1 6/5 6/5 6/5] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [6/5 -1 6/5 6/5] [ 0 1 0 0]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0] [6/5 6/5 -1 6/5]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],\n",
|
|
"\n",
|
|
"[ 1 0 0 0]\n",
|
|
"[ 0 1 0 0]\n",
|
|
"[ 0 0 1 0]\n",
|
|
"[6/5 6/5 6/5 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"execution_count": 31,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"S_i = weyl_generators(m.inverse(), standard_basis(4))\n",
|
|
"S_i"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 32,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[ 1 0 0 0 1/2 1/2 1/2 -1/2]\n",
|
|
"[ 0 1 0 0 1/2 1/2 -1/2 1/2]\n",
|
|
"[ 0 0 1 0 1/2 -1/2 1/2 1/2]\n",
|
|
"[ 0 0 0 1 -1/2 1/2 1/2 1/2]\n",
|
|
"[ 0 0 0 0 0 0 0 0]\n",
|
|
"[ 0 0 0 0 0 0 0 0]\n",
|
|
"[ 0 0 0 0 0 0 0 0]\n",
|
|
"[ 0 0 0 0 0 0 0 0]"
|
|
]
|
|
},
|
|
"execution_count": 32,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"(Wc * P * Wc.transpose()).rref()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Generalization of this method\n",
|
|
"This method seems remarkably general. Given a matrix representing a root unit of a packing, we can find the linear relation between the coordinates, and thus represent the packing with only four coordinates, allowing us to find the quadratic form."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 33,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def quadform_from_root(root_matrix):\n",
|
|
" n = root_matrix.dimensions()[1]\n",
|
|
" P = matrix([\n",
|
|
" [0, -1/2, 0, 0],\n",
|
|
" [-1/2, 0, 0, 0],\n",
|
|
" [0, 0, 1, 0],\n",
|
|
" [0, 0, 0, 1],\n",
|
|
" ])\n",
|
|
" \n",
|
|
" # step 1: find linear relation between coords\n",
|
|
" relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 1)]) * root_matrix.transpose().rref()\n",
|
|
" relations = []\n",
|
|
" for i, expr in enumerate(relations_temp):\n",
|
|
" relations.append(var('b' + str(i + 1)) == expr)\n",
|
|
" \n",
|
|
" # step 2: find matrix of quadratic form\n",
|
|
" W = root_matrix[-4:]\n",
|
|
" M = W * P * W.transpose()\n",
|
|
" \n",
|
|
" # step 3: repeat with arbitrary matrix\n",
|
|
" W2 = matrix([\n",
|
|
" [\n",
|
|
" var('bt' + str(i)),\n",
|
|
" var('b' + str(i)),\n",
|
|
" var('h1' + str(i)),\n",
|
|
" var('h2' + str(i)),\n",
|
|
" ] for i in range(1, 5)\n",
|
|
" ])\n",
|
|
" D = factor(simplify(expand(W2.transpose() * M.inverse() * W2)))\n",
|
|
" \n",
|
|
" return relations[4:], M.inverse(), D[1][1]"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 34,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[2*b5 == b1 + b2 + b3 - b4,\n",
|
|
" 2*b6 == b1 + b2 - b3 + b4,\n",
|
|
" 2*b7 == b1 - b2 + b3 + b4,\n",
|
|
" 2*b8 == -b1 + b2 + b3 + b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"5 & -3 & -3 & -3 \\\\\n",
|
|
"-3 & 5 & -3 & -3 \\\\\n",
|
|
"-3 & -3 & 5 & -3 \\\\\n",
|
|
"-3 & -3 & -3 & 5\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"5 & -3 & -3 & -3 \\\\\n",
|
|
"-3 & 5 & -3 & -3 \\\\\n",
|
|
"-3 & -3 & 5 & -3 \\\\\n",
|
|
"-3 & -3 & -3 & 5\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ 5 -3 -3 -3]\n",
|
|
"[-3 5 -3 -3]\n",
|
|
"[-3 -3 5 -3]\n",
|
|
"[-3 -3 -3 5]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, b_{1}^{2} - 6 \\, b_{1} b_{2} + 5 \\, b_{2}^{2} - 6 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 5 \\, b_{3}^{2} - 6 \\, b_{1} b_{4} - 6 \\, b_{2} b_{4} - 6 \\, b_{3} b_{4} + 5 \\, b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, b_{1}^{2} - 6 \\, b_{1} b_{2} + 5 \\, b_{2}^{2} - 6 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 5 \\, b_{3}^{2} - 6 \\, b_{1} b_{4} - 6 \\, b_{2} b_{4} - 6 \\, b_{3} b_{4} + 5 \\, b_{4}^{2}$$"
|
|
],
|
|
"text/plain": [
|
|
"5*b1^2 - 6*b1*b2 + 5*b2^2 - 6*b1*b3 - 6*b2*b3 + 5*b3^2 - 6*b1*b4 - 6*b2*b4 - 6*b3*b4 + 5*b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
|
|
"\\end{array}\\right)\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
|
|
"\\end{array}\\right)\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[\n",
|
|
"[ -1 6/5 6/5 6/5] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [6/5 -1 6/5 6/5] [ 0 1 0 0]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0] [6/5 6/5 -1 6/5]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],\n",
|
|
"\n",
|
|
"[ 1 0 0 0]\n",
|
|
"[ 0 1 0 0]\n",
|
|
"[ 0 0 1 0]\n",
|
|
"[6/5 6/5 6/5 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"# cubical\n",
|
|
"relation, mat, equation = quadform_from_root(Wc)\n",
|
|
"show([2 * eq for eq in relation])\n",
|
|
"show(32 * mat)\n",
|
|
"show(32 * equation)\n",
|
|
"show(weyl_generators(mat, standard_basis(4)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 35,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{1} + b_{3} + b_{4}, b_{6} = b_{2} + b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{1} + b_{3} + b_{4}, b_{6} = b_{2} + b_{3} + b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b5 == b1 + b3 + b4, b6 == b2 + b3 + b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"17 & 15 & -6 & -6 \\\\\n",
|
|
"15 & 17 & -6 & -6 \\\\\n",
|
|
"-6 & -6 & 4 & 0 \\\\\n",
|
|
"-6 & -6 & 0 & 4\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"17 & 15 & -6 & -6 \\\\\n",
|
|
"15 & 17 & -6 & -6 \\\\\n",
|
|
"-6 & -6 & 4 & 0 \\\\\n",
|
|
"-6 & -6 & 0 & 4\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[17 15 -6 -6]\n",
|
|
"[15 17 -6 -6]\n",
|
|
"[-6 -6 4 0]\n",
|
|
"[-6 -6 0 4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}17 \\, b_{1}^{2} + 30 \\, b_{1} b_{2} + 17 \\, b_{2}^{2} - 12 \\, b_{1} b_{3} - 12 \\, b_{2} b_{3} + 4 \\, b_{3}^{2} - 12 \\, b_{1} b_{4} - 12 \\, b_{2} b_{4} + 4 \\, b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}17 \\, b_{1}^{2} + 30 \\, b_{1} b_{2} + 17 \\, b_{2}^{2} - 12 \\, b_{1} b_{3} - 12 \\, b_{2} b_{3} + 4 \\, b_{3}^{2} - 12 \\, b_{1} b_{4} - 12 \\, b_{2} b_{4} + 4 \\, b_{4}^{2}$$"
|
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],
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"text/plain": [
|
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"17*b1^2 + 30*b1*b2 + 17*b2^2 - 12*b1*b3 - 12*b2*b3 + 4*b3^2 - 12*b1*b4 - 12*b2*b4 + 4*b4^2"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & -\\frac{30}{17} & \\frac{12}{17} & \\frac{12}{17} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"-\\frac{30}{17} & -1 & \\frac{12}{17} & \\frac{12}{17} \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"3 & 3 & -1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"3 & 3 & 0 & -1\n",
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"\\end{array}\\right)\\right]</script></html>"
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],
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & -\\frac{30}{17} & \\frac{12}{17} & \\frac{12}{17} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"-\\frac{30}{17} & -1 & \\frac{12}{17} & \\frac{12}{17} \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"3 & 3 & -1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"3 & 3 & 0 & -1\n",
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"\\end{array}\\right)\\right]$$"
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],
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"text/plain": [
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"[\n",
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"[ -1 -30/17 12/17 12/17] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [-30/17 -1 12/17 12/17]\n",
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"[ 0 0 1 0] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1],\n",
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"\n",
|
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"[ 1 0 0 0] [ 1 0 0 0]\n",
|
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"[ 0 1 0 0] [ 0 1 0 0]\n",
|
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"[ 3 3 -1 0] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 3 3 0 -1]\n",
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"]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
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"# octahedral\n",
|
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"relation, mat, equation = quadform_from_root(matrix([\n",
|
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" [2, 0, 0, 1],\n",
|
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" [2, 0, 0, -1],\n",
|
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" [1, 1, sqrt(2), 0],\n",
|
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" [1, 1, -sqrt(2), 0],\n",
|
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" [4, 2, 0, 1],\n",
|
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" [4, 2, 0, -1],\n",
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"]))\n",
|
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"show(relation)\n",
|
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"show(8 * mat)\n",
|
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"show(8 * equation)\n",
|
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"show(weyl_generators(8 * mat, standard_basis(4)))"
|
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]
|
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},
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{
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"cell_type": "code",
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"execution_count": 36,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]</script></html>"
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]$$"
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],
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"text/plain": [
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"[]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"1 & -1 & -1 & -1 \\\\\n",
