2171 lines
133 KiB
Text
2171 lines
133 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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"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": 2,
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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": 2,
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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": 3,
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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": 4,
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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": 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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"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": 5,
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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": 5,
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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": 6,
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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": 7,
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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": 8,
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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": 8,
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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": 9,
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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": 9,
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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": 10,
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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": 11,
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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": 12,
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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": 12,
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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": 13,
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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": 13,
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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": 14,
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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": [
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"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)."
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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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"(7, 2, 4, 6)"
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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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"root = vector([-1, 2, 4, 6])\n",
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"S1 * root"
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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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{
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"data": {
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"text/plain": [
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"(-1, 4, 4, 6)"
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]
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},
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"execution_count": 16,
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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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"S2 * root"
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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": 17,
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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, 2, 2, 6)"
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]
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},
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"execution_count": 17,
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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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"S3 * root"
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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": 18,
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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, 2, 4, 14)"
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]
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},
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"execution_count": 18,
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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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"S4 * root"
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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 curvatures of the dual circles of the octahedral packing\n",
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"\n",
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"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."
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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": 19,
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"metadata": {},
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"outputs": [],
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"source": [
|
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"def circle_from_points(pta, ptb, ptc):\n",
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" a = var('a')\n",
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" b = var('b')\n",
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" r = var('r')\n",
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" x = var('x')\n",
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" y = var('y')\n",
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" \n",
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" circle_func = (x - a)^2 + (y - b)^2 == r^2\n",
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" \n",
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" eq1 = circle_func.subs(x == pta[0]).subs(y == pta[1])\n",
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" eq2 = circle_func.subs(x == ptb[0]).subs(y == ptb[1])\n",
