diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..be426cc --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +./**/.ipynb_checkpoints/* diff --git a/.ipynb_checkpoints/Apollonian Circle Packings-checkpoint.ipynb b/.ipynb_checkpoints/Apollonian Circle Packings-checkpoint.ipynb deleted file mode 100644 index 2c90e60..0000000 --- a/.ipynb_checkpoints/Apollonian Circle Packings-checkpoint.ipynb +++ /dev/null @@ -1,2171 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Calculations to find the quadratic form of the octahedral packing" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$P$ is the matrix of the quadratic form corresponding to $h_1^2 + h_2^2 - \\tilde{b}b = 1$." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": {}, - "outputs": [], - "source": [ - "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", - "])" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[ 0 -2 0 0]\n", - "[-2 0 0 0]\n", - "[ 0 0 1 0]\n", - "[ 0 0 0 1]" - ] - }, - "execution_count": 2, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P.inverse()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "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", - "\n", - "Here $W$ is such a matrix computed for the $(0, 0, 1, 1, 2, 2)$ root sextuple." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [], - "source": [ - "W = matrix([\n", - " [2, 0, 0, 1],\n", - " [2, 0, 0, -1],\n", - " [-1, 1, 0, 0],\n", - " [6, 2, 2*sqrt(2), 0]\n", - "])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have $$\n", - " WPW^T = \\left(\\begin{matrix}\n", - " 1 & -1 & -1 & -1\\\\\n", - " -1 & 1 & -1 & -1\\\\\n", - " -1 & -1 & 1 & -1\\\\\n", - " -1 & -1 & -1 & -1\n", - " \\end{matrix}\\right) = M\n", - "$$\n", - "So, inverting both sides, we get $$\n", - " (WPW^T)^{-1} = M^{-1}\n", - "$$ $$\n", - " (W^T)^{-1}P^{-1}W^{-1} = M^{-1}\n", - "$$ $$\n", - " P^{-1} = W^TM^{-1}W\n", - ".$$\n", - "\n", - "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." - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "scrolled": true - }, - "outputs": [ - { - "data": { - "text/plain": [ - "[ 1 -1 -1 -2]\n", - "[-1 1 -1 -2]\n", - "[-1 -1 1 -2]\n", - "[-2 -2 -2 -4]" - ] - }, - "execution_count": 4, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "M = W * P * W.transpose()\n", - "M" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$M^{-1}$ 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", - "$$\n", - "\\left(\\begin{matrix}\n", - " 1 & 0 & 0 & -1/2\\\\\n", - " 0 & 1 & 0 & -1/2\\\\\n", - " 0 & 0 & 1 & -1/2\\\\\n", - " -1/2 & -1/2 & -1/2 & 1/4\n", - "\\end{matrix}\\right)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "scrolled": true - }, - "outputs": [ - { - "data": { - "text/plain": [ - "[ 1/2 0 0 -1/4]\n", - "[ 0 1/2 0 -1/4]\n", - "[ 0 0 1/2 -1/4]\n", - "[-1/4 -1/4 -1/4 1/8]" - ] - }, - "execution_count": 5, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "M.inverse()" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": {}, - "outputs": [], - "source": [ - "bt1 = var('bt1')\n", - "b1 = var('b1')\n", - "h11 = var('h11')\n", - "h12 = var('h12')\n", - "bt2 = var('bt2')\n", - "b2 = var('b2')\n", - "h21 = var('h21')\n", - "h22 = var('h22')\n", - "bt3 = var('bt3')\n", - "b3 = var('b3')\n", - "h31 = var('h31')\n", - "h32 = var('h32')\n", - "b5_avg = var('b5_avg')\n", - "b_avg = var('b_avg')\n", - "h1_avg = var('h1_avg')\n", - "h2_avg = var('h2_avg')\n", - "\n", - "\n", - "W2 = matrix([\n", - " [bt1, b1, h11, h12],\n", - " [bt2, b2, h21, h22],\n", - " [bt3, b3, h31, h32],\n", - " [b5_avg, b_avg, h1_avg, h2_avg],\n", - "])" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": {}, - "outputs": [], - "source": [ - "D = W2.transpose() * M.inverse() * W2" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[ 0 -2 0 0]\n", - "[-2 0 0 0]\n", - "[ 0 0 1 0]\n", - "[ 0 0 0 1]" - ] - }, - "execution_count": 8, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "P.inverse()" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "b1^2 + b2^2 + b3^2 - b1*b_avg - b2*b_avg - b3*b_avg + 1/4*b_avg^2" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "2 * factor(simplify(D[1][1]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "So, we end up deriving the quadratic form $$\n", - " b_1^2 + b_2^2 + b_3^2 + b_{\\text{sum}}^2/4 - b_{\\text{sum}}(b_1 + b_2 + b_3) = 0\n", - ".$$\n", - "\n", - "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." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Calculations to find the generators for the \"Apollonian\" Group for the octahedral packing\n", - "\n", - "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." - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [], - "source": [ - "def weyl_generators(matrix, alphas):\n", - " retval = []\n", - " for alpha in alphas:\n", - " scale_factor = (alpha.transpose() * matrix * alpha)[0][0]\n", - " retval.append(identity_matrix(len(alphas)) - 2 * alpha * alpha.transpose() * matrix / scale_factor)\n", - " return retval" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": {}, - "outputs": [], - "source": [ - "def standard_basis(dim):\n", - " return [ matrix(dim, 1, [0] * i + [1] + [0] * (dim - i - 1)) for i in range(dim) ]" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[ 2 0 0 -1]\n", - "[ 0 2 0 -1]\n", - "[ 0 0 2 -1]\n", - "[ -1 -1 -1 1/2]" - ] - }, - "execution_count": 12, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "4 * M.inverse()" - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[\n", - "[-1 0 0 1] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n", - "[ 0 1 0 0] [ 0 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n", - "[ 0 0 1 0] [ 0 0 1 0] [ 0 0 -1 1] [ 0 0 1 0]\n", - "[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 4 4 4 -1]\n", - "]" - ] - }, - "execution_count": 13, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "S_i = weyl_generators(4 * M.inverse(), standard_basis(4))\n", - "S_i" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": {}, - "outputs": [], - "source": [ - "S1 = S_i[0]\n", - "S2 = S_i[1]\n", - "S3 = S_i[2]\n", - "S4 = S_i[3]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can test this out on the packing on page 3 of the Guettler and Mallows. The root sextuple is (-1, 2, 2, 4, 4 7), so the coordinates would be (-1, 2, 4, 6). After applying $s_1$, it should swap out $s_1$ for its pair, resulting in (7, 2, 4, 6). Likewise for $s_2$ and $s_3$. Then, for $s_4$, it should give the average between the triple (-1, 2, 4) and the other tangent triple, i.e. (-1, 2, 4, 14)." - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(7, 2, 4, 6)" - ] - }, - "execution_count": 15, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "root = vector([-1, 2, 4, 6])\n", - "S1 * root" - ] - }, - { - "cell_type": "code", - "execution_count": 16, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(-1, 4, 4, 6)" - ] - }, - "execution_count": 16, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "S2 * root" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(-1, 2, 2, 6)" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "S3 * root" - ] - }, - { - "cell_type": "code", - "execution_count": 18, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "(-1, 2, 4, 14)" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "S4 * root" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Calculations to find the curvatures of the dual circles of the octahedral packing\n", - "\n", - "Here the basic idea was to work out the intersection points of the circles and let the computer algebraically find the radii of the circles. The end up coming out, rather anticlimactically, to $(0, 0, \\sqrt 2, \\sqrt 2, \\sqrt 2, \\sqrt 2, 2\\sqrt 2, 2\\sqrt 2)$. It's rather satisfying and a bit surprising that the dual of the root sextuple of the octahedral packing is the root of the cubic packing, since the dual polyhedron of the octahedron is the cube." - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "metadata": {}, - "outputs": [], - "source": [ - "def circle_from_points(pta, ptb, ptc):\n", - " a = var('a')\n", - " b = var('b')\n", - " r = var('r')\n", - " x = var('x')\n", - " y = var('y')\n", - " \n", - " circle_func = (x - a)^2 + (y - b)^2 == r^2\n", - " \n", - " eq1 = circle_func.subs(x == pta[0]).subs(y == pta[1])\n", - " eq2 = circle_func.subs(x == ptb[0]).subs(y == ptb[1])\n", - " eq3 = circle_func.subs(x == ptc[0]).subs(y == ptc[1])\n", - " \n", - " res = solve([eq1, eq2, eq3], a, b, r)[1]\n", - " \n", - " return (res[0].rhs(), res[1].rhs(), res[2].rhs())" - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "metadata": { - "scrolled": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "(1/2*sqrt(2), 1)\n", - "sqrt(2)\n", - "(1/4*sqrt(2), 0)\n", - "2*sqrt(2)\n" - ] - }, - { - "data": { - "image/png": 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\n", - "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" - ] - }, - { - "cell_type": "code", - "execution_count": 25, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[ 2 -6/5 -6/5 -6/5]\n", - "[-6/5 2 -6/5 -6/5]\n", - "[-6/5 -6/5 2 -6/5]\n", - "[-6/5 -6/5 -6/5 2]" - ] - }, - "execution_count": 25, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "32 * m.inverse() * 2/5" - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "[bt1 b1 h11 h21]\n", - "[bt2 b2 h12 h22]\n", - "[bt3 b3 h13 h23]\n", - "[bt4 b4 h14 h24]" - ] - }, - "execution_count": 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": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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" - }, - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\begin{array}{rrrr}\n", - "-1 & 4 & 4 & 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", - "1 & -1 & 0 & 1 \\\\\n", - "0 & 0 & 1 & 0 \\\\\n", - "0 & 0 & 0 & 1\n", - "\\end{array}\\right), \\left(\\begin{array}{rrrr}\n", - "1 & 0 & 0 & 0 \\\\\n", - "0 & 1 & 0 & 0 \\\\\n", - "1 & 0 & -1 & 1 \\\\\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 & 4 & 4 & -1\n", - "\\end{array}\\right)\\right]\n", - "\\end{math}" - ], - "text/plain": [ - "[\n", - "[-1 4 4 2] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n", - "[ 0 1 0 0] [ 1 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n", - "[ 0 0 1 0] [ 0 0 1 0] [ 1 0 -1 1] [ 0 0 1 0]\n", - "[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 4 4 -1]\n", - "]" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "# octahedral\n", - "relation, mat, equation = quadform_from_root(matrix([\n", - " [2, 0, 0, 1],\n", - " [2, 0, 0, -1],\n", - " [-1, 1, 0, 0],\n", - " [4, 2, 2*sqrt(2), -1],\n", - " [4, 2, 2*sqrt(2), 1],\n", - " [7, 1, 2*sqrt(2), 0],\n", - "]))\n", - "show(relation)\n", - "show(8 * mat)\n", - "show(8 * equation)\n", - "show(weyl_generators(8 * mat, standard_basis(4)))" - ] - }, - { - "cell_type": "code", - "execution_count": 34, - "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\right]\n", - "\\end{math}" - ], - "text/plain": [ - "[]" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", - "1 & -1 & -1 & -1 \\\\\n", - "-1 & 1 & -1 & -1 \\\\\n", - "-1 & -1 & 1 & -1 \\\\\n", - "-1 & -1 & -1 & 1\n", - "\\end{array}\\right)\n", - "\\end{math}" - ], - "text/plain": [ - "[ 1 -1 -1 -1]\n", - "[-1 1 -1 -1]\n", - "[-1 -1 1 -1]\n", - "[-1 -1 -1 1]" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "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", - "\\end{math}" - ], - "text/plain": [ - "b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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": [ - "" - ], - "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", - "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 2/9 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", - "[ 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": [ - "relations, Q, nqform = linear_relations_and_quadratic_form_from_wtpw(wmatrix(6), [1, 3, 5, 7, 2, 4, 6])\n", - "print(12 * Q)\n", - "show(12 * nqform)\n", - "show([3 * rel for rel in relations])\n", - "show(weyl_generators(12 * Q, standard_basis(4)))" - ] - }, - { - "cell_type": "code", - "execution_count": 39, - "metadata": {}, - "outputs": [], - "source": [ - "# function to compute the linear relations and quadratic formula for an ngon base pyramid\n", - "# the circles will be numbered 1 in the center, then 2 through n winding around the circle by step, so\n", - "# for n = 4 with step = 1, we have\n", - "# b3\n", - "# b4 b1 b2\n", - "# b5\n", - "# and the quadratic form is in terms of b1, b2, b3, and b4\n", - "# and for n = 6 with step = 2, we have\n", - "# b4 b3\n", - "# b5 b1 b2\n", - "# b6 b7\n", - "# and the quadratic form is in terms of b1, b3, b5, and b7\n", - "def ngon_linear_relations_and_quadratic_form(n, step=1):\n", - " mat = wmatrix(n)\n", - " \n", - " # work out initial relations\n", - " relations_temp = vector([ var('b' + str(i)) for i in range(1, n + 2) ]) * mat.transpose().rref()\n", - " relations = []\n", - " for i in range(4, n + 1):\n", - " relations.append(var('b' + str(i + 1)) == relations_temp[i])\n", - " \n", - " # rewrite the relations in terms of the variables we care about, depends on the step\n", - " shuffled = (list(range(1, n + 2)) * step)[::step]\n", - " targets = [ var('b' + str(i)) for i in shuffled[4:] ]\n", - " relations = solve(relations, *targets)[0]\n", - " \n", - " # find the matrix corresponding to the quadratic form, picking the appropriate rows from the matrix\n", - " Q = mat[:4*step:step,:4*step:step].inverse()\n", - " \n", - " # find the quadratic form in variables; proper units will satisfy this being equal to zero\n", - " nqform = qform(Q, vector([ var('b' + str(i)) for i in range(1, 5) ]))\n", - " \n", - " return relations, Q, expand(nqform)" - ] - }, - { - "cell_type": "code", - "execution_count": 120, - "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\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", - "\\end{math}" - ], - "text/plain": [ - "[b5 == -b3*(sqrt(2) + 1) + b4*(sqrt(2) + 1) + b2,\n", - " b6 == b4*(sqrt(2) + 2) + b2*(sqrt(2) + 1) - 2*b3*(sqrt(2) + 1),\n", - " b7 == -b3*(2*sqrt(2) + 3) + b2*(sqrt(2) + 2) + b4*(sqrt(2) + 2),\n", - " b8 == b2*(sqrt(2) + 2) - 2*b3*(sqrt(2) + 1) + b4*(sqrt(2) + 1),\n", - " b9 == b2*(sqrt(2) + 1) - b3*(sqrt(2) + 1) + b4]" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", - "-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 3 & -2 \\\\\n", - "-2 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -2 & \\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} \\\\\n", - "-\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 3 & -2 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -2 \\\\\n", - "-2 & \\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1} & -2 & -\\frac{4}{\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} - 1}\n", - "\\end{array}\\right)\n", - "\\end{math}" - ], - "text/plain": [ - "[ -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -2 -(sqrt(2) + 2)/(sqrt(2) - 2) - 3 -2]\n", - "[ -2 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2 4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)]\n", - "[ -(sqrt(2) + 2)/(sqrt(2) - 2) - 3 -2 -(sqrt(2) + 2)/(sqrt(2) - 2) + 1 -2]\n", - "[ -2 4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1) -2 -4/((sqrt(2) + 2)/(sqrt(2) - 2) - 1)]" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/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)" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\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", - "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{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & -1 & -\\frac{\\sqrt{2} + 2}{\\sqrt{2} - 2} + 1 & 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", - "-\\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", - "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{\\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": [ - "[\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", - "[ 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", - "]" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "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)))" - ] - }, - { - "cell_type": "code", - "execution_count": 163, - "metadata": {}, - "outputs": [ - { - "data": { - "text/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": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n", - "9 & -1 & -1 & -1 \\\\\n", - "-1 & 1 & -1 & -1 \\\\\n", - "-1 & -1 & 1 & -1 \\\\\n", - "-1 & -1 & -1 & 1\n", - "\\end{array}\\right)\n", - "\\end{math}" - ], - "text/plain": [ - "[ 9 -1 -1 -1]\n", - "[-1 1 -1 -1]\n", - "[-1 -1 1 -1]\n", - "[-1 -1 -1 1]" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "" - ], - "text/latex": [ - "\\begin{math}\n", - "\\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", - "\\end{math}" - ], - "text/plain": [ - "9*b1^2 - 2*b1*b2 + b2^2 - 2*b1*b3 - 2*b2*b3 + b3^2 - 2*b1*b4 - 2*b2*b4 - 2*b3*b4 + b4^2" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/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", - "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 2/9 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", - "[ 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" - }, - { - "data": { - "text/html": [ - "" - ], - "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", - "\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": [ - "" - ], - "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": [ - "" - ], - "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", - "\\end{math}" - ], - "text/plain": [ - "6*b1^2 - 4*b1*b2 + 2/3*b2^2 + 4*b1*b3 - 4*b2*b3 + 6*b3^2 - 4*b1*b4 - 4/3*b2*b4 - 4*b3*b4 + 2/3*b4^2" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "relations, Q, nqform = ngon_linear_relations_and_quadratic_form(6)\n", - "show(relations)\n", - "show(simplify(8 * nqform))" - ] - }, - { - "cell_type": "code", - "execution_count": 42, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[b5 == -1/2*b3*(sqrt(5) + 3) + 1/2*b4*(sqrt(5) + 3) + b2, b6 == -1/2*b3*(3*sqrt(5) + 5) + 1/2*b2*(sqrt(5) + 3) + b4*(sqrt(5) + 2), b7 == b4*(sqrt(5) + 3) + b2*(sqrt(5) + 2) - 2*b3*(sqrt(5) + 2), b8 == -b3*(2*sqrt(5) + 5) + b2*(sqrt(5) + 3) + b4*(sqrt(5) + 3), b9 == b2*(sqrt(5) + 3) - 2*b3*(sqrt(5) + 2) + b4*(sqrt(5) + 2), b10 == -1/2*b3*(3*sqrt(5) + 5) + 1/2*b4*(sqrt(5) + 3) + b2*(sqrt(5) + 2), b11 == 1/2*b2*(sqrt(5) + 3) - 1/2*b3*(sqrt(5) + 3) + b4]\n" - ] - } - ], - "source": [ - "relations, Q, nqform = ngon_linear_relations_and_quadratic_form(10)\n", - "print(relations)" - ] - }, - { - "cell_type": "code", - "execution_count": 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": [ - "" - ], - "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 -} diff --git a/.ipynb_checkpoints/Homework-checkpoint.ipynb b/.ipynb_checkpoints/Homework-checkpoint.ipynb deleted file mode 100644 index 56de4a4..0000000 --- a/.ipynb_checkpoints/Homework-checkpoint.ipynb +++ /dev/null @@ -1,557 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "scrolled": true - }, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - "\n" - ], - "text/plain": [ - "Graphics3d Object" - ] - }, - "execution_count": 4, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "var('x,y,z')\n", - "implicit_plot3d(x^2 + y^2 - z^2 == 0, (x,-3,3), (y,-3,3), (z,-3,3))" - ] - } - ], - "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" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/.ipynb_checkpoints/Koebe-Andreev-Thurston-checkpoint.ipynb b/.ipynb_checkpoints/Koebe-Andreev-Thurston-checkpoint.ipynb deleted file mode 100644 index 7fec515..0000000 --- a/.ipynb_checkpoints/Koebe-Andreev-Thurston-checkpoint.ipynb +++ /dev/null @@ -1,6 +0,0 @@ -{ - "cells": [], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/fractal_dimension.nb b/fractal_dimension/fractal_dimension.nb similarity index 100% rename from fractal_dimension.nb rename to fractal_dimension/fractal_dimension.nb diff --git a/fractaldimension.ipynb b/fractal_dimension/fractaldimension.ipynb similarity index 100% rename from fractaldimension.ipynb rename to fractal_dimension/fractaldimension.ipynb diff --git a/7v7f5.nb b/graham_matrix/7v7f5.nb similarity index 100% rename from 7v7f5.nb rename to graham_matrix/7v7f5.nb diff --git a/7v7f7.nb b/graham_matrix/7v7f7.nb similarity index 100% rename from 7v7f7.nb rename to graham_matrix/7v7f7.nb diff --git a/7v8f1.nb b/graham_matrix/7v8f1.nb similarity index 100% rename from 7v8f1.nb rename to graham_matrix/7v8f1.nb diff --git a/7v8f11.nb b/graham_matrix/7v8f11.nb similarity index 100% rename from 7v8f11.nb rename to graham_matrix/7v8f11.nb diff --git a/a bitch is learning graphs.nb b/graham_matrix/a bitch is learning graphs.nb similarity index 100% rename from a bitch is learning graphs.nb rename to graham_matrix/a bitch is learning graphs.nb diff --git a/gram_matrix_to_quadform.nb b/graham_matrix/gram_matrix_to_quadform.nb similarity index 100% rename from gram_matrix_to_quadform.nb rename to graham_matrix/gram_matrix_to_quadform.nb diff --git a/put_together.nb b/graham_matrix/put_together.nb similarity index 70% rename from put_together.nb rename to graham_matrix/put_together.nb index 70113ee..d86c4de 100644 --- a/put_together.nb +++ b/graham_matrix/put_together.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 39681, 984] -NotebookOptionsPosition[ 35729, 927] -NotebookOutlinePosition[ 36130, 943] -CellTagsIndexPosition[ 36087, 940] +NotebookDataLength[ 47999, 1161] +NotebookOptionsPosition[ 43133, 1093] +NotebookOutlinePosition[ 43534, 1109] +CellTagsIndexPosition[ 43491, 1106] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -37,10 +37,10 @@ Cell[BoxData[ "The currently installed versions of IGraph/M are: ", {"0.5.1"}], Editable->False]], "Print", CellChangeTimes->{3.8321516294929905`*^9, 3.8321792446538467`*^9, - 3.8321888483607073`*^9}, + 3.8321888483607073`*^9, 3.83223375355267*^9}, CellLabel-> "During evaluation of \ -In[1]:=",ExpressionUUID->"1a23cd39-3c6b-46e9-af91-4872dc234669"], +In[1]:=",ExpressionUUID->"fa2b838b-e3df-4e8d-9350-d8f470c0acc1"], Cell[BoxData[ TemplateBox[{ @@ -50,12 +50,13 @@ with the same version number \ (\\!\\(\\*RowBox[{\\\"\\\\\\\"0.5.1\\\\\\\"\\\"}]\\)) is already installed. \ Use PacletUninstall to remove the existing version first, or call \ PacletInstall with ForceVersionInstall -> True.\"", 2, 1, 1, - 26205863351082809247, "Local"}, + 26206157531447520602, "Local"}, "MessageTemplate"]], "Message", "MSG", - CellChangeTimes->{3.83217924731254*^9, 3.832188855090274*^9}, + CellChangeTimes->{3.83217924731254*^9, 3.832188855090274*^9, + 3.832233757585519*^9}, CellLabel-> "During evaluation of \ -In[1]:=",ExpressionUUID->"bae63da8-d063-461c-b011-1d4ac9d01047"], +In[1]:=",ExpressionUUID->"7cedf722-9dff-466e-8c29-7474221bbcff"], Cell[BoxData[ InterpretationBox[ @@ -69,10 +70,10 @@ Cell[BoxData[ Hyperlink["https://github.com/szhorvat/IGraphM#installation"]], Editable->False]], "Print", CellChangeTimes->{3.8321516294929905`*^9, 3.8321792446538467`*^9, - 3.832188855122637*^9}, + 3.8321888483607073`*^9, 3.832233757594738*^9}, CellLabel-> "During evaluation of \ -In[1]:=",ExpressionUUID->"f46f4636-68b0-4b87-b086-72d92dec882d"] +In[1]:=",ExpressionUUID->"bc4bba27-1aaf-4b2a-b377-2ee419fb494f"] }, Open ]], Cell[CellGroupData[{ @@ -93,15 +94,10 @@ Cell[BoxData[ GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Column"]], "Output", CellChangeTimes->{3.8321516533863616`*^9, 3.832179248194338*^9, - 3.832188856442923*^9}, - CellLabel->"Out[2]=",ExpressionUUID->"3c9be348-6ed8-4a06-91b2-f14112004d3f"] + 3.832188856442923*^9, 3.832233758550943*^9}, + CellLabel->"Out[2]=",ExpressionUUID->"eb6e8d8d-44bf-42ef-b37b-df6773c2cd46"] }, Open ]], -Cell[BoxData[ - RowBox[{"IGDocumentation", "[", "]"}]], "Input", - CellChangeTimes->{{3.8321516567464046`*^9, 3.8321516618620677`*^9}}, - CellLabel->"In[3]:=",ExpressionUUID->"736a6582-d513-4fda-84ed-908eb2a7766f"], - Cell["Takes adjacency matrix and spits out graph", "Text", CellChangeTimes->{{3.832168668844531*^9, 3.832168674438282*^9}},ExpressionUUID->"9fb785ee-e560-44c5-a5b3-\ @@ -496,10 +492,23 @@ Cell[BoxData[ RowBox[{"Flatten", "[", RowBox[{"Table", "[", RowBox[{ - RowBox[{"eqs", "[", + RowBox[{"If", "[", RowBox[{ - "i", ",", "j", ",", "c1s", ",", "c2s", ",", "ds", ",", - "Gext"}], "]"}], ",", + RowBox[{ + RowBox[{"SameQ", "[", + RowBox[{ + RowBox[{"Gext", "\[LeftDoubleBracket]", + RowBox[{"i", ",", "j"}], "\[RightDoubleBracket]"}], ",", + RowBox[{"-", "1"}]}], "]"}], "||", + RowBox[{"SameQ", "[", + RowBox[{ + RowBox[{"Gext", "\[LeftDoubleBracket]", + RowBox[{"i", ",", "j"}], "\[RightDoubleBracket]"}], ",", + "0"}], "]"}]}], ",", + RowBox[{"eqs", "[", + RowBox[{ + "i", ",", "j", ",", "c1s", ",", "c2s", ",", "ds", ",", + "Gext"}], "]"}], ",", "True"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"n", "+", "m"}]}], "}"}], ",", @@ -526,8 +535,10 @@ Cell[BoxData[ 3.832186052360086*^9}, {3.832186094120708*^9, 3.8321861420031347`*^9}, { 3.832186175260569*^9, 3.832186201772645*^9}, {3.832186235325078*^9, 3.8321862618199997`*^9}, {3.8321863613697853`*^9, - 3.8321863669835577`*^9}, {3.832186452371193*^9, 3.832186456250992*^9}}, - CellLabel->"In[12]:=",ExpressionUUID->"f4f2f39c-a979-492f-a025-beb69389c674"], + 3.8321863669835577`*^9}, {3.832186452371193*^9, 3.832186456250992*^9}, { + 3.832236951287407*^9, 3.8322369831425056`*^9}, {3.832237035633595*^9, + 3.832237063324972*^9}}, + CellLabel->"In[26]:=",ExpressionUUID->"f4f2f39c-a979-492f-a025-beb69389c674"], Cell["Same as above, but numerically approximates it.", "Text", CellChangeTimes->{{3.832188787533435*^9, @@ -616,8 +627,10 @@ Cell[BoxData[ 3.832186612097026*^9, 3.832186638550942*^9}, {3.832186914043337*^9, 3.832186922217325*^9}, {3.832187020117702*^9, 3.832187021912196*^9}, { 3.8321871707702923`*^9, 3.8321871736438*^9}, {3.8321877632435617`*^9, - 3.832187771381138*^9}, {3.83218786469829*^9, 3.832187868963749*^9}}, - CellLabel->"In[13]:=",ExpressionUUID->"e9aabedc-e11e-4cce-a76e-c710d3384ca9"], + 3.832187771381138*^9}, {3.83218786469829*^9, 3.832187868963749*^9}, { + 3.8322340786657343`*^9, + 3.8322341092778482`*^9}},ExpressionUUID->"e9aabedc-e11e-4cce-a76e-\ +c710d3384ca9"], Cell["Example usage. This is the square pyramid.", "Text", CellChangeTimes->{{3.832188805201276*^9, @@ -646,13 +659,13 @@ Cell[BoxData[ 3.8321835213661003`*^9}, {3.832183609148136*^9, 3.8321836229774103`*^9}, { 3.8321862070609922`*^9, 3.832186218032998*^9}, {3.832186249390174*^9, 3.832186256142555*^9}, {3.8321863924121113`*^9, 3.8321864066698523`*^9}}, - CellLabel->"In[14]:=",ExpressionUUID->"43731109-f22e-4624-b7f9-248becc79bba"], + CellLabel->"In[27]:=",ExpressionUUID->"43731109-f22e-4624-b7f9-248becc79bba"], Cell[BoxData[ TemplateBox[{ "Power", "infy", "\"Infinite expression \\!\\(\\*FractionBox[\\\"1\\\", \\\"0\\\"]\\) \ -encountered.\"", 2, 14, 2, 26205863351082809247, "Local"}, +encountered.\"", 2, 27, 79, 26206157531447520602, "Local"}, "MessageTemplate"]], "Message", "MSG", CellChangeTimes->{ 3.832182763093258*^9, 3.832182841541271*^9, 3.832182981310091*^9, { @@ -661,17 +674,18 @@ encountered.\"", 2, 14, 2, 26205863351082809247, "Local"}, 3.832183610774363*^9, 3.832183624231565*^9}, 3.8321860021042147`*^9, 3.8321860748628817`*^9, 3.832186146263523*^9, {3.832186194392118*^9, 3.832186256895705*^9}, {3.83218639372544*^9, 3.832186459711775*^9}, - 3.832188871475355*^9}, + 3.832188871475355*^9, 3.832233759272257*^9, 3.8322357555615997`*^9, + 3.8322369958376913`*^9, 3.832237070276142*^9}, CellLabel-> "During evaluation of \ -In[14]:=",ExpressionUUID->"abf371ca-58c8-4795-ab0f-e48019017a57"], +In[27]:=",ExpressionUUID->"c7c9db08-bc46-4183-ad6b-6d4b18afab25"], Cell[BoxData[ TemplateBox[{ "Infinity", "indet", "\"Indeterminate expression \\!\\(\\*RowBox[{\\\"0\\\", \\\" \\\", \ -\\\"ComplexInfinity\\\"}]\\) encountered.\"", 2, 14, 3, 26205863351082809247, - "Local"}, +\\\"ComplexInfinity\\\"}]\\) encountered.\"", 2, 27, 80, 26206157531447520602, + "Local"}, "MessageTemplate"]], "Message", "MSG", CellChangeTimes->{ 3.832182763093258*^9, 3.832182841541271*^9, 3.832182981310091*^9, { @@ -680,16 +694,17 @@ Cell[BoxData[ 3.832183610774363*^9, 3.832183624231565*^9}, 3.8321860021042147`*^9, 3.8321860748628817`*^9, 3.832186146263523*^9, {3.832186194392118*^9, 3.832186256895705*^9}, {3.83218639372544*^9, 3.832186459711775*^9}, - 3.832188871525887*^9}, + 3.832188871475355*^9, 3.832233759272257*^9, 3.8322357555615997`*^9, + 3.8322369958376913`*^9, 3.832237070280217*^9}, CellLabel-> "During evaluation of \ -In[14]:=",ExpressionUUID->"6add42d7-9b06-4c10-b557-2a4f82586ba9"], +In[27]:=",ExpressionUUID->"f993da93-45d0-49f8-83fd-06a87a8acf55"], Cell[BoxData[ TemplateBox[{ "Power", "infy", "\"Infinite expression \\!\\(\\*FractionBox[\\\"1\\\", \\\"0\\\"]\\) \ -encountered.\"", 2, 14, 4, 26205863351082809247, "Local"}, +encountered.\"", 2, 27, 81, 26206157531447520602, "Local"}, "MessageTemplate"]], "Message", "MSG", CellChangeTimes->{ 3.832182763093258*^9, 3.832182841541271*^9, 3.832182981310091*^9, { @@ -698,17 +713,18 @@ encountered.