|
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"-1 & 1 & -1 & -1 \\\\\n",
|
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"-1 & -1 & 1 & -1 \\\\\n",
|
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"-1 & -1 & -1 & 1\n",
|
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"\\end{array}\\right)</script></html>"
|
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"1 & -1 & -1 & -1 \\\\\n",
|
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"-1 & 1 & -1 & -1 \\\\\n",
|
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"-1 & -1 & 1 & -1 \\\\\n",
|
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"-1 & -1 & -1 & 1\n",
|
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"\\end{array}\\right)$$"
|
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],
|
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"text/plain": [
|
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"[ 1 -1 -1 -1]\n",
|
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"[-1 1 -1 -1]\n",
|
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"[-1 -1 1 -1]\n",
|
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"[-1 -1 -1 1]"
|
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]
|
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},
|
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"metadata": {},
|
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"output_type": "display_data"
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},
|
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{
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
|
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],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}$$"
|
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],
|
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"text/plain": [
|
|
"b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
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"-1 & 2 & 2 & 2 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"2 & -1 & 2 & 2 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"2 & 2 & -1 & 2 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"2 & 2 & 2 & -1\n",
|
|
"\\end{array}\\right)\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & 2 & 2 & 2 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"2 & -1 & 2 & 2 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"2 & 2 & -1 & 2 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"2 & 2 & 2 & -1\n",
|
|
"\\end{array}\\right)\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[\n",
|
|
"[-1 2 2 2] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [ 2 -1 2 2] [ 0 1 0 0] [ 0 1 0 0]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0] [ 2 2 -1 2] [ 0 0 1 0]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 2 2 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"# tetrahedral\n",
|
|
"relation, mat, equation = quadform_from_root(matrix([\n",
|
|
" [2, 0, 0, 1],\n",
|
|
" [2, 0, 0, -1],\n",
|
|
" [-1, 1, 0, 0],\n",
|
|
" [3, 1, 2, 0]\n",
|
|
"]))\n",
|
|
"show(relation)\n",
|
|
"show(4 * mat)\n",
|
|
"show(4 * equation)\n",
|
|
"show(weyl_generators(4 * mat, standard_basis(4)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# $n$-Gon Base Pyramid\n",
|
|
"\n",
|
|
"The goal here is to find the quadratic form for an arbitrary $n$-gon base pyramid. The key to the whole process is a magic formula Dylan found for the bilinear form between two circles in an $n$-gon base pyramidal packing, namely $$\n",
|
|
" \\frac{1 - \\cos\\left(\\frac{2\\pi}{n}\\right) - 4\\sin^2\\left(\\frac{p\\pi}{n}\\right)}{1-\\cos\\left(\\frac{2\\pi}{n}\\right)}\n",
|
|
"$$\n",
|
|
"\n",
|
|
"where $p$ is how many circles are between the two circles in question. If we are finding the bilinear form between the central circle and any other circle it will always be $-1$ so we don't need to worry about that case."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 37,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"# function to compute W^T*P*W for n-gon pyramidal packing\n",
|
|
"def wmatrix(n):\n",
|
|
" vals = []\n",
|
|
" for i in range(n+1):\n",
|
|
" row = []\n",
|
|
" for j in range(n+1):\n",
|
|
" if i == j: # same vertex bilinear form'd with itself, so 1\n",
|
|
" row.append(1)\n",
|
|
" elif i ==0 or j == 0: # vertex bilinear form'd with special point, so tangent and therefore -1\n",
|
|
" row.append(-1)\n",
|
|
" else:\n",
|
|
" p = abs(i - j) # otherwise Dylan's crazy formula\n",
|
|
" row.append(\n",
|
|
" (1 - cos(2 * pi / n) - 4 * sin(pi * p / n)^2) / (1 - cos(2 * pi / n))\n",
|
|
" )\n",
|
|
" vals.append(row)\n",
|
|
" return matrix(vals)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 38,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"# just convenience function for quadratic forms\n",
|
|
"def qform(matrix, vector):\n",
|
|
" return vector * matrix * vector"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 39,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def linear_relations_and_quadratic_form_from_wtpw(mat, indices=None):\n",
|
|
" if indices is None:\n",
|
|
" indices = range(mat.dimensions()[0] + 1)\n",
|
|
" n = mat.dimensions()[0]\n",
|
|
" \n",
|
|
" # work out initial relations\n",
|
|
" relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 1) ]) * mat.transpose().rref()\n",
|
|
" relations = []\n",
|
|
" for i in range(n):\n",
|
|
" relations.append(var('b' + str(i + 1)) == relations_temp[i])\n",
|
|
" \n",
|
|
" # rewrite the relations in terms of the variables we care about, depends on the step\n",
|
|
" targets = [ var('b' + str(i)) for i in indices[4:] ]\n",
|
|
" show(relations)\n",
|
|
" show(targets)\n",
|
|
" #relations = solve(relations, *targets)[0]\n",
|
|
" \n",
|
|
" # find the matrix corresponding to the quadratic form, picking the appropriate rows from the matrix\n",
|
|
" mat = matrix([\n",
|
|
" [ mat[i - 1][j - 1] for j in indices[:4] ] for i in indices[:4]\n",
|
|
" ])\n",
|
|
"\n",
|
|
" Q = mat.inverse()\n",
|
|
" \n",
|
|
" # find the quadratic form in variables; proper units will satisfy this being equal to zero\n",
|
|
" vec = vector([ var('b' + str(i)) for i in indices[:4] ])\n",
|
|
" nqform = vec * Q * vec\n",
|
|
" \n",
|
|
" return relations, Q, expand(nqform)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 40,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{4}, b_{5} = b_{2} - 2 \\, b_{3} + 2 \\, b_{4}, b_{6} = 2 \\, b_{2} - 3 \\, b_{3} + 2 \\, b_{4}, b_{7} = 2 \\, b_{2} - 2 \\, b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{4}, b_{5} = b_{2} - 2 \\, b_{3} + 2 \\, b_{4}, b_{6} = 2 \\, b_{2} - 3 \\, b_{3} + 2 \\, b_{4}, b_{7} = 2 \\, b_{2} - 2 \\, b_{3} + b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b1 == b1,\n",
|
|
" b2 == b2,\n",
|
|
" b3 == b3,\n",
|
|
" b4 == b4,\n",
|
|
" b5 == b2 - 2*b3 + 2*b4,\n",
|
|
" b6 == 2*b2 - 3*b3 + 2*b4,\n",
|
|
" b7 == 2*b2 - 2*b3 + b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{2}, b_{4}, b_{6}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{2}, b_{4}, b_{6}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b2, b4, b6]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"[ 9 -1 -1 -1]\n",
|
|
"[-1 1 -1 -1]\n",
|
|
"[-1 -1 1 -1]\n",
|
|
"[-1 -1 -1 1]\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{5} - 2 \\, b_{3} b_{5} + b_{5}^{2} - 2 \\, b_{1} b_{7} - 2 \\, b_{3} b_{7} - 2 \\, b_{5} b_{7} + b_{7}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{5} - 2 \\, b_{3} b_{5} + b_{5}^{2} - 2 \\, b_{1} b_{7} - 2 \\, b_{3} b_{7} - 2 \\, b_{5} b_{7} + b_{7}^{2}$$"
|
|
],
|
|
"text/plain": [
|
|
"9*b1^2 - 2*b1*b3 + b3^2 - 2*b1*b5 - 2*b3*b5 + b5^2 - 2*b1*b7 - 2*b3*b7 - 2*b5*b7 + b7^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{1} = 3 \\, b_{1}, 3 \\, b_{2} = 3 \\, b_{2}, 3 \\, b_{3} = 3 \\, b_{3}, 3 \\, b_{4} = 3 \\, b_{4}, 3 \\, b_{5} = 3 \\, b_{2} - 6 \\, b_{3} + 6 \\, b_{4}, 3 \\, b_{6} = 6 \\, b_{2} - 9 \\, b_{3} + 6 \\, b_{4}, 3 \\, b_{7} = 6 \\, b_{2} - 6 \\, b_{3} + 3 \\, b_{4}\\right]</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{1} = 3 \\, b_{1}, 3 \\, b_{2} = 3 \\, b_{2}, 3 \\, b_{3} = 3 \\, b_{3}, 3 \\, b_{4} = 3 \\, b_{4}, 3 \\, b_{5} = 3 \\, b_{2} - 6 \\, b_{3} + 6 \\, b_{4}, 3 \\, b_{6} = 6 \\, b_{2} - 9 \\, b_{3} + 6 \\, b_{4}, 3 \\, b_{7} = 6 \\, b_{2} - 6 \\, b_{3} + 3 \\, b_{4}\\right]$$"
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],
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"text/plain": [
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"[3*b1 == 3*b1,\n",
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" 3*b2 == 3*b2,\n",
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" 3*b3 == 3*b3,\n",
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" 3*b4 == 3*b4,\n",
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" 3*b5 == 3*b2 - 6*b3 + 6*b4,\n",
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" 3*b6 == 6*b2 - 9*b3 + 6*b4,\n",
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" 3*b7 == 6*b2 - 6*b3 + 3*b4]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"2 & -1 & 2 & 2 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"2 & 2 & 2 & -1\n",
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"\\end{array}\\right)\\right]</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"2 & -1 & 2 & 2 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"2 & 2 & -1 & 2 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"2 & 2 & 2 & -1\n",
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"\\end{array}\\right)\\right]$$"
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],
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"text/plain": [
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"[\n",
