3510 lines
184 KiB
Text
3510 lines
184 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": 23,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(1/2*sqrt(2), 1)\n",
|
|
"sqrt(2)\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"image/png": 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\n",
|
|
"text/plain": [
|
|
"Graphics object consisting of 4 graphics primitives"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(1/4*sqrt(2), 0)\n",
|
|
"2*sqrt(2)\n"
|
|
]
|
|
}
|
|
],
|
|
"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",
|
|
"\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."
|
|
]
|
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},
|
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{
|
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"cell_type": "markdown",
|
|
"metadata": {},
|
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"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."
|
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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": 24,
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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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"[ 1 0 0 0 1/2 1/2 1/2 -1/2]\n",
|
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"[ 0 1 0 0 1/2 1/2 -1/2 1/2]\n",
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"[ 0 0 1 0 1/2 -1/2 1/2 1/2]\n",
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"[ 0 0 0 1 -1/2 1/2 1/2 1/2]"
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]
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},
|
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"execution_count": 24,
|
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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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"Wc = matrix([\n",
|
|
" [4, 0, 0, 1],\n",
|
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" [0, 2, 0, 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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" [2, 1, sqrt(2), 1],\n",
|
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" [4, 0, 0, -1],\n",
|
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" [0, 2, 0, -1],\n",
|
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"])\n",
|
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"\n",
|
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"Wc.transpose().rref()"
|
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]
|
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},
|
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{
|
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"cell_type": "markdown",
|
|
"metadata": {},
|
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"source": [
|
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"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."
|
|
]
|
|
},
|
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{
|
|
"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."
|
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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": 25,
|
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"metadata": {},
|
|
"outputs": [],
|
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"source": [
|
|
"W = matrix([\n",
|
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" [4, 0, 0, 1],\n",
|
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" [0, 2, 0, 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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"])"
|
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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": 26,
|
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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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"[ 1 -3 -3 -3]\n",
|
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"[-3 1 -3 -3]\n",
|
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"[-3 -3 1 -3]\n",
|
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"[-3 -3 -3 1]"
|
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]
|
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},
|
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"execution_count": 26,
|
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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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"cell_type": "code",
|
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"execution_count": 27,
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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": [
|
|
"[ 2 -6/5 -6/5 -6/5]\n",
|
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"[-6/5 2 -6/5 -6/5]\n",
|
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"[-6/5 -6/5 2 -6/5]\n",
|
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"[-6/5 -6/5 -6/5 2]"
|
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]
|
|
},
|
|
"execution_count": 27,
|
|
"metadata": {},
|
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"output_type": "execute_result"
|
|
}
|
|
],
|
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"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>"
|
|
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|
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|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]$$"
|
|
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|
|
"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]"
|
|
]
|
|
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|
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|
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|
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|
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|
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|
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|
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|
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"<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>"
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, b_{1}^{2} - 6 \\, b_{1} b_{2} + 5 \\, b_{2}^{2} - 6 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 5 \\, b_{3}^{2} - 6 \\, b_{1} b_{4} - 6 \\, b_{2} b_{4} - 6 \\, b_{3} b_{4} + 5 \\, b_{4}^{2}$$"