|
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" eq3 = circle_func.subs(x == ptc[0]).subs(y == ptc[1])\n",
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" \n",
|
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" res = solve([eq1, eq2, eq3], a, b, r)[1]\n",
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" \n",
|
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" return (res[0].rhs(), res[1].rhs(), res[2].rhs())"
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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": 20,
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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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"name": "stdout",
|
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"output_type": "stream",
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"text": [
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"(1/2*sqrt(2), 1)\n",
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"sqrt(2)\n",
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"(1/4*sqrt(2), 0)\n",
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"2*sqrt(2)\n"
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]
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},
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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": [
|
|
"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",
|
|
"b3 = circle(( x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b4 = circle((-x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b5 = circle(( x, -y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b6 = circle((-x, -y), r, rgbcolor=(1, 0, 0))\n",
|
|
"\n",
|
|
"x, y, r = circle_from_points((sqrt(2) / 3, 1 / 3), (0, 0), (sqrt(2) / 3, -1 / 3))\n",
|
|
"print('({}, {})'.format(x, y))\n",
|
|
"print(1 / r)\n",
|
|
"\n",
|
|
"b7 = circle(( x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"b8 = circle((-x, y), r, rgbcolor=(1, 0, 0))\n",
|
|
"\n",
|
|
"show(c1 + c2 + c3 + c4 + c5 + c6 + b1 + b2 + b3 + b4 + b5 + b6 + b7 + b8)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# The quadratic form for the cubical packing\n",
|
|
"\n",
|
|
"We first made a matrix, $W_c$, whose rows are the abbc coordinates of the circles in the root octuple. Then we found the row echelon form of that matrix, resulting in a system of linear relations the coordinates must satisfy. Then, from there, we could derive the rest of the coordinates from the first four (we chose the first four to be the \"cubicle\" from the Stange), allowing us to derive the quadratic form."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 21,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def abbc_coords(b, h1, h2):\n",
|
|
" return [(h1^2 + h2^2 - 1) / b, b, h1, h2]"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"The idea here is that by multiplying $W_c$ by an arbitrary vector in $\\mathbf{R}^8$ and setting that equal to $\\vec{0}$, we can find a basis for the null space of $W_c$, which will end up giving us a bunch of linear relations the curvatures must satisfy. The thing to notice is that $W_c\\vec{v} = 0$ for some $\\vec{v}\\in\\mathbf{R}^8$ gives the system of equations with coefficient matrix $W^T$. So finding the row echelon form of $W^T$ will give us the coefficients in the simplified system of linear relations, with each row equal to 0."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 22,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[ 1 0 0 0 1/2 1/2 1/2 -1/2]\n",
|
|
"[ 0 1 0 0 1/2 1/2 -1/2 1/2]\n",
|
|
"[ 0 0 1 0 1/2 -1/2 1/2 1/2]\n",
|
|
"[ 0 0 0 1 -1/2 1/2 1/2 1/2]"
|
|
]
|
|
},
|
|
"execution_count": 22,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"Wc = matrix([\n",
|
|
" [4, 0, 0, 1],\n",
|
|
" [0, 2, 0, 1],\n",
|
|
" [2, 1, -sqrt(2), -1],\n",
|
|
" [2, 1, sqrt(2), -1],\n",
|
|
" [2, 1, -sqrt(2), 1],\n",
|
|
" [2, 1, sqrt(2), 1],\n",
|
|
" [4, 0, 0, -1],\n",
|
|
" [0, 2, 0, -1],\n",
|
|
"])\n",
|
|
"\n",
|
|
"Wc.transpose().rref()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"From here we can conclude that $$\n",
|
|
"\\begin{align*}\n",
|
|
" 2b_1 &= -b_5 - b_6 - b_7 + b_8\\\\\n",
|
|
" 2b_2 &= -b_5 - b_6 + b_7 - b_8\\\\\n",
|
|
" 2b_3 &= -b_5 + b_6 - b_7 - b_8\\\\\n",
|
|
" 2b_4 &= b_5 - b_6 - b_7 - b_8\n",
|
|
"\\end{align*}\n",
|
|
"$$\n",
|
|
"This differs slightly from Stange's system of equations because we put the circles in a slightly different order."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"In any case, this gives us the tools to derive the full octuple from just four coordinates, letting us use those four coordinates to represent the entire octuple, and finally making $WPW^T$ nonsingular, allowing us to find the quadratic form the same way we did for the Descartes quadratic form and the octahedral quadratic form."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 23,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"W = matrix([\n",
|
|
" [4, 0, 0, 1],\n",
|
|
" [0, 2, 0, 1],\n",
|
|
" [2, 1, -sqrt(2), -1],\n",
|
|
" [2, 1, sqrt(2), -1],\n",
|
|
"])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 24,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"[ 1 -3 -3 -3]\n",
|
|
"[-3 1 -3 -3]\n",
|
|
"[-3 -3 1 -3]\n",
|
|
"[-3 -3 -3 1]"
|
|
]
|
|
},
|
|
"execution_count": 24,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"m = W * P * W.transpose()\n",
|
|
"m"
|
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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": {},
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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 -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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]
|
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},
|
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"execution_count": 25,
|
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"metadata": {},
|
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"output_type": "execute_result"
|
|
}
|
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],
|
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"source": [
|
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"32 * m.inverse() * 2/5"
|
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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": [
|
|
"[bt1 b1 h11 h21]\n",
|
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"[bt2 b2 h12 h22]\n",