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"[ -1 2/9 2/9 2/9] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [ 2 -1 2 2] [ 0 1 0 0] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [ 2 2 -1 2] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 2 2 -1]\n",
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"]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"relations, Q, nqform = linear_relations_and_quadratic_form_from_wtpw(wmatrix(6), [1, 3, 5, 7, 2, 4, 6])\n",
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"print(12 * Q)\n",
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"show(12 * nqform)\n",
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"show([3 * rel for rel in relations])\n",
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"show(weyl_generators(12 * Q, standard_basis(4)))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 41,
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"metadata": {},
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"outputs": [],
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"source": [
|
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"# function to compute the linear relations and quadratic formula for an ngon base pyramid\n",
|
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"# the circles will be numbered 1 in the center, then 2 through n winding around the circle by step, so\n",
|
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"# for n = 4 with step = 1, we have\n",
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"# b3\n",
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"# b4 b1 b2\n",
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"# b5\n",
|
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"# and the quadratic form is in terms of b1, b2, b3, and b4\n",
|
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"# and for n = 6 with step = 2, we have\n",
|
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"# b4 b3\n",
|
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"# b5 b1 b2\n",
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"# b6 b7\n",
|
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"# and the quadratic form is in terms of b1, b3, b5, and b7\n",
|
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"def ngon_linear_relations_and_quadratic_form(n, step=1):\n",
|
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" mat = wmatrix(n)\n",
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" \n",
|
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" # work out initial relations\n",
|
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" relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 2) ]) * mat.transpose().rref()\n",
|
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" relations = []\n",
|
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" for i in range(4, n + 1):\n",
|
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" relations.append(var('b' + str(i + 1)) == relations_temp[i])\n",
|
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" \n",
|
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" # rewrite the relations in terms of the variables we care about, depends on the step\n",
|
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" shuffled = (list(range(1, n + 2)) * step)[::step]\n",
|
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" targets = [ var('b' + str(i)) for i in shuffled[4:] ]\n",
|
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" relations = solve(relations, *targets)[0]\n",
|
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" \n",
|
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" # find the matrix corresponding to the quadratic form, picking the appropriate rows from the matrix\n",
|
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" Q = mat[:4*step:step,:4*step:step].inverse()\n",
|
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" \n",
|
|
" # find the quadratic form in variables; proper units will satisfy this being equal to zero\n",
|
|
" nqform = qform(Q, vector([ var('b' + str(i)) for i in range(1, 5) ]))\n",
|
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" \n",
|
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" return relations, Q, expand(nqform)"
|
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]
|
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},
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{
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"cell_type": "code",
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"execution_count": 42,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = -b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4} {\\left(\\sqrt{2} + 1\\right)} + b_{2}, b_{6} = b_{4} {\\left(\\sqrt{2} + 2\\right)} + b_{2} {\\left(\\sqrt{2} + 1\\right)} - 2 \\, b_{3} {\\left(\\sqrt{2} + 1\\right)}, b_{7} = -b_{3} {\\left(2 \\, \\sqrt{2} + 3\\right)} + b_{2} {\\left(\\sqrt{2} + 2\\right)} + b_{4} {\\left(\\sqrt{2} + 2\\right)}, b_{8} = b_{2} {\\left(\\sqrt{2} + 2\\right)} - 2 \\, b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4} {\\left(\\sqrt{2} + 1\\right)}, b_{9} = b_{2} {\\left(\\sqrt{2} + 1\\right)} - b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4}\\right]</script></html>"
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = -b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4} {\\left(\\sqrt{2} + 1\\right)} + b_{2}, b_{6} = b_{4} {\\left(\\sqrt{2} + 2\\right)} + b_{2} {\\left(\\sqrt{2} + 1\\right)} - 2 \\, b_{3} {\\left(\\sqrt{2} + 1\\right)}, b_{7} = -b_{3} {\\left(2 \\, \\sqrt{2} + 3\\right)} + b_{2} {\\left(\\sqrt{2} + 2\\right)} + b_{4} {\\left(\\sqrt{2} + 2\\right)}, b_{8} = b_{2} {\\left(\\sqrt{2} + 2\\right)} - 2 \\, b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4} {\\left(\\sqrt{2} + 1\\right)}, b_{9} = b_{2} {\\left(\\sqrt{2} + 1\\right)} - b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4}\\right]$$"
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],
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"text/plain": [
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"[b5 == -b3*(sqrt(2) + 1) + b4*(sqrt(2) + 1) + b2,\n",
|
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" b6 == b4*(sqrt(2) + 2) + b2*(sqrt(2) + 1) - 2*b3*(sqrt(2) + 1),\n",
|
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" b7 == -b3*(2*sqrt(2) + 3) + b2*(sqrt(2) + 2) + b4*(sqrt(2) + 2),\n",
|
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" b8 == b2*(sqrt(2) + 2) - 2*b3*(sqrt(2) + 1) + b4*(sqrt(2) + 1),\n",
|
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" b9 == b2*(sqrt(2) + 1) - b3*(sqrt(2) + 1) + b4]"
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]
|
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 3 & -2 \\\\\n",
|
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"-2 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -2 & \\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n",
|
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 3 & -2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -2 \\\\\n",
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"-2 & \\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -2 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1}\n",
|
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"\\end{array}\\right)</script></html>"
|
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 3 & -2 \\\\\n",
|
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"-2 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -2 & \\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n",
|
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 3 & -2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -2 \\\\\n",
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"-2 & \\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -2 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1}\n",
|
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"\\end{array}\\right)$$"
|
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],
|
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"text/plain": [
|
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"[ -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -2 -(sqrt(2) + 2)/(sqrt(2) - 2) - 3 -2]\n",
|
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"[ -2 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2 4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)]\n",
|
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"[ -(sqrt(2) + 2)/(sqrt(2) - 2) - 3 -2 -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -2]\n",
|
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"[ -2 4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)]"
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]
|
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
|
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 4 \\, b_{1} b_{2} - 6 \\, b_{1} b_{3} - 4 \\, b_{2} b_{3} + b_{3}^{2} - 4 \\, b_{1} b_{4} - 4 \\, b_{3} b_{4} - \\frac{\\sqrt{2} b_{1}^{2}}{\\sqrt{2} - 2} - \\frac{2 \\, \\sqrt{2} b_{1} b_{3}}{\\sqrt{2} - 2} - \\frac{\\sqrt{2} b_{3}^{2}}{\\sqrt{2} - 2} - \\frac{2 \\, b_{1}^{2}}{\\sqrt{2} - 2} - \\frac{4 \\, b_{1} b_{3}}{\\sqrt{2} - 2} - \\frac{2 \\, b_{3}^{2}}{\\sqrt{2} - 2} - \\frac{4 \\, b_{2}^{2}}{\\frac{\\sqrt{2}}{\\sqrt{2} - 2} + \\frac{2}{\\sqrt{2} - 2} - 1} + \\frac{8 \\, b_{2} b_{4}}{\\frac{\\sqrt{2}}{\\sqrt{2} - 2} + \\frac{2}{\\sqrt{2} - 2} - 1} - \\frac{4 \\, b_{4}^{2}}{\\frac{\\sqrt{2}}{\\sqrt{2} - 2} + \\frac{2}{\\sqrt{2} - 2} - 1}</script></html>"
|