\"", 2, 14, 4, 26205863351082809247, "Local"}, 3.832183610774363*^9, 3.832183624231565*^9}, 3.8321860021042147`*^9, 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"Message",ExpressionUUID->"74f88af0-674b-40de-8074-8ed6f95ace99"], +Cell[32386, 800, 996, 18, 22, "Message",ExpressionUUID->"4194f6a4-417e-4717-9a7e-ad06fb2fd8fc"], +Cell[33385, 820, 3865, 125, 235, "Output",ExpressionUUID->"fdc7f6af-fd0d-4929-9e08-6b6c1e05f6e9"] +}, Open ]], +Cell[CellGroupData[{ +Cell[37287, 950, 833, 17, 113, "Input",ExpressionUUID->"02de66da-bc38-41f4-8a77-b770b877dfca"], +Cell[38123, 969, 547, 12, 39, "Message",ExpressionUUID->"d00c981d-e8e3-45ab-b623-d93a843ce6a8"], +Cell[38673, 983, 582, 13, 22, "Message",ExpressionUUID->"05ac3f26-ef03-4ab1-8497-67737d7312c5"], +Cell[39258, 998, 549, 12, 39, "Message",ExpressionUUID->"74c26ce0-ced0-4e13-a3d5-a4a026eb0e83"], +Cell[39810, 1012, 584, 13, 22, "Message",ExpressionUUID->"ea19ebab-bd10-406a-834f-3c7be4208e03"], +Cell[40397, 1027, 547, 12, 39, "Message",ExpressionUUID->"a0e31f72-3b86-40f7-9eec-6263fa172fe4"], +Cell[40947, 1041, 628, 13, 22, "Message",ExpressionUUID->"89eace12-6b46-40eb-aadd-50e5aaa3bd9f"], +Cell[41578, 1056, 583, 13, 22, "Message",ExpressionUUID->"5375cd8f-e536-4493-a5bd-41e24879aff7"], +Cell[42164, 1071, 632, 13, 22, "Message",ExpressionUUID->"be185428-a448-4979-81b7-42b96c1c043f"], +Cell[42799, 1086, 318, 4, 33, "Output",ExpressionUUID->"6989777a-5d19-4e97-ad8e-85324d649c62"] }, Open ]] } ] diff --git a/Apollonian Circle Packings.ipynb b/old_stuff/Apollonian Circle Packings.ipynb similarity index 100% rename from Apollonian Circle Packings.ipynb rename to old_stuff/Apollonian Circle Packings.ipynb diff --git a/polyhedra/eight.txt b/polyhedra/eight.txt new file mode 100644 index 0000000..3aad8e0 --- /dev/null +++ b/polyhedra/eight.txt @@ -0,0 +1,257 @@ +1: G|mnMC +2: GtmnJC +3: G|eJo[ +4: GtfMJw +5: Gte^jC +6: GsvJpc +7: G~uIO[ +8: G|dogs +9: GtuIZo +10: GspZhc +11: G|bBXo +12: GspZHc +13: G|bAXo +14: GtnmJG +15: GtuMZo +16: GtfAZs +17: GtpYHk +18: GtvIPs +19: G|eK^c +20: GtmmJC +21: G|eJ_[ +22: GtfMJo +23: Gte^JC +24: GsvJ`c +25: G~`?w[ +26: G~eIO[ +27: GtuIZ_ +28: GspZ`c +29: GtnMJG +30: GtrJ_s +31: GtuMZ_ +32: GtmiJC +33: G|`oo[ +34: GtfAZo +35: GtvIPc +36: GttH_[ +37: G|eK]s +38: GtenJC +39: G|eIo[ +40: GtjJPo +41: G{dPgw +42: GtdbHW +43: G{Ogws +44: GtfMJW +45: GtnB@[ +46: G{d`g[ +47: GtmmJG +48: G|eJ_k +49: GtuMJo +50: GtejJC +51: G|`Ow[ +52: GtfAZS +53: G|e\MC +54: GtemJC +55: GtbJPo +56: GtdBHW +57: GufJ@c +58: GtfMJO +59: GtfIHS +60: GtnB?[ +61: GtpHGs +62: GtPHGs +63: G~eK]s +64: GvuMZ_ +65: GtmiJK +66: GutPHk +67: GtvAZo +68: GtpYhc +69: GsnjHo +70: GtvIpc +71: GttH_{ +72: GtmmJK +73: G|eJ_{ +74: GtnMJW +75: GtnBH[ +76: G|nmIK +77: GtniJK +78: GutpHk +79: Gut`Hk +80: GtvIZo +81: GttIXs +82: GtpZhc +83: G{dJHw +84: G{fJHw +85: GtvJpc +86: G~e]mC +87: Gte]jc +88: GteIZs +89: GtbjHw +90: GtbJHw +91: GtdBgw +92: GtdRgw +93: GsvJPs +94: GsfJPs +95: Gt`Jg{ +96: Gt`JG{ +97: GteMZs +98: GtmJZC +99: G|``w[ +100: GtfAzc +101: GtmNZC +102: G|mJO[ +103: Gtf]Jg +104: G|e[Nc +105: Gte]jC +106: Gte]JC +107: GsvJPc +108: GsvJ@c +109: GteMZo +110: GtmJJC +111: G|`_w[ +112: GtfAZc +113: GtmNJC +114: G|eJO[ +115: GtfMJg +116: G{eNHo +117: GtfIHc +118: G|e[NC +119: G|e[NK +120: Gte]Jc +121: GteIZc +122: GsvJ@s +123: GtmMZC +124: G|mJ?[ +125: G|mKNS +126: GteMjc +127: GteMJc +128: GteMZS +129: GtfB_{ +130: G|nmHW +131: GtvJos +132: GtpYJk +133: GtnBZC +134: G|nAZC +135: GtvIRs +136: G|mmJG +137: GtvJOs +138: G{fh_[ +139: GttHGs +140: GtpBgw +141: GtpBg{ +142: G|fB@[ +143: GtpBG{ +144: GtpBGw +145: GtfIRs +146: G|}K]S +147: Gt}MZO +148: GtfBw[ +149: G{f@xc +150: G{fAxc +151: GtdBg[ +152: GtdBG[ +153: GsfzHg +154: Gspxgw +155: G|j@Xo +156: Gsppgw +157: G|b@Xo +158: Gsp`gw +159: G|b`Xo +160: Gsphgw +161: Gt~IPS +162: G{e}Hg +163: G{f]Hg +164: G|mJ@K +165: G|eJ@K +166: Gve]j_ +167: GvfY`c +168: GvtP_k +169: GvSogs +170: GteIJ{ +171: GstyHK +172: G{bJpc +173: GstqHK +174: G|bBWs +175: G|bAWs +176: G|b?o[ +177: GtbzGw +178: GvdY`c +179: GsdJxc +180: GsdJhc +181: GsdJHc +182: GsP`ws +183: GvfB_w +184: GtdY`c +185: G{eIXo +186: G|a}Gw +187: GvfI`c +188: G|bmGw +189: G|bMGw +190: G|aMGw +191: GteMJ{ +192: G|eMJ{ +193: GtfJzC +194: G|fAw[ +195: G|f@w[ +196: G{fJXc +197: GttYHK +198: GttIHK +199: G~f]Ig +200: GtfZgk +201: GtfZ_k +202: GtfJ_k +203: GtdJGk +204: Gstho[ +205: Gst`o[ +206: Gst`_[ +207: GsT`_[ +208: G|mIHS +209: G|eJXS +210: G~eMI{ +211: GtfIzc +212: GtdJH{ +213: GttYHk +214: GtfZjC +215: GtdIzc +216: G|tIJ{ +217: GttzHK +218: G|pow[ +219: G|oow[ +220: G|pgw[ +221: G|twg[ +222: G|pwg[ +223: G|tog[ +224: G|sog[ +225: G|t_g[ +226: G|tow[ +227: G|txgw +228: G|t`xc +229: G|t`_{ +230: G{t`_{ +231: Gtt`_{ +232: G|dJjC +233: G|tIjC +234: GvPhG{ +235: G|t`_[ +236: G{t`_[ +237: Gtt`Xc +238: G|PHWs +239: G|Ogg[ +240: G|`HWs +241: G|dJWs +242: G|tZHg +243: G|tJxc +244: G|dJxc +245: G|dIxc +246: G|dIXc +247: G|dHxc +248: G|dH_{ +249: G{dJxc +250: G{dIxc +251: G{dH_{ +252: G{dIhc +253: G|trGw +254: G|fZjC +255: G|tZHk +256: G|t`g{ +257: G{t`g{ diff --git a/polyhedra/eight_edge.txt b/polyhedra/eight_edge.txt new file mode 100644 index 0000000..04189b6 --- /dev/null +++ b/polyhedra/eight_edge.txt @@ -0,0 +1,257 @@ +1: 1,2; 1,3; 1,4; 1,5; 1,6; 1,7; 1,8; 2,8; 2,7; 2,3; 3,7; 3,6; 3,5; 3,4; 4,5; 5,6; 6,7; 7,8 +2: 1,2; 1,3; 1,4; 1,5; 1,6; 1,7; 2,7; 2,8; 3,8; 3,7; 3,6; 3,5; 3,4; 4,5; 5,6; 6,7; 7,8 +3: 1,2; 1,3; 1,4; 1,5; 1,6; 2,7; 2,3; 3,7; 3,4; 4,7; 4,5; 5,7; 5,8; 5,6; 6,8; 7,8 +4: 1,2; 1,3; 1,4; 1,5; 1,6; 1,7; 2,7; 2,6; 2,8; 3,8; 3,4; 4,8; 4,5; 5,8; 5,6; 6,8; 6,7 +5: 1,2; 1,3; 1,4; 1,5; 1,6; 1,7; 2,7; 2,8; 3,8; 3,7; 3,4; 4,7; 4,6; 4,5; 5,6; 6,7; 7,8 +6: 1,2; 1,3; 1,4; 1,5; 1,6; 2,6; 2,5; 2,7; 3,7; 3,8; 4,8; 4,7; 4,5; 5,7; 5,6; 7,8 +7: 1,2; 1,3; 1,4; 1,5; 1,6; 2,7; 2,5; 2,4; 2,3; 3,4; 4,5; 5,7; 5,8; 5,6; 6,8; 7,8 +8: 1,2; 1,3; 1,4; 1,5; 2,6; 2,3; 3,6; 3,4; 4,6; 4,7; 4,8; 4,5; 5,8; 6,7; 7,8 +9: 1,2; 1,3; 1,4; 1,5; 1,6; 2,7; 2,5; 2,8; 3,8; 3,4; 4,8; 4,5; 5,8; 5,7; 5,6; 6,7 +10: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,7; 3,7; 3,8; 4,8; 4,7; 4,6; 5,6; 6,7; 7,8 +11: 1,2; 1,3; 1,4; 1,5; 1,6; 2,6; 2,7; 2,3; 3,7; 3,8; 3,4; 4,8; 5,8; 5,7; 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7,8 +232: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,8; 2,3; 3,8; 3,7; 3,4; 4,7; 4,5; 5,6; 6,7; 7,8 +233: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,7; 2,8; 2,3; 3,8; 3,4; 4,7; 4,5; 5,6; 6,7; 7,8 +234: 1,2; 1,3; 1,4; 2,4; 2,5; 2,6; 3,6; 3,7; 3,4; 4,8; 5,8; 5,6; 6,8; 6,7; 7,8 +235: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,3; 3,6; 3,7; 3,4; 4,7; 4,5; 5,8; 6,8; 7,8 +236: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,3; 3,6; 3,7; 4,7; 4,5; 5,8; 6,8; 7,8 +237: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 3,6; 3,7; 3,8; 3,4; 4,8; 4,5; 5,7; 6,7; 7,8 +238: 1,2; 1,3; 1,4; 2,5; 2,6; 2,3; 3,7; 3,4; 4,8; 5,8; 5,7; 5,6; 6,7; 7,8 +239: 1,2; 1,3; 1,4; 2,5; 2,3; 3,6; 3,4; 4,7; 5,8; 5,6; 6,8; 6,7; 7,8 +240: 1,2; 1,3; 1,4; 1,5; 2,6; 2,3; 3,7; 3,4; 4,8; 5,8; 5,7; 5,6; 6,7; 7,8 +241: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,7; 3,4; 4,8; 4,5; 5,8; 5,7; 5,6; 6,7; 7,8 +242: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,7; 2,3; 3,7; 3,8; 3,4; 4,8; 4,6; 4,5; 5,6; 6,8; 6,7 +243: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,7; 2,3; 3,7; 3,8; 3,4; 4,8; 4,7; 4,5; 5,7; 5,6; 6,7; 7,8 +244: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,7; 3,8; 3,4; 4,8; 4,7; 4,5; 5,7; 5,6; 6,7; 7,8 +245: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,8; 3,4; 4,8; 4,7; 4,5; 5,7; 5,6; 6,7; 7,8 +246: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,8; 3,4; 4,8; 4,5; 5,7; 5,6; 6,7; 7,8 +247: 1,2; 1,3; 1,4; 1,5; 2,6; 2,3; 3,7; 3,8; 3,4; 4,8; 4,7; 4,5; 5,7; 5,6; 6,7; 7,8 +248: 1,2; 1,3; 1,4; 1,5; 2,6; 2,3; 3,7; 3,4; 4,7; 4,8; 4,5; 5,8; 5,6; 6,8; 7,8 +249: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,7; 3,8; 4,8; 4,7; 4,5; 5,7; 5,6; 6,7; 7,8 +250: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,8; 4,8; 4,7; 4,5; 5,7; 5,6; 6,7; 7,8 +251: 1,2; 1,3; 1,4; 1,5; 2,6; 2,3; 3,7; 4,7; 4,8; 4,5; 5,8; 5,6; 6,8; 7,8 +252: 1,2; 1,3; 1,4; 1,5; 2,6; 2,7; 2,3; 3,8; 4,8; 4,7; 4,5; 5,6; 6,7; 7,8 +253: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,7; 2,3; 3,7; 3,6; 3,4; 4,6; 4,8; 4,5; 5,8; 6,8; 6,7 +254: 1,2; 1,3; 1,4; 1,5; 1,6; 2,6; 2,7; 2,8; 2,3; 3,8; 3,7; 3,4; 4,7; 4,6; 4,5; 5,6; 6,7; 7,8 +255: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,7; 2,3; 3,7; 3,8; 3,4; 4,8; 4,6; 4,5; 5,6; 6,8; 6,7; 7,8 +256: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,3; 3,6; 3,7; 3,4; 4,7; 4,8; 4,5; 5,8; 6,8; 6,7; 7,8 +257: 1,2; 1,3; 1,4; 1,5; 2,5; 2,6; 2,3; 3,6; 3,7; 4,7; 4,8; 4,5; 5,8; 6,8; 6,7; 7,8 diff --git a/polyhedra/nine.txt b/polyhedra/nine.txt new file mode 100644 index 0000000..4d051ae --- /dev/null +++ b/polyhedra/nine.txt @@ -0,0 +1,2606 @@ +1: H~e[mf@ +2: Hte[mdp +3: HteKIL| +4: HteFjC] +5: HseIZsp +6: HseIZSp +7: HseIZCp +8: HseIZop +9: HseIZ_p +10: H{bB_{L +11: H{`B_{L +12: HsPhgcJ +13: H{uIRoM +14: HtdRggX +15: HsuIZOx +16: Hs`Jhc] +17: Hs`Jhc[ +18: Hs`JHc] +19: Hs`JHc[ +20: Hs`J@c[ +21: Hs`J`c] +22: Hs`J`c[ +23: Hs`J@c] +24: H|aBH[w +25: H|aBHoM +26: H|aAHoM +27: H|iAH[w +28: H|aAH[w +29: H|iAHoM +30: H|iBHoM +31: H|ikIKw +32: H{uIR_M +33: HtdRGgX +34: HteUJC[ +35: H{eIR_M +36: HtdBGgX +37: HteUJc[ +38: H{eIRoM +39: HtdBggX +40: HteEjc[ +41: HteEJc[ +42: HteEJC[ +43: HtbmJG{ +44: HtaIZsX +45: HtaIZcX +46: H|mcIKk +47: H|mCIKk +48: H|mCJKk +49: HsuMZOx +50: HteAJsN +51: HteAJcN +52: HspYHKt +53: HsvIPSx +54: H|aFGsN +55: HteKML| +56: Hte[Jl` +57: H~eAA{F +58: H|``wSF +59: HtmMB\` +60: HteEI|p +61: HtfAyCr +62: Hte[Nl` +63: H~f[IGF +64: H|mJOSL +65: Htf[ILs +66: HsbJhc] +67: HsbJHc] +68: H|aFH[w +69: H|aFHoM +70: H|aEHoM +71: H|iFHoM +72: HtmN]Da +73: Htf[MLs +74: Htf]IGt +75: H|eK[Fx +76: Hte[md` +77: HteKIL{ +78: HsbIzCp +79: HspYHKp +80: H{bBX_X +81: HspYHCp +82: H|eKBcM +83: HsuMZOp +84: HsvIPSp +85: H|aFgoL +86: HsvIPCp +87: H|aFGoL +88: H|eKFcM +89: H|eKFCM +90: HteKML{ +91: Hte[Jd` +92: H~eAA[F +93: H|`_wSF +94: HteAI\p +95: HteAI\o +96: H|mKADF +97: H~eA?