|
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"[bt3 b3 h13 h23]\n",
|
|
"[bt4 b4 h14 h24]"
|
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]
|
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},
|
|
"execution_count": 26,
|
|
"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": 27,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"D = W2.transpose() * m.inverse() * W2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 28,
|
|
"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": 28,
|
|
"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": 29,
|
|
"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": 29,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"S_i = weyl_generators(m.inverse(), standard_basis(4))\n",
|
|
"S_i"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 30,
|
|
"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": 30,
|
|
"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": 31,
|
|
"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": 122,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 \\, b_{5} = b_{1} + b_{2} + b_{3} - b_{4}, 2 \\, b_{6} = b_{1} + b_{2} - b_{3} + b_{4}, 2 \\, b_{7} = b_{1} - b_{2} + b_{3} + b_{4}, 2 \\, b_{8} = -b_{1} + b_{2} + b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\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]\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"[2*b5 == b1 + b2 + b3 - b4,\n",
|
|
" 2*b6 == b1 + b2 - b3 + b4,\n",
|
|
" 2*b7 == b1 - b2 + b3 + b4,\n",
|
|
" 2*b8 == -b1 + b2 + b3 + b4]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"5 & -3 & -3 & -3 \\\\\n",
|
|
"-3 & 5 & -3 & -3 \\\\\n",
|
|
"-3 & -3 & 5 & -3 \\\\\n",
|
|
"-3 & -3 & -3 & 5\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"5 & -3 & -3 & -3 \\\\\n",
|
|
"-3 & 5 & -3 & -3 \\\\\n",
|
|
"-3 & -3 & 5 & -3 \\\\\n",
|
|
"-3 & -3 & -3 & 5\n",
|
|
"\\end{array}\\right)\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"[ 5 -3 -3 -3]\n",
|
|
"[-3 5 -3 -3]\n",
|
|
"[-3 -3 5 -3]\n",
|
|
"[-3 -3 -3 5]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, b_{1}^{2} - 6 \\, b_{1} b_{2} + 5 \\, b_{2}^{2} - 6 \\, b_{1} b_{3} - 6 \\, b_{2} b_{3} + 5 \\, b_{3}^{2} - 6 \\, b_{1} b_{4} - 6 \\, b_{2} b_{4} - 6 \\, b_{3} b_{4} + 5 \\, b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\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}\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"5*b1^2 - 6*b1*b2 + 5*b2^2 - 6*b1*b3 - 6*b2*b3 + 5*b3^2 - 6*b1*b4 - 6*b2*b4 - 6*b3*b4 + 5*b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
|
|
"\\end{array}\\right)\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & \\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & -1 & \\frac{6}{5} & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & -1 & \\frac{6}{5} \\\\\n",
|
|
"0 & 0 & 0 & 1\n",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"\\frac{6}{5} & \\frac{6}{5} & \\frac{6}{5} & -1\n",
|
|
"\\end{array}\\right)\\right]\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"[\n",
|
|
"[ -1 6/5 6/5 6/5] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [6/5 -1 6/5 6/5] [ 0 1 0 0]\n",
|
|
"[ 0 0 1 0] [ 0 0 1 0] [6/5 6/5 -1 6/5]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],\n",
|
|
"\n",
|
|
"[ 1 0 0 0]\n",
|
|
"[ 0 1 0 0]\n",
|
|
"[ 0 0 1 0]\n",
|
|
"[6/5 6/5 6/5 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"# cubical\n",
|
|
"relation, mat, equation = quadform_from_root(Wc)\n",
|
|
"show([2 * eq for eq in relation])\n",
|
|
"show(32 * mat)\n",
|
|
"show(32 * equation)\n",
|
|
"show(weyl_generators(mat, standard_basis(4)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 33,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{1} - b_{2} + b_{4}, b_{6} = b_{1} - b_{3} + b_{4}\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = b_{1} - b_{2} + b_{4}, b_{6} = b_{1} - b_{3} + b_{4}\\right]\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"[b5 == b1 - b2 + b4, b6 == b1 - 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",
|
|
"1 & -2 & -2 & -1 \\\\\n",
|
|
"-2 & 4 & 0 & -2 \\\\\n",
|
|
"-2 & 0 & 4 & -2 \\\\\n",
|
|
"-1 & -2 & -2 & 1\n",
|
|
"\\end{array}\\right)</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
|
"1 & -2 & -2 & -1 \\\\\n",
|
|
"-2 & 4 & 0 & -2 \\\\\n",
|
|
"-2 & 0 & 4 & -2 \\\\\n",
|
|
"-1 & -2 & -2 & 1\n",
|
|
"\\end{array}\\right)\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"[ 1 -2 -2 -1]\n",
|
|
"[-2 4 0 -2]\n",
|
|
"[-2 0 4 -2]\n",
|
|
"[-1 -2 -2 1]"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 4 \\, b_{1} b_{2} + 4 \\, b_{2}^{2} - 4 \\, b_{1} b_{3} + 4 \\, b_{3}^{2} - 2 \\, b_{1} b_{4} - 4 \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 4 \\, b_{1} b_{2} + 4 \\, b_{2}^{2} - 4 \\, b_{1} b_{3} + 4 \\, b_{3}^{2} - 2 \\, b_{1} b_{4} - 4 \\, b_{2} b_{4} - 4 \\, b_{3} b_{4} + b_{4}^{2}\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"b1^2 - 4*b1*b2 + 4*b2^2 - 4*b1*b3 + 4*b3^2 - 2*b1*b4 - 4*b2*b4 - 4*b3*b4 + b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"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 & 4 & 4 & 2 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
|
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"1 & -1 & 0 & 1 \\\\\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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"1 & 0 & -1 & 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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"2 & 4 & 4 & -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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"\\begin{math}\n",
|
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"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & 4 & 4 & 2 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"1 & -1 & 0 & 1 \\\\\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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"1 & 0 & -1 & 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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"2 & 4 & 4 & -1\n",
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"\\end{array}\\right)\\right]\n",
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"\\end{math}"
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],
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"text/plain": [
|
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"[\n",
|
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"[-1 4 4 2] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