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|
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|
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|
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|
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|
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|
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|
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|
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"\\end{array}\\right)\\right]$$"
|
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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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"[ 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",
|
|
"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)))"
|
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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": 35,
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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_{1} + b_{3} + b_{4}, b_{6} = b_{2} + b_{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[b_{5} = b_{1} + b_{3} + b_{4}, b_{6} = b_{2} + b_{3} + b_{4}\\right]$$"
|
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],
|
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"text/plain": [
|
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"[b5 == b1 + b3 + b4, b6 == b2 + b3 + b4]"
|
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]
|
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},
|
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"metadata": {},
|
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"17 & 15 & -6 & -6 \\\\\n",
|
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"15 & 17 & -6 & -6 \\\\\n",
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"-6 & -6 & 4 & 0 \\\\\n",
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"-6 & -6 & 0 & 4\n",
|
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"\\end{array}\\right)</script></html>"
|
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
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"17 & 15 & -6 & -6 \\\\\n",
|
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"15 & 17 & -6 & -6 \\\\\n",
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"-6 & -6 & 4 & 0 \\\\\n",
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"-6 & -6 & 0 & 4\n",
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"\\end{array}\\right)$$"
|
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],
|
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"text/plain": [
|
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"[17 15 -6 -6]\n",
|
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"[15 17 -6 -6]\n",
|
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"[-6 -6 4 0]\n",
|
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"[-6 -6 0 4]"
|
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]
|
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},
|
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}17 \\, b_{1}^{2} + 30 \\, b_{1} b_{2} + 17 \\, b_{2}^{2} - 12 \\, b_{1} b_{3} - 12 \\, b_{2} b_{3} + 4 \\, b_{3}^{2} - 12 \\, b_{1} b_{4} - 12 \\, b_{2} b_{4} + 4 \\, b_{4}^{2}</script></html>"
|
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],
|
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"text/latex": [
|
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"$$\\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",
|
|
"-\\frac{30}{17} & -1 & \\frac{12}{17} & \\frac{12}{17} \\\\\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",
|
|
"3 & 3 & -1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
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"\\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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"3 & 3 & 0 & -1\n",
|
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"\\end{array}\\right)\\right]$$"
|
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],
|
|
"text/plain": [
|
|
"[\n",
|
|
"[ -1 -30/17 12/17 12/17] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [-30/17 -1 12/17 12/17]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1],\n",
|
|
"\n",
|
|
"[ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [ 0 1 0 0]\n",
|
|
"[ 3 3 -1 0] [ 0 0 1 0]\n",
|
|
"[ 0 0 0 1], [ 3 3 0 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"# octahedral\n",
|
|
"relation, mat, equation = quadform_from_root(matrix([\n",
|
|
" [2, 0, 0, 1],\n",
|
|
" [2, 0, 0, -1],\n",
|
|
" [1, 1, sqrt(2), 0],\n",
|
|
" [1, 1, -sqrt(2), 0],\n",
|
|
" [4, 2, 0, 1],\n",
|
|
" [4, 2, 0, -1],\n",
|
|
"]))\n",
|
|
"show(relation)\n",
|
|
"show(8 * mat)\n",
|
|
"show(8 * equation)\n",
|
|
"show(weyl_generators(8 * mat, standard_basis(4)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 36,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]$$"
|
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],
|
|
"text/plain": [
|
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"[]"
|
|
]
|
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},
|
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"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
|
|
"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"1 & -1 & -1 & -1 \\\\\n",
|
|
"-1 & 1 & -1 & -1 \\\\\n",
|
|
"-1 & -1 & 1 & -1 \\\\\n",
|
|
"-1 & -1 & -1 & 1\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"1 & -1 & -1 & -1 \\\\\n",
|
|
"-1 & 1 & -1 & -1 \\\\\n",
|
|
"-1 & -1 & 1 & -1 \\\\\n",
|
|
"-1 & -1 & -1 & 1\n",
|
|
"\\end{array}\\right)$$"
|
|
],
|
|
"text/plain": [
|
|
"[ 1 -1 -1 -1]\n",
|
|
"[-1 1 -1 -1]\n",
|
|
"[-1 -1 1 -1]\n",
|
|
"[-1 -1 -1 1]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\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}$$"
|
|
],
|
|
"text/plain": [
|
|
"b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & 2 & 2 & 2 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"2 & -1 & 2 & 2 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"2 & 2 & -1 & 2 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"2 & 2 & 2 & -1\n",
|
|
"\\end{array}\\right)\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\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]$$"