[F +98: HtmMBL` +99: HteEI\p +100: HtfAYCr +101: H|aKJc[ +102: HtfAYCp +103: HtdIH_F +104: H|eK[Dp +105: Hte[Nd` +106: H~fKIGF +107: H|eJOSL +108: HtfKILs +109: HtfKILo +110: H|mMKDB +111: H~eKIGF +112: HtmNMDa +113: HtfKMLs +114: HtfMIGt +115: H|aKNc[ +116: HtfMIGp +117: HtfIH_F +118: H|eK[Fp +119: H|eK[F` +120: H|eK[Fr +121: Hte[mDp +122: HteKILx +123: HsbIz?r +124: HspYHCt +125: HsuMZ?x +126: HsvIPCx +127: H|aFGoN +128: HteKMLx +129: Hte[JL` +130: H~eA?{F +131: HtmMA\` +132: HteEI|o +133: HtfAyCp +134: HtdYH_F +135: Hte[NL` +136: H~e[IGF +137: HtmM]Da +138: HtmMMDa +139: Htf[MLo +140: Htf]IGp +141: HtfYH_F +142: H|eK[Fj +143: Hte[Mdp +144: HteKILt +145: HteKILp +146: H|mKKDb +147: HteKMLt +148: Hte[Ml` +149: HseMZSp +150: H|aKNKw +151: Hte[MLs +152: HteEJcN +153: H|mLKFB +154: Hte[Md` +155: HteKILs +156: HteEjC[ +157: HseMZOp +158: HteAJ_N +159: HteKMLs +160: HteEJ_N +161: H|mK[F` +162: H|mKKVh +163: HteKmdp +164: HteKMdp +165: HteKMDp +166: HteKILl +167: HtbB@[x +168: HsdBHkL +169: HsbIrCr +170: HsbJ@cZ +171: HtamJG{ +172: HteIbcU +173: HsuMJOx +174: HseMJOx +175: HteKMLl +176: HtmMB?n +177: H|aJ@sF +178: HteEIxp +179: HteEJ_\ +180: HtmL]Da +181: H|mKJOJ +182: Htf[MDs +183: H{eK]Cy +184: H|e[Lf@ +185: HteKmd` +186: HteKmD` +187: HteKILk +188: HteKILg +189: HtbB@[w +190: HtbB@oM +191: HsdBH_L +192: HsdBHgL +193: HsbIrCp +194: HtamJGw +195: HtaMJGw +196: HteIBCU +197: HteIbCU +198: HsuMJOp +199: HsuMJ?p +200: HteKMLk +201: HtmMB?f +202: H|aJ?sF +203: HteEIXp +204: HtmLMDa +205: H|eKJOJ +206: HtfKMDs +207: H{eLMCw +208: H|e[LF@ +209: H|e[Kn@ +210: H|mlNF@ +211: HtmlND` +212: H|eKJwF +213: HteMMD} +214: HtmEJ\` +215: HteK^t` +216: HsfMJwp +217: H|mAB[F +218: H|nkIGF +219: H~uIOSL +220: H|dogcX +221: HtnkILc +222: HsrJpcY +223: H|jEHWw +224: HsrJ`cY +225: HtuMRoM +226: HsfAZsp +227: Ht`AzcM +228: Ht`IyCr +229: HtuIRoM +230: HtuIYOx +231: Hte^mDa +232: HtbMJwX +233: H~uKYOJ +234: HtuKYTw +235: HtbAZsX +236: HtnkMLc +237: HtuK]Tw +238: HteMAL| +239: HtbIyCr +240: HtpYHgF +241: HtuMYOx +242: HtvIPoJ +243: H|eKKNx +244: HtmlMD` +245: H|eKJoF +246: H|eKN_F +247: H|eKJ_F +248: HteMMD{ +249: HteK^d` +250: HsfMJop +251: H|mAA[F +252: H~`?wSF +253: H|nKIGF +254: H~eIOSL +255: HtuMR_M +256: HsfAZop +257: Ht`Az_M +258: Ht`IyCp +259: HtuIR_M +260: HtuIYOp +261: Hte^MDa +262: HtbMJoX +263: H~aI?{F +264: HtuKYTo +265: HtbAZoX +266: HtnKMLc +267: HtfEJoL +268: HtnMIGd +269: HtuK]To +270: HtmlID` +271: H|eBoWF +272: HufY`Cp +273: H|`ogSL +274: HteMAL{ +275: HtbIyCp +276: HtpYH_F +277: HtTH_SL +278: HtuMYOp +279: HtvIP_J +280: HttH_SL +281: H|eKKNt +282: HtmkND` +283: H|eKJgF +284: HteMMDy +285: HteK^T` +286: HtiNJCw +287: HteJRCM +288: HsfMJgp +289: H|mE?[F +290: H|mKMGF +291: Hte]mDa +292: HtmFJCk +293: HtmEJCk +294: H{eJ@cM +295: H{eJPcM +296: HtfQHK[ +297: HtfAHK[ +298: HtdQHK[ +299: HtdAHK[ +300: HtbMJgX +301: H{eFHoL +302: HteYICb +303: HteYiCb +304: Hte]iCb +305: H{eK^Cp +306: Hte]ICb +307: HteK]Tw +308: HtmkJD` +309: HtmKJD` +310: H|eB?[F +311: H|eB_[F +312: HteMALx +313: HtaNJC] +314: HsfAZcp +315: HteIRoM +316: HteIR_M +317: HteMR_M +318: HsfMJG{ +319: HtbAZcX +320: HsdIHK{ +321: HsdIHKw +322: H{aNHc[ +323: H|eCNcM +324: H|eKKNp +325: H|eKKNl +326: HtmLND` +327: H|eKJWF +328: HteMMDu +329: HtmEJ?n +330: H{eN@cF +331: H{eJ@cF +332: H{eJ@sF +333: HteK]t` +334: HtanJCw +335: HtaNJCw +336: HtaMJCw +337: HteIrCM +338: HsfMJWp +339: HtmAJ?n +340: H{``gsM +341: HtmkILc +342: H|eB_k[ +343: HtuMBoM +344: HsfAZSp +345: H{eAHoN +346: H{eKIK{ +347: H{eKIKw +348: H|e[kDB +349: H|e[KDB +350: HtmkMLc +351: H|eKN_R +352: H|eKJ_R +353: H|e[J_R +354: HtuK]Dw +355: HtmLJD` +356: H|eBO[F +357: HteMALt +358: H{eKMK{ +359: H|e\KFB +360: HtmLMD` +361: H|eKJOF +362: HteMMDs +363: HteK]d` +364: HtamJCw +365: HteIbCM +366: HsfMJOp +367: HtmKILc +368: HsrJ@cY +369: HtuMB_M +370: HtaiJCw +371: Ht`RGoL +372: HsfAZOp +373: Ht`I@c] +374: HteIYOp +375: HtmKMLc +376: HtmMBCe +377: H|aJ?sM +378: HtmIBCe +379: H|`_gSL +380: HtfEJ_L +381: H|e[J?R +382: HtuK]Do +383: HtmLID` +384: H|eBOWF +385: HteMALs +386: HtbI@c] +387: H|e^kF@ +388: HtelND` +389: H|eKIwF +390: H|eKMWF +391: H|eKIWF +392: HteMMDm +393: HtmEJX` +394: H|mCMDk +395: Hte\mDa +396: HtbMJWX +397: H~aIAsF +398: H{d`gSF +399: H~uKIOJ +400: H|eJ_cT +401: HtuKITw +402: HtbAZSX +403: HtnkMDc +404: HtuKMTw +405: HtelJD` +406: HteLJD` +407: H|eAO[F +408: H|eAo[F +409: HufI`cp +410: H|`Og[L +411: HteMALl +412: H{eKJCr +413: HtbIqCr +414: HtuMIOx +415: H|aKmC] +416: H|e^KF@ +417: HtelMD` +418: HteLMD` +419: H|eKIoF +420: HteMMDk +421: Hte\MDa +422: HteeJCk +423: H{eI`cM +424: H{dB?kL +425: HtPHGcJ +426: H{eIb_U +427: HtdBHGh +428: HtbMJOX +429: H~aI?sF +430: H{d`_SF +431: H~eKIOJ +432: H|eJ?cT +433: HtnKMDc +434: HtemBCe +435: H|aI_sM +436: H|aM?sM +437: HteIBCe +438: HtfEJOL +439: H{eN@_F +440: HtfAHCL +441: HtdAHCL +442: H|eKN?R +443: H|eKJ?R +444: HtuKMTo +445: HtelID` +446: H|eAoWF +447: HstQHCd +448: HufI`Cp +449: H|aUGoL +450: HteMALk +451: HtbIqCp +452: HsemJGw +453: HtbAZOX +454: HtuMIOp +455: H|eK[f@ +456: H|eK[v@ +457: H~vKKND +458: HtfKML{ +459: HteEI\x +460: HtfAYCz +461: HtfMIG| +462: Hte[nd` +463: HsuMZop +464: HteAI\x +465: HsvIPsp +466: H~vKIGF +467: H|eJoSL +468: HtfKIL{ +469: HteVjC[ +470: HsuIZop +471: Hs`Izcp +472: HtmnMDa +473: H|e[lf@ +474: HtfKMD{ +475: HtemBL` +476: HteKnd` +477: HsuMJop +478: H|uAA[F +479: H|`OwSF +480: H|vKIGF +481: H|eIoSL +482: HtfKILk +483: HsbIrcp +484: HtimJGw +485: HteJbCU +486: HtenMDa +487: HtmeJGk +488: H{eJ`cU +489: HtfKMLk +490: HteEIXx +491: HtfAYCj +492: HtfMIGl +493: HtmmB?f +494: H|aJ_sF +495: HtrIPSX +496: H{d`GkL +497: HtmlMDa +498: H|eKJoJ +499: H|eKKVp +500: HtfKMDw +501: HteKnD` +502: HsuMJ_p +503: H|uA?[F +504: H|uKIGF +505: HtfKILg +506: HsbIr_p +507: HtjB@oM +508: HsdbH_L +509: HtiMJGw +510: HteJBCU +511: H|`A_[L +512: H|aIQoM +513: HtemMDa +514: HtqMJ_X +515: HtmEJGk +516: H{eJ@cU +517: HtfKMLg +518: HteEIXw +519: HtfAYCh +520: HtdIHOF +521: HtfMIGh +522: HtfIHOF +523: HtmmB?F +524: H|aJ_oF +525: HtpIHCT +526: HtPH?sM +527: HtrIPCX +528: HtpH?sM +529: HtmkMDa +530: H|eKJ_J +531: H|eKKVX +532: HtfKMDk +533: HtemB?f +534: H|aI_sF +535: HteEIXh +536: HtelMDa +537: H|eKIoJ +538: HteeJGk +539: H{eI`cU +540: H{d@GkL +541: H|eKKVP +542: HtfKMDg +543: HtemB?F +544: HteEIXg +545: HtfA@SM +546: Htp@GcL +547: HteMIGh +548: H|e[nF@ +549: HtfKMLw +550: HtmmAL` +551: HtrIPcX +552: HteEI\w +553: HtfAYCx +554: HtpIHcT +555: HtfMIGx +556: Hte[nD` +557: HsuMZ_p +558: H~uA?[F +559: H|`ooSF +560: HteAI\w +561: HsvIPcp +562: H~uKIGF +563: H|eJ_SL +564: HtfKILw +565: HteVJC[ +566: HsuIZ_p +567: Hs`Iz_p +568: HtjMJGw +569: HtmmMDa +570: HtnEJGk +571: H|e[kFr +572: HtmkNL` +573: H|e[JgF +574: HteM]Dy +575: HtmKZT` +576: H|mB@[F +577: HtmK^T` +578: HtiNZCw +579: HtmJRCM +580: H|iJPoM +581: HtdQHk[ +582: H|mkJGF +583: Hte]mDq +584: HtmFJCm +585: H{eJPsM +586: HtbMJg\ +587: HteK]Tx +588: HtmkJL` +589: H|eB_{F +590: HteMA\x +591: HtaJJC} +592: Ht`Bg{L +593: HtaNJC} +594: HteIRsM +595: HsfMJG| +596: HtbAZcZ +597: H|e[KNp +598: HtmkML` +599: H|e[J_F +600: HtmNBCM +601: HtmJBCM +602: H|aJPoM +603: HtdAHk[ +604: HteM]Dw +605: HtmKZD` +606: H|mB?[F +607: HtmK^D` +608: H|mKJGF +609: Hte]MDq +610: HtmEJCm +611: H{eJ@sM +612: HtbMJ_\ +613: HteK]Tp +614: HtmkIL` +615: H|eB_wF +616: HteMA\w +617: HtaJJC{ +618: Ht`BgwL +619: HsfAZCx +620: HtaNJC{ +621: HteIRcM +622: HsfMJGx +623: HtmAJCm +624: H{``gsL +625: H{b`_sL +626: H~uK]SM +627: HtuMZoM +628: HtfAZsp +629: HtvIYOx +630: HttIXoJ +631: H|aNJC] +632: H|aMjC] +633: HtfAZSp +634: HtfIYOx +635: HtfMJwp +636: Hte^iDb +637: HtuIZoM +638: HspZhCr +639: H{jBXop +640: HspYhCr +641: H|jAXop +642: HspZ`Cr +643: H|jAX_X +644: H|bBXop +645: HspZHCr +646: H|bAXop +647: H|jBX_X +648: HspZhCp +649: H|bBX_X +650: HsvJpCr +651: H~eMO[L +652: HsvJ`Cr +653: H~uM?[L +654: H~eM?[L +655: H~uMOWF +656: HsvJpCp +657: H~uM?WF +658: H~eM?WF +659: H~uEO[L +660: HsvIpCr +661: HtnmILc +662: HtrJpcY +663: H{neHWw +664: HtrIpcY +665: H|e|NF@ +666: Hte|ND` +667: H~eKIwF +668: H|eJ@sL +669: HvvKILo +670: HtmMMDm +671: HtnEJX` +672: Hte\mDb +673: HtefJC{ +674: H{eIrcM +675: H}eIb_] +676: H{dRGk[ +677: HtdbHGx +678: HtbMJWx +679: H~aIAsN +680: H{d`gsF +681: H~uKION +682: H|eJ_c\ +683: HtuKI\w +684: HtbAZSx +685: HtnmMDc +686: H|eK{f` +687: Hte|MD` +688: H~eKIoF +689: H|eJ@cL +690: HtmMMDk +691: Hte]BCf +692: H~aI?sN +693: HtnEIX` +694: HvuKMLo +695: HsmmJCw +696: HtfMIop +697: Hte\MDb +698: HteeJC{ +699: H{eIbcM +700: HtdBHKX +701: HueIZ`` +702: H{eIb_] +703: H{dBGk[ +704: HueMZ`` +705: HsvI`Cr +706: H~eAO[L +707: Hut@HKT +708: H~eKION +709: HtnMMDc +710: Hte}BCe +711: H~aI_sM +712: HvaMYpo +713: HtrI`cY +714: HtuKM\o +715: Hte|ID` +716: H~eAoWF +717: HtmMALk +718: HsfBH[w +719: HvuEJ_M +720: HsmiJCw +721: HtfAYop +722: HtdIH_Z +723: HtvAYOp +724: HtdIX_J +725: HsemJKw +726: HteaJC{ +727: H{`RgoL +728: HubJbC[ +729: HspY`Cr +730: HtvMIOp +731: HtfIX_J +732: H|eK[f` +733: Hte\MD` +734: H~eKIOF +735: H|eJ?cL +736: HtuKILo +737: HteeJC[ +738: H{eIb_M +739: HtdBHGX +740: HueIZ?p +741: Hu`J@c[ +742: HueMZ?p +743: H~eAOWF +744: HtmMMDc +745: Hte]BCe +746: H~aI?sM +747: H{d`_oF +748: HtaIYpo +749: HtnEJ?L +750: H~uKKNL +751: HtvKMLw +752: HtuEI\w +753: HtfAYSx +754: HtpIHcV +755: HtfMIWx +756: Hte\nD` +757: HuuMZ_p +758: H~uAO[F +759: HtuAI\w +760: HsvIpcp +761: H~uKIWF +762: H|eJ_sL +763: HtvKILw +764: HtevJC[ +765: HuuIZ_p +766: H}eIr_M +767: HtjMJWw +768: H~aIQsM +769: HtqIYpp +770: HtmmMDe +771: HtqMYpp +772: HtrIPcZ +773: HtnEJGl +774: H~eK[vP +775: HveK]to +776: HtmkIDf +777: HufZaCb +778: HutP@kF +779: HtvMALw +780: HvbIy_p +781: HtpYh_R +782: Hta~JC[ +783: HtiIJ[w +784: HsdRjC[ +785: HveIr_M +786: HvuIY_p +787: HsnMJWw +788: HtmAJKl +789: HvuMY_p +790: HtvIp_R +791: HtmkMDf +792: HtvMMDw +793: HteMI[x +794: HteL~D` +795: HtiMJ[w +796: H|aIRsM +797: H{dBgkX +798: HuvMJ_p +799: H|}AO[F +800: H~`OgsJ +801: H|}KIWF +802: H~eI_sL +803: H|d`HcX +804: Hte}MDe +805: HtmEJKl +806: HtrMIpp +807: Ht~KMLg +808: HtmmBKF +809: HtrIPCz +810: HtvEIXw +811: HtnAYSh +812: Hse~JCs +813: HtjBXoM +814: HtnMIWh +815: H|eK~F@ +816: HveK]To +817: HtmkIDb +818: HufY@Cr +819: HtfMALw +820: HteIIKx +821: HtbIy_p +822: Hta^JC[ +823: HtiIJKw +824: HudY@Cr +825: HveIR_M +826: HtuMY_p +827: HtvIP_R +828: HttH_cL +829: HtmkMDb +830: H|eKJ_N +831: Hte}MDa +832: HveKYTo +833: HtmAJKk +834: HtrAZ_X +835: HtmEJKk +836: H{eJ@c] +837: HtrMJ_X +838: H~aI_wF +839: H~eKY_J +840: H|eK[vP +841: HtmkMDe +842: H|eKJ_Z +843: HtvKMDw +844: HteLnD` +845: HtiMJWw +846: HteJBCu +847: H|`Ag[L +848: H|aIQsM +849: HuuMJ_p +850: H|uAO[F +851: H|uKIWF +852: HtvKILg +853: HtjBPoM +854: HubIr_p +855: HtemMDe +856: HtmEJGl +857: H{eJ@cV +858: HtqMIpp +859: HtvKMLg +860: HtmmBGF +861: H|aJ_kX +862: HtpIHCV +863: HtrIPCZ +864: HtpH?sN +865: HtuEIXw +866: HtfAYSh +867: HtdIHON +868: HtfMIWh +869: HtfIHON +870: H~e]mF@ +871: Hte[mDr +872: HteKI\x +873: HteBJC} +874: H{`Bg{L +875: HtbAZCz +876: HteFJC} +877: H{eIRsM +878: HtdBgkX +879: HsuIZp` +880: HsuIZ`` +881: HseIZp` +882: HtbMJG| +883: HsuMZp` +884: HsvIPCz +885: Hte{NL` +886: H~e[IgF +887: H|mJ@SL +888: Hvf[ILo +889: HtmM]Di +890: H|e{LN@ +891: HteKmDr +892: HtaMJG| +893: HsuMJp` +894: HsfIPCz +895: HteKM\h +896: HseMJKt +897: HtbB@sN +898: HsbIrd` +899: Hsp@Hk\ +900: HtmMBON +901: HveEIxo +902: HtfA`c] +903: HtmK]Di +904: Hvf[MDo +905: Hte]Igp +906: H|e{LF@ +907: HteKmDb +908: HtaMJGx +909: HsuMJ`` +910: HsfIPCr +911: HteKM\g +912: HseMJKs +913: HtbB@sM +914: HsbIr`` +915: HsbIr?p +916: Hsp@Hk[ +917: H|e[M?b +918: HtmMBOF +919: HtmMB?F +920: H|aJ?oF +921: HveEIXo +922: HteEIXo +923: HtfA`cM +924: HtmKMDi +925: HtmKMDa +926: H|eKJ?J +927: HvfKMDo +928: HteMIgp +929: H|e[LN@ +930: HteKmDp +931: HteKILh +932: HtbB@oN +933: HsdBHcL +934: HsbIr?r +935: HtaMJG{ +936: HteIBcU +937: HsuMJ?x +938: HteKMLh +939: HtmMB?N +940: H|aJ@oF +941: HteEIxo +942: HtmK]Da +943: H|mKJ?J +944: Htf[MDo +945: H|e{NF@ +946: Hte[mDb +947: HteKI\w +948: HteBJC{ +949: H{`BgwL +950: HtbAZCx +951: HteFJC{ +952: H{eIRcM +953: HtdBGkX +954: HtbMJGx +955: HtbMJ?x +956: HsuMZ`` +957: HsvIPCr +958: Hte{ND` +959: H~eKIgF +960: H|eJ@SL +961: HvfKILo +962: HtmMMDi +963: H|mN]F@ +964: HtmL]Db +965: H|mKJON +966: Htf]MDs +967: HvbMIxo +968: HteYIKt +969: HstHHK\ +970: HtbIhc] +971: HteK}dp +972: HtamJK{ +973: HteIbc] +974: HsvMJOx +975: HtmKILl +976: HtbBH[x +977: HsrIrCr +978: HvuMA\o +979: HteIYsp +980: HtmKMLl +981: HtmMBCn +982: H|aJ@sN +983: HtfEIxp +984: HvuM]Do +985: HteMYsp +986: H|e|KFB +987: HtmLMDb +988: H|aK]Dw +989: HtfMMDs +990: HteMIKt +991: HteK}d` +992: HtamJKw +993: HteIbC] +994: H|aMBoM +995: HsvMJOp +996: HtmKILk +997: HtbBH[w +998: HtbBHoM +999: HsrIrCp +1000: HvuMB_M +1001: HtaiJKw +1002: HtaIJKw +1003: HsdY`Cr +1004: HsvAZOp +1005: Ht`I`c] +1006: HteIYop +1007: HtmKMLk +1008: HtmMBCf +1009: HtfEIXp +1010: HtfAHcN +1011: HvuK]Do +1012: HtmLIDb +1013: HsfY`Cr +1014: HtfMALs +1015: HteIIKt +1016: HtbI`c] +1017: HteMYop +1018: H|e\KFb +1019: HtmL]D` +1020: H|mKJOF +1021: Huf]J?p +1022: H|mBOWF +1023: HstHHC\ +1024: Hte]MDs +1025: HtmEJCN +1026: HtbMIxo +1027: HtbIHc] +1028: HteK]dp +1029: HtamJC{ +1030: HteIbcM +1031: HufAz?p +1032: Ht`RgoL +1033: H{aJX_X +1034: HsfMJOx +1035: HtmAJCN +1036: HtbAYto +1037: Ht`IHc] +1038: H|nmILD +1039: HtniJKF +1040: HutpHkF +1041: Hut`HkF +1042: HutpHcF +1043: Hut`HcF +1044: H|rHpCr +1045: H|zH@cZ +1046: H|rH@cZ +1047: H|zHpCR +1048: H|zH?cZ +1049: H{zHpCr +1050: HutPHkF +1051: H{zH@cZ +1052: HtvJpcR +1053: HtvJPcR +1054: H|nMHOx +1055: H|fmHOx +1056: H|fMHOx +1057: H|nmHOX +1058: HtvJp_R +1059: HtliBKF +1060: HteZj?b +1061: H|eYj?b +1062: H|e]j?b +1063: HtfIITw +1064: H|e^ggX +1065: HtniJGF +1066: HutpHgF +1067: Hut`HgF +1068: HutpH_F +1069: H{wogoF +1070: H|fo_sM +1071: H|f__sM +1072: H|bo_sM +1073: H|`o_sM +1074: HutPH_F +1075: Hut@H_F +1076: HtvIRoM +1077: HttIPsM +1078: HtpZhCR +1079: H{dIHK{ +1080: H{dIHKw +1081: H{fiHKw +1082: HttIPcM +1083: H{fIHK{ +1084: H|fAXOp +1085: H{fBXOp +1086: HtpYHCR +1087: H{nBXOp +1088: HtpYhCR +1089: HtvJpCR +1090: HtvJ`CR +1091: HtvJPCR +1092: HtvJ@CR +1093: H|nMHOp +1094: H|fmHOp +1095: H|fMHOp +1096: H{nmHOp +1097: HtvIpCR +1098: H|nmHop +1099: HtvJpCr +1100: HtvIZoM +1101: HtpZhCr +1102: H|nAXop +1103: H{nAXop +1104: HtpY`Cr +1105: HtdIYSp +1106: H|fAXop +1107: H|nBX_X +1108: HtpZhCp +1109: HtpZHCp +1110: H|fBX_X +1111: H{nBXop +1112: HtpYhCr +1113: HtdIYSx +1114: HuTHHKl +1115: H|fBXop +1116: H{fBXop +1117: H|e^IDb +1118: HtvJ`Cr +1119: H|nmH_X +1120: HtvJpCp +1121: H|nMH_X +1122: H|fmH_X +1123: H|fMH_X +1124: H{nmHop +1125: HtvIpCr +1126: H|fmHop +1127: HtvJPCr +1128: HtvJ@Cr +1129: H|fMHop +1130: H|nmHoX +1131: HtvJpcp +1132: H|eYjD` +1133: HteZjD` +1134: HtniIDf +1135: H|nmHWw +1136: HtvJpcY +1137: HtvJ`cY +1138: H{eZJCq +1139: HtlIIKd +1140: H|e^JCq +1141: H|fmHWw +1142: HtvJPcY +1143: HtpIBc^ +1144: HtnAJ?n +1145: H|nAJ?N +1146: H|mMILc +1147: HtfIYPp +1148: H|e\jCq +1149: HtvIpcY +1150: H|nmJGw +1151: HtvIRsp +1152: HtvIRcp +1153: HtvIZoX +1154: HtpYJkp +1155: HtpYJcp +1156: H{dJHwL +1157: H{fbHwL +1158: H{dbHwL +1159: H{fBHwL +1160: H{fjHoL +1161: H{fJHwL +1162: H|fAZCw +1163: H{nBZCw +1164: HtpYIkp +1165: H{nAZCw +1166: H{fBZCw +1167: H|mmJGw +1168: HtfIRsp +1169: HtfIRcp +1170: HtfIRSp +1171: HtvIZOX +1172: HtpBhc] +1173: HtpBHc] +1174: H{djH_L +1175: H{dJH_L +1176: HtTIXCX +1177: H{ejHgL +1178: HtpAhc] +1179: H{nB@oM +1180: HtpAhc[ +1181: HtdIXCX +1182: H{p`_cL +1183: HtnB@[w +1184: H{dbHgL +1185: HtpIXSX +1186: HtPIXSX +1187: H{fjH_L +1188: HttIXCX +1189: H{fJH_L +1190: H{fJHgL +1191: H|eNJCU +1192: H|eMjCU +1193: H|eMJCU +1194: H|eLjCU +1195: HtfIQsp +1196: HtnmJGw +1197: H|eZbCU +1198: H|e^JCU +1199: H|fmJGw +1200: H{nmJGw +1201: HtvIQsp +1202: H~e]mC[ +1203: Hte]jc[ +1204: HteIZsp +1205: HteIZcp +1206: HteIZSp +1207: HteIZCp +1208: HtbiJK{ +1209: HsdZhCr +1210: HsdZhCp +1211: HsdZHCp +1212: HtbRgwX +1213: HsdYhCr +1214: H|iIXop +1215: H|aIXop +1216: H|aJXop +1217: HsdZHCr +1218: H|iJX_X +1219: H|aJX_X +1220: HsvIZOx +1221: Ht`Jhc] +1222: Ht`JHc] +1223: HstIXSx +1224: H{ajHg\ +1225: Ht`Ihc] +1226: H|mAH[w +1227: Ht`J`c] +1228: H{bJHgX +1229: H{bjH_X +1230: H{bJH_X +1231: H{bjH_\ +1232: HstIXCx +1233: H{bJHg\ +1234: H{bJH_\ +1235: H|eFjC[ +1236: H|eFJC[ +1237: H|eFJC] +1238: H|eEjC] +1239: H|eEJC] +1240: HteMZsp +1241: HtmNYDb +1242: HsfZhCr +1243: HsfZ`Cr +1244: HsfZHCr +1245: H|iMXop +1246: H|aMXop +1247: H|aNXop +1248: H|iNX_X +1249: H|aNX_X +1250: H{iNXop +1251: HsfYhCr +1252: Htf]ILs +1253: HtbJhc] +1254: HtdYIKt +1255: HtfYIKt +1256: H{eMZCy +1257: HtbJHc] +1258: HtfYICt +1259: H|e[Nd` +1260: HteMZop +1261: HtfAI\p +1262: HtdIIKt +1263: HtfMILs +1264: HtfIIKt +1265: H{eNJCw +1266: HtfIIKp +1267: H|e[ND` +1268: H|e[Md` +1269: HteMjC[ +1270: HteIZOp +1271: HspJ@c] +1272: HteMZOp +1273: HtfAJ_N +1274: HtmNID` +1275: HteMILs +1276: HtfIICt +1277: H|e[NL` +1278: HteMZcp +1279: Htf]ILo +1280: HtfYIKp +1281: H|e[Ml` +1282: HteMZSp +1283: H|mK]DB +1284: H~f[MN@ +1285: Hte[MLt +1286: HteEJc^ +1287: Hte[Mlp +1288: HseMZSx +1289: HteAJc^ +1290: HteEjc] +1291: HseIZSx +1292: Hs`JHc^ +1293: Hs`JHc\ +1294: H|aBHoN +1295: H|mK[Fb +1296: Hte[MLp +1297: Hte[ILp +1298: HteEJc] +1299: H~eM{FB +1300: HteM}Dq +1301: HtmKYTh +1302: HtbBgsN +1303: Hve]Alo +1304: HteYicp +1305: HtaIJk\ +1306: HsdIHC~ +1307: H{eBHsM +1308: HsdIHC| +1309: HsdIHCx +1310: H{aJH_\ +1311: H{eFHsM +1312: HteYiCp +1313: H{bh_cL +1314: H{bH_cL +1315: H{eFHoM +1316: Hve]mDo +1317: Hte]icp +1318: HtmFZCM +1319: H{mJPoM +1320: HtdQhc[ +1321: HseYJkp +1322: H{``xcM +1323: H{b`_{L +1324: H{``_{L +1325: H{mN@oM +1326: HtfAhc[ +1327: H{eN@oM +1328: HteK]Dz +1329: HteMA|p +1330: HsfMJx` +1331: HsfIHC~ +1332: H|e[KNd +1333: HteM]Dq +1334: HtmK]T` +1335: Hsf]JGp +1336: Hsf]J?p +1337: H|mkJ?F +1338: HtaNZCw +1339: HtmIRCM +1340: H|iJ@oM +1341: HtdQHc[ +1342: HtdAHc[ +1343: HtfAHc[ +1344: HtmK]D` +1345: H|mKJ?F +1346: H|eKJ?F +1347: Hte]mDo +1348: H|e[FCF +1349: HteK]Dx +1350: HteMA\p +1351: HtaMJC} +1352: HsfAzCp +1353: H|eCJK[ +1354: HteIBcM +1355: HteIBsM +1356: HsfMJ?| +1357: H{aNH_\ +1358: HtmKJL` +1359: H|eB?{F +1360: H|eCNC] +1361: HtmKNL` +1362: H|e[JGF +1363: H|e[KN` +1364: H|ekKNd +1365: HteMmDq +1366: H{eKNCx +1367: H{eKNCw +1368: HteK]Th +1369: HsfMJGt +1370: H{aNHcX +1371: H{aNH_X +1372: HteMBoN +1373: HsfAZCr +1374: HtaJJCy +1375: Ht`BG{L +1376: HtaNJCy +1377: HteIRSM +1378: HteIRCM +1379: HteK]Dy +1380: HteMAlp +1381: H{eKJCx +1382: HtmKJT` +1383: H|eB@[F +1384: HtmKNT` +1385: H|ekJGF +1386: H|eKNGF +1387: H|eKJGF +1388: H|eKKNd +1389: H|eKKN` +1390: H|ekKN` +1391: HteMmDo +1392: HteMiCp +1393: HteK]Dh +1394: HsfMJ?t +1395: HteMB_N +1396: HsfAZ?r +1397: HsdIHCt +1398: HtaMJCy +1399: HteIBSM +1400: H|ekKFd +1401: HteKmDq +1402: HteKITh +1403: H|eCMCm +1404: HteK]Di +1405: HteMB_V +1406: HtaIJCZ +1407: H|nmHWL +1408: HtvJosX +1409: HtpYIDv +1410: H|fBZD` +1411: HtfBZD` +1412: HtfAZD` +1413: H|fAZD` +1414: HtnBZD` +1415: H|nBYD` +1416: HtnBYD` +1417: HtnAYD` +1418: H{fhoSL +1419: H{fho[L +1420: H{f`o[L +1421: H{fhg[L +1422: HtpAJk\ +1423: HtpAHk\ +1424: H|fBBCm +1425: HtpAJK\ +1426: HtfBBCm +1427: H|fABCm +1428: H|fBBCM +1429: H|fABCM +1430: H|nBBCM +1431: HttHGsX +1432: HttHGcX +1433: HttHGoJ +1434: HtpAJk[ +1435: H{dh_WF +1436: H{dH_WF +1437: HtvIQDz +1438: H|nMJH` +1439: H|nmJGd +1440: HtvIRsJ +1441: HtvIRcJ +1442: HtvIPsJ +1443: HtvJosJ +1444: HtpYJkF +1445: HtpYHkF +1446: H|nBYCb +1447: HtnBYCb +1448: H|nAZCb +1449: HtpYJcF +1450: HtpYHcF +1451: H|nAYCb +1452: HtnAZCb +1453: HtnBZCb +1454: H|nBZ?b +1455: HtpYBkF +1456: HtpBgsF +1457: H|fBZ?b +1458: HtpYBKF +1459: HtpBGsF +1460: HtpYBCF +1461: H|fB@Cl +1462: H|fAZ?b +1463: HtfBZ?b +1464: HtfB@Cl +1465: H{d@gWF +1466: H{f@oWF +1467: H{fhoWF +1468: HtpYBcF +1469: H{f`oWF +1470: H{f`_WF +1471: H{d`_WF +1472: HtpHGoJ +1473: HtPHGoJ +1474: H{f`wWF +1475: H{d`wWF +1476: H{f`gWF +1477: H{d`gWF +1478: H{fhgWF +1479: H|e^gcT +1480: HtvIRoJ +1481: HtfIRoJ +1482: HtfIR_J +1483: HtpZgcF +1484: HtpZ_cF +1485: H|nAXCh +1486: H|nMHGh +1487: H|nMHGl +1488: H|neHGl +1489: H|nmHGL +1490: HtvIBsJ +1491: HtfIBsJ +1492: H|nMJ?d +1493: HtvIBcJ +1494: HtfIBcJ +1495: H|mMJ?d +1496: HtnmJ?d +1497: HtmmJ?d +1498: H|eJ_cF +1499: H|eNgcF +1500: H|eNgcV +1501: HtfIRsJ +1502: HtfIPsJ +1503: HtvJOsJ +1504: HtpBg{F +1505: Htp@g{F +1506: H|fB@Cn +1507: H|fB@CN +1508: H|nA@Cn +1509: HtpB_{F +1510: HtpB_wF +1511: H|nB@CN +1512: HtpBgwF +1513: Htp@gwF +1514: H{f`g[F +1515: H|eNgcT +1516: H|eNGcT +1517: HtvJ?cJ +1518: H|eNGcV +1519: HtfIRcJ +1520: H|nmJGk +1521: HtvIRsX +1522: HtpYJkT +1523: HtpIJkT +1524: HtpIHkT +1525: H|nAZCi +1526: HtpYJcT +1527: H|mmJGk +1528: HtvIRSX +1529: HtpBgc^ +1530: H|fB@[h +1531: HtpBGc^ +1532: HtnB@[h +1533: H{d`HkL +1534: H{fhHcL +1535: HtpBgc\ +1536: HtfIRsX +1537: H|emJGk +1538: H|eMJGk +1539: HtfIRCX +1540: HtpBGc\ +1541: H|mMJGk +1542: HtfIRcX +1543: HtnmJGk +1544: HtrIRsX +1545: H{e^`cU +1546: H|e^HcU +1547: H|e[ncp +1548: HtfMIH| +1549: Hte^jCU +1550: HsvIRsp +1551: HtuIZoX +1552: HspYJkp +1553: HspYHkp +1554: HtjBZCw +1555: HsfJHwL +1556: HspYJKp +1557: HspYJCp +1558: HtbBZCw +1559: HtbAZCw +1560: H|bAZCw +1561: H|jAZCw +1562: HspYJcp +1563: HtjAZCw +1564: H|bmJGw +1565: H|bMJGw +1566: H|jMJGw +1567: HsvIRcp +1568: H~qMQoM +1569: HtuMZoX +1570: HtfAYDz +1571: HtvIPsX +1572: H{fMHwL +1573: HtfAYDr +1574: H|mmAL` +1575: HtfAYDx +1576: HtpYHcT +1577: HtvIPcX +1578: H|aN_{F +1579: HtrIPsX +1580: H|eKNsJ +1581: HteMIH| +1582: HteNjCU +1583: HsfIRsp +1584: HsfIRSp +1585: HsfIRCp +1586: HtuIZOX +1587: HspBhc] +1588: HtjB@[w +1589: HsdbHgL +1590: HsfjH_L +1591: HspBhc[ +1592: H|bB@[w +1593: HspBHc] +1594: HspBHc[ +1595: H|bB@oM +1596: H|amJGw +1597: H|aMJGw +1598: H|iMJGw +1599: HsfIRcp +1600: H|eIQoM +1601: HtfmJGk +1602: HtbIRsX +1603: HtuMZOX +1604: HtfABsN +1605: HtpYHKT +1606: HtvIPSX +1607: H|aNGsF +1608: HtfABcN +1609: H|mmB?F +1610: HtfABsM +1611: HtpYHCT +1612: HtvIPCX +1613: HttH?sM +1614: H|e[NkT +1615: HtmNZCi +1616: H|mJPSM +1617: Htf]JgT +1618: HtbJg{F +1619: HtbJgwF +1620: H{eIXsJ +1621: H{eIXoJ +1622: H{eMXsJ +1623: H{eMXoJ +1624: H{eNXoJ +1625: Hte]jcR +1626: HsvJPsJ +1627: HsvJ@sJ +1628: Ht`Jg{F +1629: Ht`JgwF +1630: Ht`JG{F +1631: Ht`JGwF +1632: H|eBHCn +1633: H|eBHCN +1634: H|mAHCn +1635: H|mAHCN +1636: HteIZsJ +1637: HteIZcJ +1638: H|ekJKd +1639: H|eKJKd +1640: H|mkIKd +1641: HteIZoJ +1642: HspZHcF +1643: HspZ@cF +1644: H|bMHG| +1645: HteMZsJ +1646: H|mkNCd +1647: HteNZCi +1648: H|mIPSM +1649: HsfJhcU +1650: H|`@H{L +1651: Htf]JGT +1652: H{eNX_J +1653: HtfYHCT +1654: HtbJgsF +1655: H{eJX_J +1656: HtdYHCT +1657: H{eIXcJ +1658: H{eMXcJ +1659: Hte]j_R +1660: HtbjHG\ +1661: HtbjHGX +1662: HtdR?k[ +1663: HtdR_k[ +1664: HtfB_k[ +1665: HtdB_k[ +1666: HsdJHcU +1667: HsP`gsM +1668: HsvJPoJ +1669: H{bhgoF +1670: H{bh_oF +1671: Ht`J_sF +1672: Ht`J_oF +1673: H{`HgoF +1674: H{bHgoF +1675: H{bH_oF +1676: H{bh_wF +1677: H{bH_wF +1678: Ht`JgsF +1679: Ht`JgoF +1680: H{`H_wF +1681: H{`HgwF +1682: H{bHgwF +1683: H|eFGcN +1684: HteIJsJ +1685: HteIJcJ +1686: HteIJSJ +1687: HteIJCJ +1688: H|ekJCd +1689: H|eKJCd +1690: H|mkICd +1691: HteIJoJ +1692: HteMJsJ +1693: H|f{KND +1694: Htf{MLc +1695: Hve[ido +1696: Hte[ido +1697: Hte[Ido +1698: HteAJsL +1699: HteAJcL +1700: HspYHKs +1701: H|aAgsM +1702: H|aAGsM +1703: HsvIPSw +1704: HteAJ{L +1705: HteAJkL +1706: HteAJKL +1707: H{b@xcY +1708: HsbzHg[ +1709: HsbjHg[ +1710: HsbJHg[ +1711: Hs`JHcY +1712: Hs`JhcY +1713: Hs`JxcY +1714: H{a}Hg[ +1715: H{b]Hg[ +1716: H{a]Hg[ +1717: HteEJ{L +1718: H{eMBkF +1719: H{eIBkF +1720: H~eKm_R +1721: H~eKi_R +1722: Ht}K]Tg +1723: Ht}MYOh +1724: Ht~IPOJ +1725: HteMB{F +1726: HtbJyCb +1727: H|aM@{X +1728: H{eEHwp +1729: H{eEH{p +1730: H{eEI{p +1731: HteIBsF +1732: HteIBcF +1733: HteIBCF +1734: HteMBcF +1735: HteMBCF +1736: HtaNzCY +1737: Ht}IYOh +1738: HsfBxcM +1739: H{eAI{p +1740: HsdBhcM +1741: HsdBHcM +1742: HsP@H{X +1743: Ht`B@{L +1744: Ht`BzCM +1745: HtbBrCM +1746: HtbB@sL +1747: Ht}IROM +1748: Ht}MBOM +1749: HtmMBOM +1750: HtuMROM +1751: HtuIYOh +1752: HsfBpcM +1753: Ht`IyCb +1754: Ht`AzCM +1755: HtbArCM +1756: Ht`I@C^ +1757: HtuIROM +1758: HtuMBOM +1759: HteIYOh +1760: H{fKIKs +1761: HteIROM +1762: HtmMROM +1763: H|iJ@Kw +1764: HtdQ`c[ +1765: HtfA`c[ +1766: HtmIROM +1767: Hsf}JGs +1768: HtbBwsM +1769: H|ef@Kk +1770: HtbBWsM +1771: H|mF@Kk +1772: HtbBgsM +1773: H|elN?b +1774: H{e{MCs +1775: H{e{MKs +1776: H{f[MKs +1777: H|mkJ?b +1778: H|mKJ?b +1779: H|eKJ?b +1780: H|mKN?b +1781: H|eKN?b +1782: Hve[mdo +1783: Hve]i_p +1784: HvfY`_R +1785: HvtP_cT +1786: HvSogcX +1787: HteKID~ +1788: HsfJyCb +1789: HstyHGF +1790: H|aEJwX +1791: HsfJiCb +1792: HstqHGF +1793: H|b?wcJ +1794: H|aEJ[X +1795: H|aEI[X +1796: H|aEGoN +1797: H|aEGoL +1798: HteFzCM +1799: HvfQ`c[ +1800: HvdY`_R +1801: HvdQ`c[ +1802: HseIJ{p +1803: HspyHK[ +1804: H{bBpcM +1805: HspqHK[ +1806: HsT@@{F +1807: H|aAG{X +1808: H|aAGoN +1809: H|aAH[X +1810: H|aAI{X +1811: H{}M@oM +1812: HvfA`c[ +1813: HtdY`_R +1814: H|akIK[ +1815: H|a[IK[ +1816: H|aKIK[ +1817: H|b[IK[ +1818: H|eD}CM +1819: HveMi_p +1820: HvfI`_R +1821: H|a]GgL +1822: H|bkMK[ +1823: H|bKMK[ +1824: H|aKMK[ +1825: H|aKMC[ +1826: HteKMD~ +1827: H|eKMD~ +1828: HteMJ|` +1829: H|eIB{F +1830: HteIJ|` +1831: H|eAH{L +1832: H{bJ`cY +1833: H{bH`cY +1834: H{bJXcY +1835: H|`BgsM +1836: H|bB_sM +1837: HstIHKs +1838: HstAHKs +1839: H|bAwsM +1840: H|bAosM +1841: H|b?wsM +1842: HtfBzCM +1843: H|aIJsX +1844: H{dBhcM +1845: H{dBHcM +1846: H{fBHcM +1847: H{fAhcM +1848: HtdB@kL +1849: HtdB@KL +1850: H{dAhcM +1851: HtbJJCY +1852: H{f@`cM +1853: H{fAxcM +1854: HtdBjCM +1855: HtdBJCM +1856: HtdAJCM +1857: H{f@_{L +1858: H{f@_sL +1859: H{fApcM +1860: HtdAjCM +1861: HtfBZCM +1862: HtfB@[L +1863: H{d@hcM +1864: H{d@_kL +1865: H{eMJsp +1866: HtfJiCb +1867: H|eM@{F +1868: HtfJYCb +1869: H{fJ`CR +1870: H|aMJkX +1871: HtfIyCb +1872: HttYHGF +1873: H{fJHCR +1874: HttIHGF +1875: H~eKMD^ +1876: HteMI|p +1877: HteIJC~ +1878: H|`BgsN +1879: HtfIyCr +1880: HttYHgF +1881: HtfMJx` +1882: HteMZt` +1883: HtfAzcM +1884: HsdJH{p +1885: HsdJH_^ +1886: HsdJHwp +1887: HsdJHsp +1888: HsdJHkp +1889: HsdJH[p +1890: HtdIyCr +1891: H{eIJSx +1892: HtdAjcM +1893: HtdAJcM +1894: HtdIiCr +1895: HtdIICr +1896: H{eIJCx +1897: HsfJH{p +1898: H|mIB[F +1899: H|e[MDu +1900: HteMItp +1901: HtfAYt` +1902: HtdIHC^ +1903: HtpQHk[ +1904: H{eJHKp +1905: HtfIqCr +1906: HteMJCn +1907: HsfIjCr +1908: HstQHKt +1909: HtfMIx` +1910: HtfIHC^ +1911: H|e]@gF +1912: HtfIHCN +1913: Hte]J?v +1914: HteIItp +1915: HtbJH[X +1916: HsvIrCq +1917: HstHH_^ +1918: H~eIAsF +1919: H|d`gSF +1920: HtmMJX` +1921: H~uKMDN +1922: HteMI\x +1923: HteJJC} +1924: H|`Bg{L +1925: HtfIYCz +1926: HteNJC} +1927: H|eIRsM +1928: HtfMJG| +1929: HtbIZcZ +1930: HtuMZp` +1931: HtvIPCz +1932: Hte]jd` +1933: HteII\x +1934: HtbjHwX +1935: HsdIzcp +1936: HtdRjC[ +1937: HtfBjC[ +1938: H{eIZ_p +1939: HsvJPsp +1940: Ht`IZcZ +1941: H|e[MDr +1942: HteMI\p +1943: HtfIYCr +1944: HttIHgF +1945: HteMJC} +1946: HsfIzCp +1947: H|`BgsL +1948: H|`BgoL +1949: HstYHKp +1950: H|eIBsM +1951: H|dBgoF +1952: H|eEJK[ +1953: H|eIBcM +1954: H|dBGoF +1955: HtfMJ?| +1956: H|eYBcF +1957: H|e]BCF +1958: H|eYBCF +1959: HteMZp` +1960: HtfAZcM +1961: HtdIYCr +1962: H|mIBKF +1963: H|e[IDb +1964: Hte]Jd` +1965: HteII\p +1966: HtmMJL` +1967: H|e[MDb +1968: H|eKmDr +1969: HteMI\h +1970: HtfB@sN +1971: H{dBHcN +1972: HtfIYCj +1973: HttIHWF +1974: HtuMJp` +1975: HteMjd` +1976: HteII\h +1977: Hsp`hc] +1978: H|eNJC} +1979: HtfIJK| +1980: HstZHCv +1981: H|bJX_Z +1982: HtfZjC[ +1983: HstJXsp +1984: HsTZHCv +1985: H|`IXsp +1986: H|`JX_Z +1987: H|`IX_Z +1988: H|eIJ[w +1989: H|eMJ[w +1990: H~eMMD] +1991: HteMJc^ +1992: HteIZCz +1993: H|`BHsN +1994: HteMZCz +1995: H|mIBSN +1996: HtfAjc] +1997: HsdJHc^ +1998: Htf]Jh` +1999: H~eKMDV +2000: HteMJC^ +2001: HteIZ?z +2002: H|`BHsM +2003: HtfBbC] +2004: HsfIZCr +2005: HstIHKt +2006: HteMZ?z +2007: H|mIBSF +2008: Hte]Jh` +2009: HtmMJT` +2010: HteMIlp +2011: HtfIiCr +2012: H|aMJ_\ +2013: HtfIICr +2014: H|mMMDm +2015: HtmMJCn +2016: H|eJ@sN +2017: Hte]JCv +2018: HtbJH[x +2019: HsvIrCr +2020: H~eIAsN +2021: H|d`gsF +2022: HteIYtp +2023: HtnMJX` +2024: H|eMZsX +2025: HtfIyDr +2026: HtfIyDp +2027: HtfJH{L +2028: H|dBhc] +2029: HstYJKt +2030: HstYJCt +2031: H|fB`c] +2032: HstIJKt +2033: HtbJJCy +2034: H|bIJCy +2035: H|bIZCy +2036: H|dBHc] +2037: H|fAhc] +2038: HstBHc^ +2039: HstBHc\ +2040: HtbJZCy +2041: H{dBhc] +2042: HtfBHsL +2043: H{dBHc] +2044: H{fBHc[ +2045: H{dBHc[ 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HtpjHg[ +2512: HtpJ@[X +2513: HtpJ?oN +2514: HtpJ@kX +2515: HttJjCY +2516: HttJJCY +2517: H|dJzCY +2518: H|dIzCY +2519: H|dIZCY +2520: H|dJ@{X +2521: H|dJ?{X +2522: H|dJ@[X +2523: H{tJHg[ +2524: H|tjHg[ +2525: H|tbHg[ +2526: H|t`XcY +2527: H|tJHg[ +2528: H|vJrCY +2529: HtthH{X +2530: HuTJXsX +2531: HuTJXcX +2532: HuTJXCX +2533: HuTJHCX +2534: HuTHHCX +2535: HuTHXcX +2536: HuTHXsX +2537: HuTHXSX +2538: H|pJHwL +2539: H|p`XsJ +2540: H|p`oSN +2541: H|pbHwL +2542: H|pBHwL +2543: H|pbGwL +2544: HtpbHwL +2545: H|eMzd` +2546: HtnII\h +2547: Htphhc] +2548: HvfIz_M +2549: HtdJXsp +2550: HtfJXsp +2551: H|vIRsp +2552: HtthxcM +2553: Htth_{L +2554: H|pIHCz +2555: HtpIHCz +2556: HuTZHkF +2557: HuTJHkF +2558: H|pJXCx +2559: H|pIXCx +2560: HtpIXCx +2561: H|pIXCz +2562: H|fIJ[h +2563: H|fIJSh +2564: H|fIJKh +2565: H|nIJKh +2566: H|nIIKh +2567: H|tZHkF +2568: H|t`g{F +2569: H|tZHcF +2570: H|tJHcF +2571: H|tZJCu +2572: H|tbHg\ +2573: HttbHg\ +2574: HttBHg\ +2575: H|tIJCz +2576: H|t`HcZ +2577: HttZ@Kt +2578: HttZJCu +2579: H{tZJCu +2580: H|pHhc] +2581: H|ph_k\ +2582: H|pH_k\ +2583: HstZJCu +2584: H|phHc] +2585: HuThHoN +2586: H|pHHc] +2587: H|ph_wF +2588: H|pH_wF +2589: H{phHc] +2590: HuTHHoN +2591: H|t`hc] +2592: Htt`hc] +2593: HtdJJCu +2594: H{dHg{X +2595: H{tHgkX +2596: H{pHgkX +2597: H|tHgoN +2598: HttJ@Kt +2599: H|tHgoF +2600: H|pHgoF +2601: H|pHg{X +2602: H|tJgkX +2603: H{thHc] +2604: HvTHHoN +2605: H|tJHk\ +2606: HttJHk\ diff --git a/polyhedra/nine_edge.txt b/polyhedra/nine_edge.txt new file mode 100644 index 0000000..ca4b3d0 Binary files /dev/null and b/polyhedra/nine_edge.txt differ diff --git a/polyhedra/polyhedra.nb b/polyhedra/polyhedra.nb new file mode 100644 index 0000000..5aa0689 --- /dev/null +++ b/polyhedra/polyhedra.nb @@ -0,0 +1,1157 @@ +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 0, 0] +NotebookDataLength[ 49948, 1156] +NotebookOptionsPosition[ 46613, 1092] +NotebookOutlinePosition[ 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\\!