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"[ 0 1 0 0] [ 1 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n",
|
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"[ 0 0 1 0] [ 0 0 1 0] [ 1 0 -1 1] [ 0 0 1 0]\n",
|
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 4 4 -1]\n",
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"]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
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"# octahedral\n",
|
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"relation, mat, equation = quadform_from_root(matrix([\n",
|
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" [2, 0, 0, 1],\n",
|
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" [2, 0, 0, -1],\n",
|
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" [-1, 1, 0, 0],\n",
|
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" [4, 2, 2*sqrt(2), -1],\n",
|
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" [4, 2, 2*sqrt(2), 1],\n",
|
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" [7, 1, 2*sqrt(2), 0],\n",
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"]))\n",
|
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"show(relation)\n",
|
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"show(8 * mat)\n",
|
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"show(8 * equation)\n",
|
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"show(weyl_generators(8 * mat, standard_basis(4)))"
|
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]
|
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},
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{
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"cell_type": "code",
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"execution_count": 34,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]</script></html>"
|
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],
|
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"text/latex": [
|
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"\\begin{math}\n",
|
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"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]\n",
|
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"\\end{math}"
|
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],
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"text/plain": [
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"[]"
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]
|
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"1 & -1 & -1 & -1 \\\\\n",
|
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"-1 & 1 & -1 & -1 \\\\\n",
|
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"-1 & -1 & 1 & -1 \\\\\n",
|
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"-1 & -1 & -1 & 1\n",
|
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"\\end{array}\\right)</script></html>"
|
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],
|
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"text/latex": [
|
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"\\begin{math}\n",
|
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"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n",
|
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"1 & -1 & -1 & -1 \\\\\n",
|
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"-1 & 1 & -1 & -1 \\\\\n",
|
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"-1 & -1 & 1 & -1 \\\\\n",
|
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"-1 & -1 & -1 & 1\n",
|
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"\\end{array}\\right)\n",
|
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"\\end{math}"
|
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],
|
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"text/plain": [
|
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"[ 1 -1 -1 -1]\n",
|
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"[-1 1 -1 -1]\n",
|
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"[-1 -1 1 -1]\n",
|
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"[-1 -1 -1 1]"
|
|
]
|
|
},
|
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"metadata": {},
|
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"output_type": "display_data"
|
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},
|
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{
|
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"data": {
|
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"text/html": [
|
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\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}\n",
|
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"\\end{math}"
|
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],
|
|
"text/plain": [
|
|
"b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
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"output_type": "display_data"
|
|
},
|
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{
|
|
"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",
|
|
"-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": [
|
|
"\\begin{math}\n",
|
|
"\\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]\n",
|
|
"\\end{math}"
|
|
],
|
|
"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": 35,
|
|
"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": 36,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"# just convenience function for quadratic forms\n",
|
|
"def qform(matrix, vector):\n",
|
|
" return vector * matrix * vector"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 37,
|
|
"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])\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",
|
|
" 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": 123,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"[ 9 -1 -1 -1]\n",
|
|
"[-1 1 -1 -1]\n",
|
|
"[-1 -1 1 -1]\n",
|
|
"[-1 -1 -1 1]\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{5} - 2 \\, b_{3} b_{5} + b_{5}^{2} - 2 \\, b_{1} b_{7} - 2 \\, b_{3} b_{7} - 2 \\, b_{5} b_{7} + b_{7}^{2}</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\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}\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"9*b1^2 - 2*b1*b3 + b3^2 - 2*b1*b5 - 2*b3*b5 + b5^2 - 2*b1*b7 - 2*b3*b7 - 2*b5*b7 + b7^2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
},
|
|
{
|
|
"data": {
|
|
"text/html": [
|
|