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],
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"text/plain": [
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"[b1 == b1,\n",
|
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" b2 == b2,\n",
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" b3 == b3,\n",
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" b4 == b4,\n",
|
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" b5 == b2 - 2*b3 + 2*b4,\n",
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" b6 == 2*b2 - 3*b3 + 2*b4,\n",
|
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" b7 == 2*b2 - 2*b3 + b4]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{2}, b_{4}, b_{6}\\right]</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{2}, b_{4}, b_{6}\\right]$$"
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],
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"text/plain": [
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"[b2, b4, b6]"
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"output_type": "display_data"
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[ 9 -1 -1 -1]\n",
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"[-1 1 -1 -1]\n",
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"[-1 -1 1 -1]\n",
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"[-1 -1 -1 1]\n"
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]
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{5} - 2 \\, b_{3} b_{5} + b_{5}^{2} - 2 \\, b_{1} b_{7} - 2 \\, b_{3} b_{7} - 2 \\, b_{5} b_{7} + b_{7}^{2}</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{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}$$"
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],
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"text/plain": [
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"9*b1^2 - 2*b1*b3 + b3^2 - 2*b1*b5 - 2*b3*b5 + b5^2 - 2*b1*b7 - 2*b3*b7 - 2*b5*b7 + b7^2"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[3 \\, b_{1} = 3 \\, b_{1}, 3 \\, b_{2} = 3 \\, b_{2}, 3 \\, b_{3} = 3 \\, b_{3}, 3 \\, b_{4} = 3 \\, b_{4}, 3 \\, b_{5} = 3 \\, b_{2} - 6 \\, b_{3} + 6 \\, b_{4}, 3 \\, b_{6} = 6 \\, b_{2} - 9 \\, b_{3} + 6 \\, b_{4}, 3 \\, b_{7} = 6 \\, b_{2} - 6 \\, b_{3} + 3 \\, b_{4}\\right]</script></html>"
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],
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"text/latex": [
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"$$\\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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"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",
|
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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",
|
|
"print(12 * Q)\n",
|
|
"show(12 * nqform)\n",
|
|
"show([3 * rel for rel in relations])\n",
|
|
"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": [],
|
|
"source": [
|
|
"# function to compute the linear relations and quadratic formula for an ngon base pyramid\n",
|
|
"# the circles will be numbered 1 in the center, then 2 through n winding around the circle by step, so\n",
|
|
"# for n = 4 with step = 1, we have\n",
|
|
"# b3\n",
|
|
"# b4 b1 b2\n",
|
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"# b5\n",
|
|
"# and the quadratic form is in terms of b1, b2, b3, and b4\n",
|
|
"# and for n = 6 with step = 2, we have\n",
|
|
"# b4 b3\n",
|
|
"# b5 b1 b2\n",
|
|
"# b6 b7\n",
|
|
"# and the quadratic form is in terms of b1, b3, b5, and b7\n",
|
|
"def ngon_linear_relations_and_quadratic_form(n, step=1):\n",
|
|
" mat = wmatrix(n)\n",
|
|
" \n",
|
|
" # work out initial relations\n",
|
|
" relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 2) ]) * mat.transpose().rref()\n",
|
|
" relations = []\n",
|
|
" for i in range(4, n + 1):\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",
|
|
" shuffled = (list(range(1, n + 2)) * step)[::step]\n",
|
|
" targets = [ var('b' + str(i)) for i in shuffled[4:] ]\n",
|
|
" relations = solve(relations, *targets)[0]\n",
|
|
" \n",
|
|
" # find the matrix corresponding to the quadratic form, picking the appropriate rows from the matrix\n",
|
|
" Q = mat[:4*step:step,:4*step:step].inverse()\n",
|
|
" \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",
|
|
" \n",
|
|
" return relations, Q, expand(nqform)"
|
|
]
|
|
},
|
|
{
|
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"cell_type": "code",
|
|
"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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],
|
|
"text/latex": [
|
|
"$$\\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",
|
|
" b6 == b4*(sqrt(2) + 2) + b2*(sqrt(2) + 1) - 2*b3*(sqrt(2) + 1),\n",
|
|
" b7 == -b3*(2*sqrt(2) + 3) + b2*(sqrt(2) + 2) + b4*(sqrt(2) + 2),\n",
|
|
" b8 == b2*(sqrt(2) + 2) - 2*b3*(sqrt(2) + 1) + b4*(sqrt(2) + 1),\n",
|
|
" b9 == b2*(sqrt(2) + 1) - b3*(sqrt(2) + 1) + b4]"
|
|
]
|
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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",
|
|
"[ -(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": [
|
|
"$$\\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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"\\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]</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)$$"
|
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],
|
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"text/plain": [
|
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"[ 9 -1 -1 -1]\n",
|
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"[-1 1 -1 -1]\n",
|
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"[-1 -1 1 -1]\n",
|
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"[-1 -1 -1 1]"
|
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]
|
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},
|
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"metadata": {},
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"output_type": "display_data"
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},
|
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{
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
|