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 4 \\, b_{1} b_{2} - 6 \\, b_{1} b_{3} - 4 \\, b_{2} b_{3} + b_{3}^{2} - 4 \\, b_{1} b_{4} - 4 \\, b_{3} b_{4} - \\frac{\\sqrt{2} b_{1}^{2}}{\\sqrt{2} - 2} - \\frac{2 \\, \\sqrt{2} b_{1} b_{3}}{\\sqrt{2} - 2} - \\frac{\\sqrt{2} b_{3}^{2}}{\\sqrt{2} - 2} - \\frac{2 \\, b_{1}^{2}}{\\sqrt{2} - 2} - \\frac{4 \\, b_{1} b_{3}}{\\sqrt{2} - 2} - \\frac{2 \\, b_{3}^{2}}{\\sqrt{2} - 2} - \\frac{4 \\, b_{2}^{2}}{\\frac{\\sqrt{2}}{\\sqrt{2} - 2} + \\frac{2}{\\sqrt{2} - 2} - 1} + \\frac{8 \\, b_{2} b_{4}}{\\frac{\\sqrt{2}}{\\sqrt{2} - 2} + \\frac{2}{\\sqrt{2} - 2} - 1} - \\frac{4 \\, b_{4}^{2}}{\\frac{\\sqrt{2}}{\\sqrt{2} - 2} + \\frac{2}{\\sqrt{2} - 2} - 1}$$"
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],
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"text/plain": [
|
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"b1^2 - 4*b1*b2 - 6*b1*b3 - 4*b2*b3 + b3^2 - 4*b1*b4 - 4*b3*b4 - sqrt(2)*b1^2/(sqrt(2) - 2) - 2*sqrt(2)*b1*b3/(sqrt(2) - 2) - sqrt(2)*b3^2/(sqrt(2) - 2) - 2*b1^2/(sqrt(2) - 2) - 4*b1*b3/(sqrt(2) - 2) - 2*b3^2/(sqrt(2) - 2) - 4*b2^2/(sqrt(2)/(sqrt(2) - 2) + 2/(sqrt(2) - 2) - 1) + 8*b2*b4/(sqrt(2)/(sqrt(2) - 2) + 2/(sqrt(2) - 2) - 1) - 4*b4^2/(sqrt(2)/(sqrt(2) - 2) + 2/(sqrt(2) - 2) - 1)"
|
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]
|
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -\\frac{2 \\, {\\left(\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 3\\right)}}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
|
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"-\\frac{2 \\, {\\left(\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 3\\right)}}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -1 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n",
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"0 & 0 & 0 & 1\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & 2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -1\n",
|
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"\\end{array}\\right)\\right]</script></html>"
|
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
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"-1 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -\\frac{2 \\, {\\left(\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 3\\right)}}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
|
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -1 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & 2 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"-\\frac{2 \\, {\\left(\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 3\\right)}}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -1 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & 2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -1\n",
|
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"\\end{array}\\right)\\right]$$"
|
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],
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"text/plain": [
|
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"[\n",
|
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"[ -1 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2*((sqrt(2) + 2)/(sqrt(2) - 2) + 3)/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
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"[ 0 1 0 0] [-(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -1 -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 2] [ 0 1 0 0] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [-2*((sqrt(2) + 2)/(sqrt(2) - 2) + 3)/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -1 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [-(sqrt(2) + 2)/(sqrt(2) - 2) + 1 2 -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -1]\n",
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"]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"relations, Q, nqform = ngon_linear_relations_and_quadratic_form(8)\n",
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"\n",
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"show(relations)\n",
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"show(8 * Q)\n",
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"show(8 * nqform)\n",
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"show(weyl_generators(8 * Q, standard_basis(4)))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 43,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{2} = 2 \\, b_{3} - b_{5} + 2 \\, b_{7}, 3 \\, b_{4} = 2 \\, b_{3} + 2 \\, b_{5} - b_{7}, 3 \\, b_{6} = -b_{3} + 2 \\, b_{5} + 2 \\, b_{7}\\right]</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{2} = 2 \\, b_{3} - b_{5} + 2 \\, b_{7}, 3 \\, b_{4} = 2 \\, b_{3} + 2 \\, b_{5} - b_{7}, 3 \\, b_{6} = -b_{3} + 2 \\, b_{5} + 2 \\, b_{7}\\right]$$"
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],
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"text/plain": [
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"[3*b2 == 2*b3 - b5 + 2*b7, 3*b4 == 2*b3 + 2*b5 - b7, 3*b6 == -b3 + 2*b5 + 2*b7]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
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"9 & -1 & -1 & -1 \\\\\n",
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"-1 & 1 & -1 & -1 \\\\\n",
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"-1 & -1 & 1 & -1 \\\\\n",
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"-1 & -1 & -1 & 1\n",
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"\\end{array}\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
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"9 & -1 & -1 & -1 \\\\\n",
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"-1 & 1 & -1 & -1 \\\\\n",
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"-1 & -1 & 1 & -1 \\\\\n",
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"-1 & -1 & -1 & 1\n",
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"\\end{array}\\right)$$"
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],
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"text/plain": [
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"[ 9 -1 -1 -1]\n",
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"[-1 1 -1 -1]\n",
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"[-1 -1 1 -1]\n",
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"[-1 -1 -1 1]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
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],
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}$$"
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],
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"text/plain": [
|
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"9*b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"2 & -1 & 2 & 2 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
|
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"2 & 2 & -1 & 2 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"2 & 2 & 2 & -1\n",
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"\\end{array}\\right)\\right]</script></html>"
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],
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
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"-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"2 & -1 & 2 & 2 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"2 & 2 & -1 & 2 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"2 & 2 & 2 & -1\n",
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"\\end{array}\\right)\\right]$$"
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],
|
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"text/plain": [
|
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"[\n",
|
|
"[ -1 2/9 2/9 2/9] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [ 2 -1 2 2] [ 0 1 0 0] [ 0 1 0 0]\n",
|
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"[ 0 0 1 0] [ 0 0 1 0] [ 2 2 -1 2] [ 0 0 1 0]\n",
|
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 2 2 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
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"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2}{9} \\, b_{2} + \\frac{2}{9} \\, b_{3} + \\frac{2}{9} \\, b_{4}</script></html>"
|
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],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2}{9} \\, b_{2} + \\frac{2}{9} \\, b_{3} + \\frac{2}{9} \\, b_{4}$$"
|
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],
|
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"text/plain": [
|
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"2/9*b2 + 2/9*b3 + 2/9*b4"
|
|
]
|
|
},
|
|
"metadata": {},
|
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"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"relations, Q, nqform = ngon_linear_relations_and_quadratic_form(6, 2)\n",
|
|
"\n",
|
|
"show([3 * eq for eq in relations])\n",
|
|
"show(12 * Q)\n",
|
|
"show(12 * nqform)\n",
|
|
"show(weyl_generators(12 * Q, standard_basis(4)))\n",
|
|
"\n",
|
|
"sols = solve(nqform, b1)\n",
|
|
"s1 = sols[0].rhs()\n",
|
|
"s2 = sols[1].rhs()\n",
|
|
"show(s1 + s2)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 44,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"class Circle:\n",
|
|
" def __init__(self, bt, b, h1, h2):\n",
|
|
" self.bt = bt\n",
|
|
" self.b = b\n",
|
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" self.h1 = h1\n",
|
|
" self.h2 = h2\n",
|
|
" self.vec = vector(bt, b, h1, h2)\n",
|
|
" \n",
|
|
" def __mul__(self, other):\n",
|
|
" self.vec *= other\n",
|
|
" self.bt = self.vec[0]\n",
|
|
" self.b = self.vec[1]\n",
|
|
" self.h1 = self.vec[2]\n",
|
|
" self.h2 = self.vec[3]\n",
|
|
" \n",
|
|
" def draw(self, plt):\n",
|
|
" pass\n",
|
|
" \n",
|
|
" \n",
|
|
"def from_xyr(x, y, r):\n",
|
|
" b = 1/r\n",
|
|
" h1 = b * x\n",
|
|
" h2 = b * y\n",
|
|