"<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>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\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]\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"[3*b2 == 2*b3 - b5 + 2*b7, 3*b4 == 2*b3 + 2*b5 - b7, 3*b6 == -b3 + 2*b5 + 2*b7]"
|
|
]
|
|
},
|
|
"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{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\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": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\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",
|
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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]\n",
|
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"\\end{math}"
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],
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"text/plain": [
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"[\n",
|
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"[ -1 2/9 2/9 2/9] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
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"[ 0 1 0 0] [ 2 -1 2 2] [ 0 1 0 0] [ 0 1 0 0]\n",
|
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"[ 0 0 1 0] [ 0 0 1 0] [ 2 2 -1 2] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 2 2 -1]\n",
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"]"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
|
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"relations, Q, nqform = linear_relations_and_quadratic_form_from_wtpw(wmatrix(6), [1, 3, 5, 7, 2, 4, 6])\n",
|
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"print(12 * Q)\n",
|
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"show(12 * nqform)\n",
|
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"show([3 * rel for rel in relations])\n",
|
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"show(weyl_generators(12 * Q, standard_basis(4)))"
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]
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},
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{
|
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"cell_type": "code",
|
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"execution_count": 39,
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"metadata": {},
|
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"outputs": [],
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"source": [
|
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"# function to compute the linear relations and quadratic formula for an ngon base pyramid\n",
|
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"# the circles will be numbered 1 in the center, then 2 through n winding around the circle by step, so\n",
|
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"# for n = 4 with step = 1, we have\n",
|
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"# b3\n",
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"# b4 b1 b2\n",
|
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"# b5\n",
|
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"# and the quadratic form is in terms of b1, b2, b3, and b4\n",
|
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"# and for n = 6 with step = 2, we have\n",
|
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"# b4 b3\n",
|
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"# b5 b1 b2\n",
|
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"# b6 b7\n",
|
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"# and the quadratic form is in terms of b1, b3, b5, and b7\n",
|
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"def ngon_linear_relations_and_quadratic_form(n, step=1):\n",
|
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" mat = wmatrix(n)\n",
|
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" \n",
|
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" # work out initial relations\n",
|
|
" relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 2) ]) * mat.transpose().rref()\n",
|
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" relations = []\n",
|
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" for i in range(4, n + 1):\n",
|
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" relations.append(var('b' + str(i + 1)) == relations_temp[i])\n",
|
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" \n",
|
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" # rewrite the relations in terms of the variables we care about, depends on the step\n",
|
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" shuffled = (list(range(1, n + 2)) * step)[::step]\n",
|
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" targets = [ var('b' + str(i)) for i in shuffled[4:] ]\n",
|
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" relations = solve(relations, *targets)[0]\n",
|
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" \n",
|
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" # find the matrix corresponding to the quadratic form, picking the appropriate rows from the matrix\n",
|
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" Q = mat[:4*step:step,:4*step:step].inverse()\n",
|
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" \n",
|
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" # find the quadratic form in variables; proper units will satisfy this being equal to zero\n",
|
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" nqform = qform(Q, vector([ var('b' + str(i)) for i in range(1, 5) ]))\n",
|
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" \n",
|
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" return relations, Q, expand(nqform)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 120,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[b_{5} = -b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4} {\\left(\\sqrt{2} + 1\\right)} + b_{2}, b_{6} = b_{4} {\\left(\\sqrt{2} + 2\\right)} + b_{2} {\\left(\\sqrt{2} + 1\\right)} - 2 \\, b_{3} {\\left(\\sqrt{2} + 1\\right)}, b_{7} = -b_{3} {\\left(2 \\, \\sqrt{2} + 3\\right)} + b_{2} {\\left(\\sqrt{2} + 2\\right)} + b_{4} {\\left(\\sqrt{2} + 2\\right)}, b_{8} = b_{2} {\\left(\\sqrt{2} + 2\\right)} - 2 \\, b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4} {\\left(\\sqrt{2} + 1\\right)}, b_{9} = b_{2} {\\left(\\sqrt{2} + 1\\right)} - b_{3} {\\left(\\sqrt{2} + 1\\right)} + b_{4}\\right]</script></html>"
|
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],
|