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],
|
|
"text/latex": [
|
|
"$$\\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": [
|
|
"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>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\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",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"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]$$"
|
|
],
|
|
"text/plain": [
|
|
"[\n",
|
|
"[ -1 2/9 2/9 2/9] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
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"[ 0 1 0 0] [ 2 -1 2 2] [ 0 1 0 0] [ 0 1 0 0]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0] [ 2 2 -1 2] [ 0 0 1 0]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 2 2 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2}{9} \\, b_{2} + \\frac{2}{9} \\, b_{3} + \\frac{2}{9} \\, b_{4}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"$$\\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"
|
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]
|
|
},
|
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"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",
|
|
" 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)"
|
|
]
|
|
},
|
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{
|
|
"cell_type": "code",
|
|
"execution_count": 45,
|
|
"metadata": {},
|
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"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",
|
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"(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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"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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"[6/5 6/5 6/5 -1]\n",
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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",
|
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"Wc = matrix([\n",
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" [4, 0, 0, 1],\n",
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" [0, 2, 0, 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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" [2, 1, sqrt(2), 1],\n",
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" [4, 0, 0, -1],\n",
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" [0, 2, 0, -1],\n",
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"])\n",
|
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"relation, mat, equation = linear_relations_and_quadratic_form_from_wtpw(Wc * P * Wc.transpose(), [1, 2, 3, 4, 5, 6, 7, 8])\n",
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"show([2 * eq for eq in 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(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": 68,
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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}}-\\frac{1}{2 \\, {\\left(2 \\, \\sqrt{6} - 5\\right)}}</script></html>"
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],
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"text/latex": [
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"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2 \\, {\\left(2 \\, \\sqrt{6} - 5\\right)}}$$"
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],
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"text/plain": [
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"-1/2/(2*sqrt(6) - 5)"
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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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"def abbc_coords_3d(b, h1, h2, h3):\n",
|
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" return (1-h1^2-h2^2-h3^2)/b\n",
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"\n",
|
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"show(simplify(abbc_coords_3d(10-4*sqrt(6), 0, 0, 0)))"
|
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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": 64,
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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(\\begin{array}{rrrrr}\n",
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"-\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & \\sqrt{6} + \\frac{5}{2} \\\\\n",
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"1 & 1 & 1 & 1 & -4 \\, \\sqrt{6} + 10 \\\\\n",
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"1 & -1 & 0 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & -1 & 0 \\\\\n",
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"-\\frac{1}{2} \\, \\sqrt{2} & -\\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & 0\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}{rrrrr}\n",
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"-\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & -\\frac{1}{2} & \\sqrt{6} + \\frac{5}{2} \\\\\n",
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"1 & 1 & 1 & 1 & -4 \\, \\sqrt{6} + 10 \\\\\n",
|
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"1 & -1 & 0 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & -1 & 0 \\\\\n",
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"-\\frac{1}{2} \\, \\sqrt{2} & -\\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & \\frac{1}{2} \\, \\sqrt{2} & 0\n",
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"\\end{array}\\right)$$"
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],
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"text/plain": [
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"[ -1/2 -1/2 -1/2 -1/2 sqrt(6) + 5/2]\n",
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"[ 1 1 1 1 -4*sqrt(6) + 10]\n",
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"[ 1 -1 0 0 0]\n",
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"[ 0 0 1 -1 0]\n",
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"[ -1/2*sqrt(2) -1/2*sqrt(2) 1/2*sqrt(2) 1/2*sqrt(2) 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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"source": [
|
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"\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",
|
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" [0, 0, 1, -1, 0],\n",