" return Circle((1 - h1^2 - h2^2) / b, b, h1, h2)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 45,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(-2/3*sqrt(3), -1)\t1/3*sqrt(3)\n",
|
|
"(-1/6*sqrt(3), -1)\t1/6*sqrt(3)\n",
|
|
"(0, 1)\tsqrt(3)\n"
|
|
]
|
|
},
|
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{
|
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"data": {
|
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"image/png": 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\n",
|
|
"text/plain": [
|
|
"Graphics object consisting of 14 graphics primitives"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"b1, b2, b3, b4, b5, b6, b7 = var('b1 b2 b3 b4 b5 b6 b7')\n",
|
|
"test = [relation.subs([b1 == 0, b2 == 1, b3 == 0, b4 == 1]) for relation in relations]\n",
|
|
"\n",
|
|
"plt = line([(-2*sqrt(3), 1), (2*sqrt(3), 1)])\n",
|
|
"plt += line([(-2*sqrt(3), -1), (2*sqrt(3), -1)])\n",
|
|
"plt += circle((-sqrt(3), 0), 1)\n",
|
|
"plt += circle((-1/3 * sqrt(3), -2/3), 1/3)\n",
|
|
"plt += circle((0, -3/4), 1/4)\n",
|
|
"plt += circle((1/3 * sqrt(3), -2/3), 1/3)\n",
|
|
"plt += circle((sqrt(3), 0), 1)\n",
|
|
"\n",
|
|
"plt += line([(-sqrt(3), 2*sqrt(3)), (-sqrt(3), -2*sqrt(3))], rgbcolor=(1,0,0))\n",
|
|
"plt += line([( sqrt(3), 2*sqrt(3)), ( sqrt(3), -2*sqrt(3))], rgbcolor=(1,0,0))\n",
|
|
"\n",
|
|
"x, y = var('x y')\n",
|
|
"c1 = (x + sqrt(3))^2 + y^2 == 1\n",
|
|
"c2 = (x + 1/3*sqrt(3))^2 + (y + 2/3)^2 == 1/9\n",
|
|
"c3 = x^2 + (y + 3/4)^2 == 1/16\n",
|
|
"i1 = solve([c1, c2], [x, y])[0]\n",
|
|
"x1 = i1[0].rhs()\n",
|
|
"y1 = i1[1].rhs()\n",
|
|
"i2 = solve([c2, c3], [x, y])[0]\n",
|
|
"x2 = i2[0].rhs()\n",
|
|
"y2 = i2[1].rhs()\n",
|
|
"\n",
|
|
"x, y, r = circle_from_points((-sqrt(3), -1), (-1/3*sqrt(3), -1), (x1, y1))\n",
|
|
"print('({}, {})\\t{}'.format(x, y, r))\n",
|
|
"\n",
|
|
"plt += circle((x, y), r, rgbcolor=(1,0,0))\n",
|
|
"plt += circle((-x, y), r, rgbcolor=(1,0,0))\n",
|
|
"\n",
|
|
"x, y, r = circle_from_points((-1/3*sqrt(3), -1), (0, -1), (x2, y2))\n",
|
|
"print('({}, {})\\t{}'.format(x, y, r))\n",
|
|
"\n",
|
|
"plt += circle((x, y), r, rgbcolor=(1,0,0))\n",
|
|
"plt += circle((-x, y), r, rgbcolor=(1,0,0))\n",
|
|
"\n",
|
|
"x, y, r = circle_from_points((-sqrt(3), 1), (x1, y1), (sqrt(3), 1))\n",
|
|
"print('({}, {})\\t{}'.format(x, y, r))\n",
|
|
"\n",
|
|
"plt += circle((x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"\n",
|
|
"show(plt)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 46,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{2} - 2 \\, b_{3} + 2 \\, b_{4}, b_{6} = 2 \\, b_{2} - 3 \\, b_{3} + 2 \\, b_{4}, b_{7} = 2 \\, b_{2} - 2 \\, b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{2} - 2 \\, b_{3} + 2 \\, b_{4}, b_{6} = 2 \\, b_{2} - 3 \\, b_{3} + 2 \\, b_{4}, b_{7} = 2 \\, b_{2} - 2 \\, b_{3} + b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b5 == b2 - 2*b3 + 2*b4, b6 == 2*b2 - 3*b3 + 2*b4, b7 == 2*b2 - 2*b3 + b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}6 \\, b_{1}^{2} - 4 \\, b_{1} b_{2} + \\frac{2}{3} \\, b_{2}^{2} + 4 \\, b_{1} b_{3} - 4 \\, b_{2} b_{3} + 6 \\, b_{3}^{2} - 4 \\, b_{1} b_{4} - \\frac{4}{3} \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + \\frac{2}{3} \\, b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}6 \\, b_{1}^{2} - 4 \\, b_{1} b_{2} + \\frac{2}{3} \\, b_{2}^{2} + 4 \\, b_{1} b_{3} - 4 \\, b_{2} b_{3} + 6 \\, b_{3}^{2} - 4 \\, b_{1} b_{4} - \\frac{4}{3} \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + \\frac{2}{3} \\, b_{4}^{2}$$"
|
|
],
|
|
"text/plain": [
|
|
"6*b1^2 - 4*b1*b2 + 2/3*b2^2 + 4*b1*b3 - 4*b2*b3 + 6*b3^2 - 4*b1*b4 - 4/3*b2*b4 - 4*b3*b4 + 2/3*b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"relations, Q, nqform = ngon_linear_relations_and_quadratic_form(6)\n",
|
|
"show(relations)\n",
|
|
"show(simplify(8 * nqform))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 47,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"[b5 == -1/2*b3*(sqrt(5) + 3) + 1/2*b4*(sqrt(5) + 3) + b2, b6 == -1/2*b3*(3*sqrt(5) + 5) + 1/2*b2*(sqrt(5) + 3) + b4*(sqrt(5) + 2), b7 == b4*(sqrt(5) + 3) + b2*(sqrt(5) + 2) - 2*b3*(sqrt(5) + 2), b8 == -b3*(2*sqrt(5) + 5) + b2*(sqrt(5) + 3) + b4*(sqrt(5) + 3), b9 == b2*(sqrt(5) + 3) - 2*b3*(sqrt(5) + 2) + b4*(sqrt(5) + 2), b10 == -1/2*b3*(3*sqrt(5) + 5) + 1/2*b4*(sqrt(5) + 3) + b2*(sqrt(5) + 2), b11 == 1/2*b2*(sqrt(5) + 3) - 1/2*b3*(sqrt(5) + 3) + b4]\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"relations, Q, nqform = ngon_linear_relations_and_quadratic_form(10)\n",
|
|
"print(relations)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 48,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"-1/2*b2*bt1 - 1/2*b1*bt2 + h11*h12 + h21*h22"
|
|
]
|
|
},
|
|
"execution_count": 48,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"vec1 = vector([var('bt1'), var('b1'), var('h11'), var('h21')])\n",
|
|
"vec2 = vector([var('bt2'), var('b2'), var('h12'), var('h22')])\n",
|
|
"vec1 * P * vec2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 49,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, \\sin\\left(\\frac{\\pi k}{n}\\right)^{2}}{\\sin\\left(\\frac{\\pi}{n}\\right)^{2}} + 1</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, \\sin\\left(\\frac{\\pi k}{n}\\right)^{2}}{\\sin\\left(\\frac{\\pi}{n}\\right)^{2}} + 1$$"
|
|
],
|
|
"text/plain": [
|
|
"-2*sin(pi*k/n)^2/sin(pi/n)^2 + 1"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"k = var('k', domain=RR)\n",
|
|
"n = var('n', domain=RR)\n",
|
|
"\n",
|
|
"c1 = 1\n",
|
|
"c2i = sin(2 * pi * k / n)\n",
|
|
"c2r = cos(2 * pi * k / n)\n",
|
|
"\n",
|
|
"dist = sin(pi * k / n)\n",
|
|
"r = dist.subs(k==1) / 2\n",
|
|
"dist /= r\n",
|
|
"\n",
|
|
"bf = 1/2 * (1 + 1 * (1 - dist^2))\n",
|
|
"show(bf)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 50,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"0 \t 0\n",
|
|
"1 \t x - 1\n",
|
|
"2 \t 2*x^2 - 2\n",
|
|
"3 \t 4*x^3 - 3*x - 1\n",
|
|
"4 \t 8*x^4 - 8*x^2\n",
|
|
"5 \t 16*x^5 - 20*x^3 + 5*x - 1\n",
|
|
"6 \t 32*x^6 - 48*x^4 + 18*x^2 - 2\n",
|
|
"7 \t 64*x^7 - 112*x^5 + 56*x^3 - 7*x - 1\n",
|
|
"8 \t 128*x^8 - 256*x^6 + 160*x^4 - 32*x^2\n",
|
|
"9 \t 256*x^9 - 576*x^7 + 432*x^5 - 120*x^3 + 9*x - 1\n",
|
|
"10 \t 512*x^10 - 1280*x^8 + 1120*x^6 - 400*x^4 + 50*x^2 - 2\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"T = []\n",
|
|
"x = var('x')\n",
|
|
"T.append(1)\n",
|
|
"T.append(x)\n",
|
|
"\n",
|
|
"def nextT():\n",
|
|
" i = len(T)\n",
|
|
" T.append(expand(2 * x * T[i - 1] - T[i - 2]))\n",
|
|
" #print(i, '\\t', T[i] - 1)\n",
|
|
" \n",
|
|
"for _ in range(9):\n",
|
|
" nextT()\n",
|
|
"\n",
|
|
"for i, poly in enumerate(T):\n",
|
|
" print(i, '\\t', poly - 1)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 51,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"-x^6 + x^5 + x^4 - x^3 - 2*x^2 + 2*x"
|
|
]
|
|
},
|
|
"execution_count": 51,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"expand(x * (1 - x) * (x^4 - x^2 + 2))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 52,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"S1 = matrix([\n",
|
|
" [-1, 0, 0, 2],\n",
|
|
" [0, 1, 0, 0],\n",
|
|
" [0, 0, 1, 0],\n",
|
|
" [0, 0, 0, 1],\n",
|
|
"])\n",
|
|
"S2 = matrix([\n",
|
|
" [1, 0, 0, 0],\n",
|
|
" [0, -1, 0, 2],\n",
|
|
" [0, 0, 1, 0],\n",
|
|
" [0, 0, 0, 1],\n",
|
|
"])\n",
|
|
"S3 = matrix([\n",
|
|
" [1, 0, 0, 0],\n",
|
|
" [0, 1, 0, 0],\n",
|
|
" [0, 0, -1, 2],\n",
|
|
" [0, 0, 0, 1],\n",
|
|
"])\n",
|
|
"S4 = matrix([\n",
|
|
" [1, 0, 0, 0],\n",
|
|
" [0, 1, 0, 0],\n",
|
|
" [0, 0, 1, 0],\n",
|
|
" [2, 2, 2, -1],\n",
|
|
"])\n",
|
|
"Ss = [S4, S4 * S1, S4 * S1 * S2, S4 * S1 * S3, S4 * S1 * S2 * S3, S4 * S2, S4 * S2 * S3, S4 * S3]\n",
|
|
"v1 = vector(var('b' + str(i + 1)) for i in range(4))\n",
|
|
"v2 = vector(var('b' + str(i + 1)) for i in range(6))\n",
|
|
"temp = [matrix * v1 for matrix in Ss]\n",
|
|
"new = []\n",
|
|
"for blah in temp:\n",
|
|
" new.append([var('b' + str(i+1) + 'p') == blah[i] for i in range(len(blah))])\n",
|
|
"\n",
|
|
"for eqs in new:\n",
|
|
" for i in range(len(eqs)):\n",
|
|
" eqs[i] = eqs[i].subs(var('b4') == var('b1') + var('b6'))\n",
|
|
" eqs[i] = eqs[i].subs(var('b4p') == var('b1p') + var('b6p'))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 53,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{1} + b_{2} - b_{3}, b_{5} = b_{4}, b_{6} = b_{2} - b_{3} - b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{1} + b_{2} - b_{3}, b_{5} = b_{4}, b_{6} = b_{2} - b_{3} - b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b1 == b1,\n",
|
|
" b2 == b2,\n",
|
|
" b3 == b3,\n",
|
|
" b4 == b1 + b2 - b3,\n",
|
|
" b5 == b4,\n",
|
|
" b6 == b2 - b3 - b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{4}, b_{5}, b_{6}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{4}, b_{5}, b_{6}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b4, b5, b6]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{1} + b_{2} - b_{3}, b_{5} = b_{4}, b_{6} = b_{2} - b_{3} - b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{1} + b_{2} - b_{3}, b_{5} = b_{4}, b_{6} = b_{2} - b_{3} - b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b1 == b1,\n",
|
|
" b2 == b2,\n",
|
|
" b3 == b3,\n",
|
|
" b4 == b1 + b2 - b3,\n",
|
|
" b5 == b4,\n",
|
|
" b6 == b2 - b3 - b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"4 & 0 & -2 & 2 \\\\\n",
|
|
"0 & 4 & -2 & 2 \\\\\n",
|
|
"-2 & -2 & 1 & -3 \\\\\n",
|
|
"2 & 2 & -3 & 9\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"4 & 0 & -2 & 2 \\\\\n",
|
|
"0 & 4 & -2 & 2 \\\\\n",
|
|
"-2 & -2 & 1 & -3 \\\\\n",
|
|
"2 & 2 & -3 & 9\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ 4 0 -2 2]\n",
|
|
"[ 0 4 -2 2]\n",
|
|
"[-2 -2 1 -3]\n",
|
|