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"text/latex": [
|
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"\\begin{math}\n",
|
|
"\\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]\n",
|
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"\\end{math}"
|
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],
|
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"text/plain": [
|
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"[b5 == -b3*(sqrt(2) + 1) + b4*(sqrt(2) + 1) + b2,\n",
|
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" b6 == b4*(sqrt(2) + 2) + b2*(sqrt(2) + 1) - 2*b3*(sqrt(2) + 1),\n",
|
|
" 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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},
|
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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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"\\begin{math}\n",
|
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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)\n",
|
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"\\end{math}"
|
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],
|
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"text/plain": [
|
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"[ -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -2 -(sqrt(2) + 2)/(sqrt(2) - 2) - 3 -2]\n",
|
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"[ -2 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2 4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)]\n",
|
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"[ -(sqrt(2) + 2)/(sqrt(2) - 2) - 3 -2 -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -2]\n",
|
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"[ -2 4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)]"
|
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]
|
|
},
|
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"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>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\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}\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"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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"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": [
|
|
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
|
|
"-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",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
|
"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
|
"-\\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",
|
|
"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & 2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -1\n",
|
|
"\\end{array}\\right)\\right]</script></html>"
|
|
],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
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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",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
|
"-\\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",
|
|
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
|
"0 & 1 & 0 & 0 \\\\\n",
|
|
"0 & 0 & 1 & 0 \\\\\n",
|
|
"-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & 2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -1\n",
|
|
"\\end{array}\\right)\\right]\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
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"[\n",
|
|
"[ -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",
|
|
"[ 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",
|
|
"[ 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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"metadata": {},
|
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"output_type": "display_data"
|
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}
|
|
],
|
|
"source": [
|
|
"relations, Q, nqform = ngon_linear_relations_and_quadratic_form(8)\n",
|
|
"\n",
|
|
"show(relations)\n",
|
|
"show(8 * Q)\n",
|
|
"show(8 * nqform)\n",
|
|
"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",
|
|
"execution_count": 163,
|
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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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"\\begin{math}\n",
|
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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]\n",
|
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"\\end{math}"
|
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],
|
|
"text/plain": [
|
|
"[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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"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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"\\begin{math}\n",
|
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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)\n",
|
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"\\end{math}"
|
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],
|
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"text/plain": [
|
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"[ 9 -1 -1 -1]\n",
|
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"[-1 1 -1 -1]\n",
|
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"[-1 -1 1 -1]\n",
|
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"[-1 -1 -1 1]"
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]
|
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"data": {
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}</script></html>"
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],
|
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"text/latex": [
|
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"\\begin{math}\n",
|
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"\\newcommand{\\Bold}[1]{\\mathbf{#1}}9 \\, b_{1}^{2} - 2 \\, b_{1} b_{2} + b_{2}^{2} - 2 \\, b_{1} b_{3} - 2 \\, b_{2} b_{3} + b_{3}^{2} - 2 \\, b_{1} b_{4} - 2 \\, b_{2} b_{4} - 2 \\, b_{3} b_{4} + b_{4}^{2}\n",
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"\\end{math}"
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],
|
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"text/plain": [
|
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"9*b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2"
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]
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},