|
|
" [-1/sqrt(2), -1/sqrt(2), 1/sqrt(2), 1/sqrt(2), 0]\n",
|
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"])\n",
|
|
"show(W)"
|
|
]
|
|
},
|
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{
|
|
"cell_type": "code",
|
|
"execution_count": 62,
|
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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(\\begin{array}{rrrrr}\n",
|
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"0 & -\\frac{1}{2} & 0 & 0 & 0 \\\\\n",
|
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"-\\frac{1}{2} & 0 & 0 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 0 & 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}{rrrrr}\n",
|
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"0 & -\\frac{1}{2} & 0 & 0 & 0 \\\\\n",
|
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"-\\frac{1}{2} & 0 & 0 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right)$$"
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],
|
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"text/plain": [
|
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"[ 0 -1/2 0 0 0]\n",
|
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"[-1/2 0 0 0 0]\n",
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"[ 0 0 1 0 0]\n",
|
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"[ 0 0 0 1 0]\n",
|
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"[ 0 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": [
|
|
"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)"
|
|
]
|
|
},
|
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{
|
|
"cell_type": "code",
|
|
"execution_count": 63,
|
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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(\\begin{array}{rrrrr}\n",
|
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"2 & 0 & 0 & 0 & -\\frac{3}{2} \\, \\sqrt{6} + \\frac{5}{4} \\\\\n",
|
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"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",
|
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"-\\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",
|
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"\\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",
|
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"\\end{array}\\right)$$"
|
|
],
|
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"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": 75,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"ename": "AttributeError",
|
|
"evalue": "'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense' object has no attribute 'full_simplify'",
|
|
"output_type": "error",
|
|
"traceback": [
|
|
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
|
|
"\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)",
|
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"\u001b[0;32m<ipython-input-75-c08f2798205a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m 10\u001b[0m ])\n\u001b[1;32m 11\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 12\u001b[0;31m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mW\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtranspose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mP\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mW\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfull_simplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
|
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"\u001b[0;32m/usr/lib/python3.9/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4716)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 491\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;34m'LeftZeroSemigroup_with_category.element_class'\u001b[0m \u001b[0mobject\u001b[0m \u001b[0mhas\u001b[0m \u001b[0mno\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m'blah_blah'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 492\u001b[0m \"\"\"\n\u001b[0;32m--> 493\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 494\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 495\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mgetattr_from_category\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;32m/usr/lib/python3.9/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4828)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 504\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 505\u001b[0m \u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mP\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_abstract_element_class\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 506\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mgetattr_from_other_class\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 507\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 508\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__dir__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;32m/usr/lib/python3.9/site-packages/sage/cpython/getattr.pyx\u001b[0m in \u001b[0;36msage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2625)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 370\u001b[0m \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcls\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 371\u001b[0m \u001b[0mdummy_error_message\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 372\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mAttributeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdummy_error_message\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 373\u001b[0m \u001b[0mattribute\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m<\u001b[0m\u001b[0mobject\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mattr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 374\u001b[0m \u001b[0;31m# Check for a descriptor (__get__ in Python)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
|
|
"\u001b[0;31mAttributeError\u001b[0m: 'sage.matrix.matrix_symbolic_dense.Matrix_symbolic_dense' object has no attribute 'full_simplify'"
|
|
]
|
|
}
|
|
],
|
|
"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": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.9.5"
|
|
},
|
|
"name": "Apollonian Circle Packings.ipynb"
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|