"[ 2 2 -3 9]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, b_{0}^{2} + 4 \\, b_{1}^{2} - 4 \\, b_{0} b_{2} - 4 \\, b_{1} b_{2} + b_{2}^{2} + 4 \\, b_{0} b_{3} + 4 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 9 \\, b_{3}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, b_{0}^{2} + 4 \\, b_{1}^{2} - 4 \\, b_{0} b_{2} - 4 \\, b_{1} b_{2} + b_{2}^{2} + 4 \\, b_{0} b_{3} + 4 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 9 \\, b_{3}^{2}$$"
|
|
],
|
|
"text/plain": [
|
|
"4*b0^2 + 4*b1^2 - 4*b0*b2 - 4*b1*b2 + b2^2 + 4*b0*b3 + 4*b1*b3 - 6*b2*b3 + 9*b3^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & 0 & 1 & -1 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & -1 & 1 & -1 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"4 & 4 & -1 & 6 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"-\\frac{4}{9} & -\\frac{4}{9} & \\frac{2}{3} & -1\n",
|
|
"\\end{array}\\right)\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & 0 & 1 & -1 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & -1 & 1 & -1 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"4 & 4 & -1 & 6 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"-\\frac{4}{9} & -\\frac{4}{9} & \\frac{2}{3} & -1\n",
|
|
"\\end{array}\\right)\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[\n",
|
|
"[-1 0 1 -1] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [ 0 -1 1 -1] [ 0 1 0 0] [ 0 1 0 0]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0] [ 4 4 -1 6] [ 0 0 1 0]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [-4/9 -4/9 2/3 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"W = matrix([\n",
|
|
" [2, 0, 0, 1],\n",
|
|
" [4, 2, 0, -1],\n",
|
|
" [2, 0, 0, -1],\n",
|
|
" [4, 2, 0, 1],\n",
|
|
" [1, 1, sqrt(2), 0],\n",
|
|
" [1, 1, -sqrt(2), 0],\n",
|
|
"])\n",
|
|
"\n",
|
|
"relations, Q, nqform = linear_relations_and_quadratic_form_from_wtpw(W * P * W.transpose())#, [1, 2, 3, 4, 5, 6])\n",
|
|
"show(relations)\n",
|
|
"show(8 * Q)\n",
|
|
"show(8 * nqform)\n",
|
|
"show(weyl_generators(8 * Q, standard_basis(4)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 54,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def diagonalize(mat):\n",
|
|
" eigs = WPWT.eigenvectors_right()\n",
|
|
" eigvs = []\n",
|
|
" eigvals = []\n",
|
|
" for eig in eigs:\n",
|
|
" for eigv in eig[1]:\n",
|
|
" eigvs.append(eigv)\n",
|
|
" for _ in range(eig[2]):\n",
|
|
" eigvals.append(eig[0])\n",
|
|
" Q = matrix(eigvs)\n",
|
|
" D = diagonal_matrix(eigvals)\n",
|
|
" return (Q, D)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 55,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrrrrrr}\n",
|
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrr}\n",
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"1 & -1 & -3 & -1 & -5 & -3 & -1 & -3 \\\\\n",
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"-1 & 1 & -1 & -3 & -3 & -5 & -3 & -1 \\\\\n",
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"-3 & -1 & 1 & -1 & -1 & -3 & -5 & -3 \\\\\n",
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"-1 & -3 & -1 & 1 & -3 & -1 & -3 & -5 \\\\\n",
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"-5 & -3 & -1 & -3 & 1 & -1 & -3 & -1 \\\\\n",
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"-3 & -5 & -3 & -1 & -1 & 1 & -1 & -3 \\\\\n",
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"-1 & -3 & -5 & -3 & -3 & -1 & 1 & -1 \\\\\n",
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"-3 & -1 & -3 & -5 & -1 & -3 & -1 & 1\n",
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"\\end{array}\\right)</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrr}\n",
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"1 & -1 & -3 & -1 & -5 & -3 & -1 & -3 \\\\\n",
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"-1 & 1 & -1 & -3 & -3 & -5 & -3 & -1 \\\\\n",
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"-3 & -1 & 1 & -1 & -1 & -3 & -5 & -3 \\\\\n",
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"-1 & -3 & -1 & 1 & -3 & -1 & -3 & -5 \\\\\n",
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"-5 & -3 & -1 & -3 & 1 & -1 & -3 & -1 \\\\\n",
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"-3 & -5 & -3 & -1 & -1 & 1 & -1 & -3 \\\\\n",
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"-1 & -3 & -5 & -3 & -3 & -1 & 1 & -1 \\\\\n",
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"-3 & -1 & -3 & -5 & -1 & -3 & -1 & 1\n",
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"\\end{array}\\right)$$"
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],
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"text/plain": [
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"[ 1 -1 -3 -1 -5 -3 -1 -3]\n",
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"[-1 1 -1 -3 -3 -5 -3 -1]\n",
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"[-3 -1 1 -1 -1 -3 -5 -3]\n",
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"[-1 -3 -1 1 -3 -1 -3 -5]\n",
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"[-5 -3 -1 -3 1 -1 -3 -1]\n",
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"[-3 -5 -3 -1 -1 1 -1 -3]\n",
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"[-1 -3 -5 -3 -3 -1 1 -1]\n",
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"[-3 -1 -3 -5 -1 -3 -1 1]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
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"WPWT = matrix([\n",
|
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" [1, -1, -3, -1, -5, -3, -1, -3],\n",
|
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" [-1, 1, -1, -3, -3, -5, -3, -1],\n",
|
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" [-3, -1, 1, -1, -1, -3, -5, -3],\n",
|
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" [-1, -3, -1, 1, -3, -1, -3, -5],\n",
|
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" [-5, -3, -1, -3, 1, -1, -3, -1],\n",
|
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" [-3, -5, -3, -1, -1, 1, -1, -3],\n",
|
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" [-1, -3, -5, -3, -3, -1, 1, -1],\n",
|
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" [-3, -1, -3, -5, -1, -3, -1, 1]\n",
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"])\n",
|
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"show(weyl_generators(WPWT, standard_basis(8)))\n",
|
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"# WPWT = W * P * W.transpose()\n",
|
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"Q, D = diagonalize(WPWT) \n",
|
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"print(WPWT)\n",
|
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"show(D)\n",
|
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"show(Q)\n",
|
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"Qi = expand(Q.inverse())\n",
|
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"for i in range(6):\n",
|
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" for j in range(6):\n",
|
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" print('{}, {}\\t\\t{}'.format(i, j, expand(Qi[:,i].transpose() * Qi[:,j])))\n",
|
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"show(expand(Q.inverse() * D * Q))"
|
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]
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},
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{
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"cell_type": "code",
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"execution_count": 56,
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"metadata": {
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"scrolled": true
|
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},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,1,\\,-\\frac{1}{3},\\,-\\frac{1}{3}\\right)</script></html>"
|
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,1,\\,-\\frac{1}{3},\\,-\\frac{1}{3}\\right)$$"
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],
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"text/plain": [
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"(1, 1, -1/3, -1/3)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-\\frac{1}{3},\\,1,\\,-\\frac{1}{3}\\right)</script></html>"
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-\\frac{1}{3},\\,1,\\,-\\frac{1}{3}\\right)$$"
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],
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"text/plain": [
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"(1, -1/3, 1, -1/3)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-3,\\,-3,\\,1\\right)</script></html>"
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-3,\\,-3,\\,1\\right)$$"
|
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],
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"text/plain": [
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"(1, -3, -3, 1)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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],
|
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-\\frac{1}{3},\\,-\\frac{1}{3},\\,1\\right)$$"
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],
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"text/plain": [
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"(1, -1/3, -1/3, 1)"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-3,\\,1,\\,-3\\right)</script></html>"
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,-3,\\,1,\\,-3\\right)$$"
|
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],
|
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"text/plain": [
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"(1, -3, 1, -3)"
|
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]
|
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},
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"metadata": {},
|