|
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"metadata": {},
|
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"output_type": "display_data"
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},
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{
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"data": {
|
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"text/html": [
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"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n",
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"-1 & \\frac{2}{9} & \\frac{2}{9} & \\frac{2}{9} \\\\\n",
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"0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
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"1 & 0 & 0 & 0 \\\\\n",
|
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"2 & -1 & 2 & 2 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"2 & 2 & -1 & 2 \\\\\n",
|
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"0 & 0 & 0 & 1\n",
|
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"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
|
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"1 & 0 & 0 & 0 \\\\\n",
|
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"0 & 1 & 0 & 0 \\\\\n",
|
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"0 & 0 & 1 & 0 \\\\\n",
|
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"2 & 2 & 2 & -1\n",
|
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"\\end{array}\\right)\\right]</script></html>"
|
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],
|
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"text/latex": [
|
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"\\begin{math}\n",
|
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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]\n",
|
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"\\end{math}"
|
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],
|
|
"text/plain": [
|
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"[\n",
|
|
"[ -1 2/9 2/9 2/9] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
|
"[ 0 1 0 0] [ 2 -1 2 2] [ 0 1 0 0] [ 0 1 0 0]\n",
|
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"[ 0 0 1 0] [ 0 0 1 0] [ 2 2 -1 2] [ 0 0 1 0]\n",
|
|
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 2 2 -1]\n",
|
|
"]"
|
|
]
|
|
},
|
|
"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{2}{9} \\, b_{2} + \\frac{2}{9} \\, b_{3} + \\frac{2}{9} \\, b_{4}</script></html>"
|
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],
|
|
"text/latex": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2}{9} \\, b_{2} + \\frac{2}{9} \\, b_{3} + \\frac{2}{9} \\, b_{4}\n",
|
|
"\\end{math}"
|
|
],
|
|
"text/plain": [
|
|
"2/9*b2 + 2/9*b3 + 2/9*b4"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"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": 164,
|
|
"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",
|
|
" \n",
|
|
" \n",
|
|
"def from_xyr(x, y, r):\n",
|
|
" b = 1/r\n",
|
|
" h1 = b * x\n",
|
|
" h2 = b * y\n",
|
|
" return Circle((1 - h1^2 - h2^2) / b, b, h1, h2)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 158,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"(-2/3*sqrt(3), -1)\t1/3*sqrt(3)\n",
|
|
"(-1/6*sqrt(3), -1)\t1/6*sqrt(3)\n",
|
|
"(0, 1)\tsqrt(3)\n"
|
|
]
|
|
},
|
|
{
|
|
"data": {
|
|
"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",
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"\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": 152,
|
|
"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": [
|
|
"\\begin{math}\n",
|
|
"\\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]\n",
|
|
"\\end{math}"
|
|
],
|
|
"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": [
|
|
"\\begin{math}\n",
|
|
"\\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}\n",
|
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"\\end{math}"
|
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],
|
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"text/plain": [
|
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"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"
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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(6)\n",
|
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"show(relations)\n",
|
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"show(simplify(8 * nqform))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 42,
|
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"metadata": {},
|
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"outputs": [
|
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{
|
|
"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"
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]
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}
|
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],
|
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"source": [
|
|
"relations, Q, nqform = ngon_linear_relations_and_quadratic_form(10)\n",
|
|
"print(relations)"
|
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]
|
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},
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{
|
|
"cell_type": "code",
|
|
"execution_count": 43,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"-1/2*b2*bt1 - 1/2*b1*bt2 + h11*h12 + h21*h22"
|
|
]
|
|
},
|
|
"execution_count": 43,
|
|
"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": 44,
|
|
"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": [
|
|
"\\begin{math}\n",
|
|
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\, \\sin\\left(\\frac{\\pi k}{n}\\right)^{2}}{\\sin\\left(\\frac{\\pi}{n}\\right)^{2}} + 1\n",
|
|
"\\end{math}"
|
|
],
|
|
"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": 57,
|
|
"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": 46,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/plain": [
|
|
"-x^6 + x^5 + x^4 - x^3 - 2*x^2 + 2*x"
|
|
]
|
|
},
|
|
"execution_count": 46,
|
|
"metadata": {},
|
|
"output_type": "execute_result"
|
|
}
|
|
],
|
|
"source": [
|
|
"expand(x * (1 - x) * (x^4 - x^2 + 2))"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "SageMath 9.2",
|
|
"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": "3.9.2"
|
|
},
|
|
"name": "Apollonian Circle Packings.ipynb"
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|