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"output_type": "display_data"
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},
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{
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,1,\\,-3,\\,-3\\right)</script></html>"
|
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],
|
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,\\,1,\\,-3,\\,-3\\right)$$"
|
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],
|
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"text/plain": [
|
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"(1, 1, -3, -3)"
|
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]
|
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},
|
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"metadata": {},
|
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"output_type": "display_data"
|
|
}
|
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],
|
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"source": [
|
|
"S1 = matrix([\n",
|
|
" [1, 0, 3, 3],\n",
|
|
" [0, 1, 3, 3],\n",
|
|
" [0, 0, 0, -1],\n",
|
|
" [0, 0, -1, 0]\n",
|
|
"]).transpose()\n",
|
|
"S2 = matrix([\n",
|
|
" [1, 3, 0, 3],\n",
|
|
" [0, 0, 0, -1],\n",
|
|
" [0, 3, 1, 3],\n",
|
|
" [0, -1, 0, 0]\n",
|
|
"]).transpose()\n",
|
|
"S3 = matrix([\n",
|
|
" [0, 0, 0, -1],\n",
|
|
" [3, 1, 0, 3],\n",
|
|
" [3, 0, 1, 3],\n",
|
|
" [-1, 0, 0, 0]\n",
|
|
"]).transpose()\n",
|
|
"S4 = matrix([\n",
|
|
" [1, 3, 3, 0],\n",
|
|
" [0, 0, -1, 0],\n",
|
|
" [0, -1, 0, 0],\n",
|
|
" [0, 3, 3, 1]\n",
|
|
"]).transpose()\n",
|
|
"S5 = matrix([\n",
|
|
" [0, 0, -1, 0],\n",
|
|
" [3, 1, 3, 0],\n",
|
|
" [-1, 0, 0, 0],\n",
|
|
" [3, 0, 3, 1]\n",
|
|
"]).transpose()\n",
|
|
"S6 = matrix([\n",
|
|
" [0, -1, 0, 0],\n",
|
|
" [-1, 0, 0, 0],\n",
|
|
" [3, 3, 1, 0],\n",
|
|
" [3, 3, 0, 1]\n",
|
|
"]).transpose()\n",
|
|
"\n",
|
|
"for mat in [S1, S2, S3, S4, S5, S6]:\n",
|
|
" for eigvec in mat.eigenvectors_left():\n",
|
|
" if (eigvec[0] == -1):\n",
|
|
" show(eigvec[1][0])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 57,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{4}, b_{5} = \\frac{1}{2} \\, b_{1} + \\frac{1}{2} \\, b_{2} + \\frac{1}{2} \\, b_{3} - \\frac{1}{2} \\, b_{4}, b_{6} = \\frac{1}{2} \\, b_{1} + \\frac{1}{2} \\, b_{2} - \\frac{1}{2} \\, b_{3} + \\frac{1}{2} \\, b_{4}, b_{7} = \\frac{1}{2} \\, b_{1} - \\frac{1}{2} \\, b_{2} + \\frac{1}{2} \\, b_{3} + \\frac{1}{2} \\, b_{4}, b_{8} = -\\frac{1}{2} \\, b_{1} + \\frac{1}{2} \\, b_{2} + \\frac{1}{2} \\, b_{3} + \\frac{1}{2} \\, b_{4}\\right]</script></html>"
|
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],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{1} = b_{1}, b_{2} = b_{2}, b_{3} = b_{3}, b_{4} = b_{4}, b_{5} = \\frac{1}{2} \\, b_{1} + \\frac{1}{2} \\, b_{2} + \\frac{1}{2} \\, b_{3} - \\frac{1}{2} \\, b_{4}, b_{6} = \\frac{1}{2} \\, b_{1} + \\frac{1}{2} \\, b_{2} - \\frac{1}{2} \\, b_{3} + \\frac{1}{2} \\, b_{4}, b_{7} = \\frac{1}{2} \\, b_{1} - \\frac{1}{2} \\, b_{2} + \\frac{1}{2} \\, b_{3} + \\frac{1}{2} \\, b_{4}, b_{8} = -\\frac{1}{2} \\, b_{1} + \\frac{1}{2} \\, b_{2} + \\frac{1}{2} \\, b_{3} + \\frac{1}{2} \\, b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[b1 == b1,\n",
|
|
" b2 == b2,\n",
|
|
" b3 == b3,\n",
|
|
" b4 == b4,\n",
|
|
" b5 == 1/2*b1 + 1/2*b2 + 1/2*b3 - 1/2*b4,\n",
|
|
" b6 == 1/2*b1 + 1/2*b2 - 1/2*b3 + 1/2*b4,\n",
|
|
" b7 == 1/2*b1 - 1/2*b2 + 1/2*b3 + 1/2*b4,\n",
|
|
" b8 == -1/2*b1 + 1/2*b2 + 1/2*b3 + 1/2*b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
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},
|
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{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5}, b_{6}, b_{7}, b_{8}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5}, b_{6}, b_{7}, b_{8}\\right]$$"
|
|
],
|
|
"text/plain": [
|
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"[b5, b6, b7, b8]"
|
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]
|
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},
|
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"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{1} = 2 \\, b_{1}, 2 \\, b_{2} = 2 \\, b_{2}, 2 \\, b_{3} = 2 \\, b_{3}, 2 \\, b_{4} = 2 \\, b_{4}, 2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{1} = 2 \\, b_{1}, 2 \\, b_{2} = 2 \\, b_{2}, 2 \\, b_{3} = 2 \\, b_{3}, 2 \\, b_{4} = 2 \\, b_{4}, 2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]$$"
|
|
],
|
|
"text/plain": [
|
|
"[2*b1 == 2*b1,\n",
|
|
" 2*b2 == 2*b2,\n",
|
|
" 2*b3 == 2*b3,\n",
|
|
" 2*b4 == 2*b4,\n",
|
|
" 2*b5 == b1 + b2 + b3 - b4,\n",
|
|
" 2*b6 == b1 + b2 - b3 + b4,\n",
|
|
" 2*b7 == b1 - b2 + b3 + b4,\n",
|
|
" 2*b8 == -b1 + b2 + b3 + b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"\\frac{5}{4} & -\\frac{3}{4} & -\\frac{3}{4} & -\\frac{3}{4} \\\\\n",
|
|
"-\\frac{3}{4} & \\frac{5}{4} & -\\frac{3}{4} & -\\frac{3}{4} \\\\\n",
|
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"-\\frac{3}{4} & -\\frac{3}{4} & \\frac{5}{4} & -\\frac{3}{4} \\\\\n",
|
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"-\\frac{3}{4} & -\\frac{3}{4} & -\\frac{3}{4} & \\frac{5}{4}\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"\\frac{5}{4} & -\\frac{3}{4} & -\\frac{3}{4} & -\\frac{3}{4} \\\\\n",
|
|
"-\\frac{3}{4} & \\frac{5}{4} & -\\frac{3}{4} & -\\frac{3}{4} \\\\\n",
|
|
"-\\frac{3}{4} & -\\frac{3}{4} & \\frac{5}{4} & -\\frac{3}{4} \\\\\n",
|
|
"-\\frac{3}{4} & -\\frac{3}{4} & -\\frac{3}{4} & \\frac{5}{4}\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ 5/4 -3/4 -3/4 -3/4]\n",
|
|
"[-3/4 5/4 -3/4 -3/4]\n",
|
|
"[-3/4 -3/4 5/4 -3/4]\n",
|
|
"[-3/4 -3/4 -3/4 5/4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
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},
|
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{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{5}{4} \\, b_{1}^{2} - \\frac{3}{2} \\, b_{1} b_{2} + \\frac{5}{4} \\, b_{2}^{2} - \\frac{3}{2} \\, b_{1} b_{3} - \\frac{3}{2} \\, b_{2} b_{3} + \\frac{5}{4} \\, b_{3}^{2} - \\frac{3}{2} \\, b_{1} b_{4} - \\frac{3}{2} \\, b_{2} b_{4} - \\frac{3}{2} \\, b_{3} b_{4} + \\frac{5}{4} \\, b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{5}{4} \\, b_{1}^{2} - \\frac{3}{2} \\, b_{1} b_{2} + \\frac{5}{4} \\, b_{2}^{2} - \\frac{3}{2} \\, b_{1} b_{3} - \\frac{3}{2} \\, b_{2} b_{3} + \\frac{5}{4} \\, b_{3}^{2} - \\frac{3}{2} \\, b_{1} b_{4} - \\frac{3}{2} \\, b_{2} b_{4} - \\frac{3}{2} \\, b_{3} b_{4} + \\frac{5}{4} \\, b_{4}^{2}$$"
|
|
],
|
|
"text/plain": [
|
|
"5/4*b1^2 - 3/2*b1*b2 + 5/4*b2^2 - 3/2*b1*b3 - 3/2*b2*b3 + 5/4*b3^2 - 3/2*b1*b4 - 3/2*b2*b4 - 3/2*b3*b4 + 5/4*b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
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"\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
|
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"\\end{array}\\right)\\right]</script></html>"
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],
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"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
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"\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
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"\\end{array}\\right)\\right]$$"
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],
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"text/plain": [
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"[\n",
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"[ -1 6/5 6/5 6/5] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [6/5 -1 6/5 6/5] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [6/5 6/5 -1 6/5]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],\n",
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"\n",
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"[ 1 0 0 0]\n",
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"[ 0 1 0 0]\n",
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"[ 0 0 1 0]\n",
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"[6/5 6/5 6/5 -1]\n",
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"]"
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]
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},
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"metadata": {},
|
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"output_type": "display_data"
|
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}
|
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],
|
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"source": [
|
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"# cubical\n",
|
|
"Wc = matrix([\n",
|
|
" [4, 0, 0, 1],\n",
|
|
" [0, 2, 0, 1],\n",
|
|
" [2, 1, -sqrt(2), -1],\n",
|
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" [2, 1, sqrt(2), -1],\n",
|
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" [2, 1, -sqrt(2), 1],\n",
|
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" [2, 1, sqrt(2), 1],\n",
|
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" [4, 0, 0, -1],\n",
|
|
" [0, 2, 0, -1],\n",
|
|
"])\n",
|
|
"relation, mat, equation = linear_relations_and_quadratic_form_from_wtpw(Wc * P * Wc.transpose(), [1, 2, 3, 4, 5, 6, 7, 8])\n",
|
|
"show([2 * eq for eq in relation])\n",
|
|
"show(8 * mat)\n",
|
|
"show(8 * equation)\n",
|
|
"show(weyl_generators(mat, standard_basis(4)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 58,
|
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"metadata": {},
|
|
"outputs": [
|
|
{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2 \\, {\\left(2 \\, \\sqrt{6} - 5\\right)}}</script></html>"
|
|
],
|
|
"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2 \\, {\\left(2 \\, \\sqrt{6} - 5\\right)}}$$"
|
|
],
|
|
"text/plain": [
|
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"-1/2/(2*sqrt(6) - 5)"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"def abbc_coords_3d(b, h1, h2, h3):\n",
|
|
" return (1-h1^2-h2^2-h3^2)/b\n",
|
|
"\n",
|
|
"show(simplify(abbc_coords_3d(10-4*sqrt(6), 0, 0, 0)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 59,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
|
|
"-\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & \\sqrt{6} + \\frac{5}{2} \\\\\n",
|
|
"1 & 1 & 1 & 1 & -4 \\, \\sqrt{6} + 10 \\\\\n",
|
|
"1 & -1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & -1 & 0 \\\\\n",
|
|
"-\\frac{1}{2} \\, \\sqrt{2} & -\\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & 0\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
|
|
"-\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & \\sqrt{6} + \\frac{5}{2} \\\\\n",
|
|
"1 & 1 & 1 & 1 & -4 \\, \\sqrt{6} + 10 \\\\\n",
|
|
"1 & -1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & -1 & 0 \\\\\n",
|
|
"-\\frac{1}{2} \\, \\sqrt{2} & -\\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & 0\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ -1/2 -1/2 -1/2 -1/2 sqrt(6) + 5/2]\n",
|
|
"[ 1 1 1 1 -4*sqrt(6) + 10]\n",
|
|
"[ 1 -1 0 0 0]\n",
|
|
"[ 0 0 1 -1 0]\n",
|
|
"[ -1/2*sqrt(2) -1/2*sqrt(2) 1/2*sqrt(2) 1/2*sqrt(2) 0]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"\n",
|
|
"W = matrix([\n",
|
|
" [-1/2, -1/2, -1/2, -1/2, 5/2+sqrt(6)],\n",
|
|
" [1, 1, 1, 1, 10-4*sqrt(6)],\n",
|
|
" [1, -1, 0, 0, 0],\n",
|
|
" [0, 0, 1, -1, 0],\n",
|
|
" [-1/sqrt(2), -1/sqrt(2), 1/sqrt(2), 1/sqrt(2), 0]\n",
|
|
"])\n",
|
|
"show(W)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 60,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
|
|
"0 & -\\frac{1}{2} & 0 & 0 & 0 \\\\\n",
|
|
"-\\frac{1}{2} & 0 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
|
|
"0 & -\\frac{1}{2} & 0 & 0 & 0 \\\\\n",
|
|
"-\\frac{1}{2} & 0 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ 0 -1/2 0 0 0]\n",
|
|
"[-1/2 0 0 0 0]\n",
|
|
"[ 0 0 1 0 0]\n",
|
|
"[ 0 0 0 1 0]\n",
|
|
"[ 0 0 0 0 1]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"P = matrix([\n",
|
|
" [0, -1/2, 0, 0, 0],\n",
|
|
" [-1/2, 0, 0, 0, 0],\n",
|
|
" [0, 0, 1, 0, 0],\n",
|
|
" [0, 0, 0, 1, 0],\n",
|
|
" [0, 0, 0, 0, 1]\n",
|
|
"])\n",
|
|
"show(P)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 61,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
|
|
"2 & 0 & 0 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"0 & 2 & 0 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"0 & 0 & 2 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"0 & 0 & 0 & 2 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"-\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & {\\left(2 \\, \\sqrt{6} + 5\\right)} {\\left(2 \\, \\sqrt{6} - 5\\right)}\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrr}\n",
|
|
"2 & 0 & 0 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"0 & 2 & 0 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"0 & 0 & 2 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"0 & 0 & 0 & 2 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
|
"-\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} & {\\left(2 \\, \\sqrt{6} + 5\\right)} {\\left(2 \\, \\sqrt{6} - 5\\right)}\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ 2 0 0 0 -3/2*sqrt(6) + 5/4]\n",
|
|
"[ 0 2 0 0 -3/2*sqrt(6) + 5/4]\n",
|
|
"[ 0 0 2 0 -3/2*sqrt(6) + 5/4]\n",
|
|
"[ 0 0 0 2 -3/2*sqrt(6) + 5/4]\n",
|
|
"[ -3/2*sqrt(6) + 5/4 -3/2*sqrt(6) + 5/4 -3/2*sqrt(6) + 5/4 -3/2*sqrt(6) + 5/4 (2*sqrt(6) + 5)*(2*sqrt(6) - 5)]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"show(W.transpose() * P * W)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 62,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrrrrrrrrr}\n",
|
|
"1 & -1 & -1 & -3 & -1 & -3 & -3 & -5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 \\\\\n",
|
|
"-1 & 1 & -3 & -1 & -3 & -1 & -5 & -3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 \\\\\n",
|
|
"-1 & -3 & 1 & -1 & -3 & -5 & -1 & -3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 \\\\\n",
|
|
"-3 & -1 & -1 & 1 & -5 & -3 & -3 & -1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 \\\\\n",
|
|
"-1 & -3 & -3 & -5 & 1 & -1 & -1 & -3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 \\\\\n",
|
|
"-3 & -1 & -5 & -3 & -1 & 1 & -3 & -1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 \\\\\n",
|
|
"-3 & -5 & -1 & -3 & -1 & -3 & 1 & -1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 \\\\\n",
|
|
"-5 & -3 & -3 & -1 & -3 & -1 & -1 & 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & 1 & -1 & -1 & -3 & -1 & -3 & -3 & -5 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -1 & 1 & -3 & -1 & -3 & -1 & -5 & -3 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -1 & -3 & 1 & -1 & -3 & -5 & -1 & -3 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -3 & -1 & -1 & 1 & -5 & -3 & -3 & -1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -1 & -3 & -3 & -5 & 1 & -1 & -1 & -3 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -3 & -1 & -5 & -3 & -1 & 1 & -3 & -1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -3 & -5 & -1 & -3 & -1 & -3 & 1 & -1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -5 & -3 & -3 & -1 & -3 & -1 & -1 & 1\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrrrrrrrrrrrrrr}\n",
|
|
"1 & -1 & -1 & -3 & -1 & -3 & -3 & -5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 \\\\\n",
|
|
"-1 & 1 & -3 & -1 & -3 & -1 & -5 & -3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 \\\\\n",
|
|
"-1 & -3 & 1 & -1 & -3 & -5 & -1 & -3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 \\\\\n",
|
|
"-3 & -1 & -1 & 1 & -5 & -3 & -3 & -1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 \\\\\n",
|
|
"-1 & -3 & -3 & -5 & 1 & -1 & -1 & -3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 \\\\\n",
|
|
"-3 & -1 & -5 & -3 & -1 & 1 & -3 & -1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 \\\\\n",
|
|
"-3 & -5 & -1 & -3 & -1 & -3 & 1 & -1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 \\\\\n",
|
|
"-5 & -3 & -3 & -1 & -3 & -1 & -1 & 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & 1 & -1 & -1 & -3 & -1 & -3 & -3 & -5 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -1 & 1 & -3 & -1 & -3 & -1 & -5 & -3 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -1 & -3 & 1 & -1 & -3 & -5 & -1 & -3 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -3 & -1 & -1 & 1 & -5 & -3 & -3 & -1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -1 & -3 & -3 & -5 & 1 & -1 & -1 & -3 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -3 & -1 & -5 & -3 & -1 & 1 & -3 & -1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -3 & -5 & -1 & -3 & -1 & -3 & 1 & -1 \\\\\n",
|
|
"-\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 5 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 3 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} - 1 & -\\sqrt{3} - \\frac{1}{\\sqrt{3} + 2} + 1 & -5 & -3 & -3 & -1 & -3 & -1 & -1 & 1\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
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"text/plain": [
|
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"[ 1 -1 -1 -3 -1 -3 -3 -5 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5]\n",
|
|
"[ -1 1 -3 -1 -3 -1 -5 -3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3]\n",
|
|
"[ -1 -3 1 -1 -3 -5 -1 -3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3]\n",
|
|
"[ -3 -1 -1 1 -5 -3 -3 -1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1]\n",
|
|
"[ -1 -3 -3 -5 1 -1 -1 -3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3]\n",
|
|
"[ -3 -1 -5 -3 -1 1 -3 -1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1]\n",
|
|
"[ -3 -5 -1 -3 -1 -3 1 -1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1]\n",
|
|
"[ -5 -3 -3 -1 -3 -1 -1 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 1 -1 -1 -3 -1 -3 -3 -5]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -1 1 -3 -1 -3 -1 -5 -3]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -1 -3 1 -1 -3 -5 -1 -3]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -3 -1 -1 1 -5 -3 -3 -1]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -1 -3 -3 -5 1 -1 -1 -3]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -3 -1 -5 -3 -1 1 -3 -1]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -3 -5 -1 -3 -1 -3 1 -1]\n",
|
|
"[-sqrt(3) - 1/(sqrt(3) + 2) - 5 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 3 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) - 1 -sqrt(3) - 1/(sqrt(3) + 2) + 1 -5 -3 -3 -1 -3 -1 -1 1]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"bt = 2 * (2 + sqrt(3))\n",
|
|
"b = 1 / (2 + sqrt(3))\n",
|
|
"\n",
|
|
"W = matrix([\n",
|
|
" [2, 2, 2, 2, 2, 2, 2, 2, bt, bt, bt, bt, bt, bt, bt, bt],\n",
|
|
" [1, 1, 1, 1, 1, 1, 1, 1, b, b, b, b, b, b, b, b],\n",
|
|
" [1, 1, 1, 1, -1, -1, -1, -1] * 2,\n",
|
|
" [1, 1, -1, -1, 1, 1, -1, -1] * 2,\n",
|
|
" [1, -1, 1, -1, 1, -1, 1, -1] * 2\n",
|
|
"])\n",
|
|
"\n",
|
|
"show(W.transpose() * P * W)"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "SageMath 9.3",
|
|
"language": "sage",
|
|
"name": "sagemath"
|
|
},
|
|
"language_info": {
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"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
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"name": "python",
|
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"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.9.6"
|
|
},
|
|
"name": "Apollonian Circle Packings.ipynb"
|
|
},
|
|
"nbformat": 4,
|
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"nbformat_minor": 4
|
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}
|