\\(\\*RowBox[{\\\"\\\\\\\"IGraphM\\\\\\\"\\\"}]\\) \ +with the same version number \ +(\\!\\(\\*RowBox[{\\\"\\\\\\\"0.5.1\\\\\\\"\\\"}]\\)) is already installed. \ +Use PacletUninstall to remove the existing version first, or call \ +PacletInstall with ForceVersionInstall -> True.\"", 2, 1, 1, + 26206974750047846679, "Local"}, + "MessageTemplate"]], "Message", "MSG", + CellChangeTimes->{3.832337222565331*^9, 3.832358887306924*^9}, + CellLabel-> + "During evaluation of \ +In[1]:=",ExpressionUUID->"1bd7e6c6-08bf-40e0-ada5-1b9c296f7ca2"], + +Cell[BoxData[ + InterpretationBox[ + RowBox[{"\<\"Installation failed. Please install IGraph/M manually. \"\>", + "\[InvisibleSpace]", + TemplateBox[{ + "\"https://github.com/szhorvat/IGraphM#installation\"", + "https://github.com/szhorvat/IGraphM#installation"}, + "HyperlinkURL"]}], + SequenceForm["Installation failed. Please install IGraph/M manually. ", + Hyperlink["https://github.com/szhorvat/IGraphM#installation"]], + Editable->False]], "Print", + CellChangeTimes->{3.832337217807197*^9, 3.832358887315374*^9}, + CellLabel-> + "During evaluation of \ +In[1]:=",ExpressionUUID->"ecde106e-abb1-4475-bcc8-474beba04d23"] +}, Open ]], + +Cell[CellGroupData[{ + +Cell[BoxData[ + RowBox[{"<<", "IGraphM`"}]], "Input", + CellChangeTimes->{{3.8321516420783434`*^9, 3.832151645302351*^9}}, + CellLabel->"In[2]:=",ExpressionUUID->"8cc401a5-7736-45db-b864-33aa0faea5b6"], + +Cell[BoxData[ + TagBox[GridBox[{ + {"\<\"IGraph/M 0.5.1 (October 12, 2020)\"\>"}, + {"\<\"Evaluate \\!\\(\\*ButtonBox[\\\"IGDocumentation[]\\\",BaseStyle->\\\ +\"Link\\\",ButtonData->\\\"paclet:IGraphM\\\"]\\) to get started.\"\>"} + }, + DefaultBaseStyle->"Column", + GridBoxAlignment->{"Columns" -> {{Left}}}, + GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], + "Column"]], "Output", + CellChangeTimes->{3.832337477452114*^9, 3.832358888188767*^9}, + CellLabel->"Out[2]=",ExpressionUUID->"2bca6177-8c17-477b-b102-0112c41c8522"] +}, Open ]], + +Cell["\<\ +Function to convert some representation of a graph to mathematica\ +\[CloseCurlyQuote]s representation. You can give it the option \ +`format->graph6` if in graph6 format\ +\>", "Text", + CellChangeTimes->{{3.832358495397357*^9, + 3.8323585393912582`*^9}},ExpressionUUID->"b79ac481-66e7-4669-b8f4-\ +0cea700ed78c"], + +Cell[BoxData[{ + RowBox[{ + RowBox[{"Clear", "[", + RowBox[{"format", ",", "edgeList", ",", "graph6"}], "]"}], + ";"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{ + RowBox[{"Options", "[", "graphFromPlantri", "]"}], "=", + RowBox[{"{", + RowBox[{"format", "\[Rule]", "edgeList"}], "}"}]}], + ";"}], "\[IndentingNewLine]", + RowBox[{ + RowBox[{"graphFromPlantri", "[", + RowBox[{"data_", ",", + RowBox[{"OptionsPattern", "[", "]"}]}], "]"}], ":=", + RowBox[{"If", "[", + RowBox[{ + RowBox[{"format", "===", "edgeList"}], ",", + RowBox[{"PlanarGraph", "[", + RowBox[{ + RowBox[{"Graph", "[", + RowBox[{ + RowBox[{"Partition", "[", + RowBox[{ + RowBox[{"ToExpression", "[", + RowBox[{"StringCases", "[", + RowBox[{"data", ",", + RowBox[{"DigitCharacter", ".."}]}], "]"}], "]"}], ",", "2"}], + "]"}], "/.", + RowBox[{ + RowBox[{"{", + RowBox[{"x_", ",", "y_"}], "}"}], "\[Rule]", + RowBox[{"(", + RowBox[{"x", "\[UndirectedEdge]", "y"}], ")"}]}]}], "]"}], ",", + RowBox[{"VertexLabels", "\[Rule]", "\"\\""}]}], "]"}], ",", + "\[IndentingNewLine]", + RowBox[{"PlanarGraph", "[", + RowBox[{ + RowBox[{"ImportString", "[", + RowBox[{"data", ",", "\"\\""}], "]"}], ",", + RowBox[{"VertexLabels", "\[Rule]", "\"\\""}]}], "]"}]}], + "]"}]}]}], "Input", + CellChangeTimes->{{3.832336494011004*^9, 3.832336534340336*^9}, { + 3.8323517999289827`*^9, 3.832351947441263*^9}, {3.8323520237090063`*^9, + 3.8323520406687717`*^9}, {3.8323532427822723`*^9, 3.832353249640938*^9}}, + CellLabel->"In[3]:=",ExpressionUUID->"0491d7dc-7152-49be-9137-070980e44ae7"], + +Cell["\<\ +Function to tell if a polyhedron is prime. About O(2^n) and not likely able \ +to improve. There are about O(2^n) cycles (it\[CloseCurlyQuote]s been \ +rigorously proven to be P-hard to count the number of cycles), and I don\ +\[CloseCurlyQuote]t see a way of getting around that.\ +\>", "Text", + CellChangeTimes->{{3.83235854587254*^9, + 3.832358618254396*^9}},ExpressionUUID->"83ebdaf9-e7b9-4912-83aa-\ +66018e3b4210"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"primePolyhedronQ", "[", "graph_", "]"}], ":=", + RowBox[{"Block", "[", + RowBox[{ + RowBox[{"{", + RowBox[{ + "cycles", ",", "valid", ",", "faceDecomposable", ",", + "vertexDecomposable", ",", "dualGraph"}], "}"}], ",", + "\[IndentingNewLine]", + RowBox[{ + RowBox[{"cycles", "=", + RowBox[{ + RowBox[{"FindCycle", "[", + RowBox[{"graph", ",", "Infinity", ",", "All"}], "]"}], "/.", + RowBox[{ + RowBox[{"(", + RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", + "x"}]}]}], ";", "\[IndentingNewLine]", + RowBox[{"valid", "=", + RowBox[{"Select", "[", + RowBox[{"cycles", ",", + RowBox[{ + RowBox[{"Not", "[", + RowBox[{"IGConnectedQ", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"graph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], + "]"}]}], ";", "\[IndentingNewLine]", + RowBox[{"faceDecomposable", "=", + RowBox[{"AnyTrue", "[", + RowBox[{"valid", ",", + RowBox[{ + RowBox[{"With", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"components", "=", + RowBox[{"ConnectedComponents", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"graph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", + "\[IndentingNewLine]", + RowBox[{ + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"graph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "1", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", + "3"}], "]"}], "&&", + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"graph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "2", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", + "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}]}], ";", + "\[IndentingNewLine]", + RowBox[{"If", "[", + RowBox[{"faceDecomposable", ",", + RowBox[{"Return", "[", "False", "]"}]}], "]"}], ";", + "\[IndentingNewLine]", + RowBox[{"dualGraph", "=", + RowBox[{"IGDualGraph", "[", "graph", "]"}]}], ";", + "\[IndentingNewLine]", + RowBox[{"cycles", "=", + RowBox[{ + RowBox[{"FindCycle", "[", + RowBox[{"dualGraph", ",", "Infinity", ",", "All"}], "]"}], "/.", + RowBox[{ + RowBox[{"(", + RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", + "x"}]}]}], ";", "\[IndentingNewLine]", + RowBox[{"valid", "=", + RowBox[{"Select", "[", + RowBox[{"cycles", ",", + RowBox[{ + RowBox[{"Not", "[", + RowBox[{"IGConnectedQ", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], + "]"}]}], ";", "\[IndentingNewLine]", + RowBox[{"Not", "[", + RowBox[{"AnyTrue", "[", + RowBox[{"valid", ",", + RowBox[{ + RowBox[{"With", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"components", "=", + RowBox[{"ConnectedComponents", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", + "\[IndentingNewLine]", + RowBox[{ + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"dualGraph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "1", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", + "3"}], "]"}], "&&", + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"dualGraph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "2", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", + "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}], "]"}]}]}], + "\[IndentingNewLine]", "]"}]}]], "Input", + CellChangeTimes->{{3.8323514934530973`*^9, 3.832351646852213*^9}, { + 3.8323516804671373`*^9, 3.83235168336065*^9}}, + CellLabel->"In[6]:=",ExpressionUUID->"1445a032-9806-4ee7-8079-9849d6ac9896"], + +Cell["\<\ +Function to find the possible decompositions of a graph. Returns a list where \ +the first element of the list is the number of possible face decompositions \ +and the second is the number of possible vertex decompositions. Should be \ +easy to modify to actually return the decompositions. I wonder if we could \ +extend this recursively to \[OpenCurlyDoubleQuote]prime factorize\ +\[CloseCurlyDoubleQuote] polyhedra?\ +\>", "Text", + CellChangeTimes->{{3.832358623451818*^9, + 3.832358720650798*^9}},ExpressionUUID->"a601bc96-4656-4520-9aeb-\ +158895d9368a"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"decomposable", "[", "graph_", "]"}], ":=", + RowBox[{"Block", "[", + RowBox[{ + RowBox[{"{", + RowBox[{ + "cycles", ",", "valid", ",", "faceDecomposable", ",", + "vertexDecomposable", ",", "dualGraph"}], "}"}], ",", + "\[IndentingNewLine]", + RowBox[{ + RowBox[{"cycles", "=", + RowBox[{ + RowBox[{"FindCycle", "[", + RowBox[{"graph", ",", "Infinity", ",", "All"}], "]"}], "/.", + RowBox[{ + RowBox[{"(", + RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", + "x"}]}]}], ";", "\[IndentingNewLine]", + RowBox[{"valid", "=", + RowBox[{"Select", "[", + RowBox[{"cycles", ",", + RowBox[{ + RowBox[{"Not", "[", + RowBox[{"IGConnectedQ", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"graph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], + "]"}]}], ";", "\[IndentingNewLine]", + RowBox[{"faceDecomposable", "=", + RowBox[{"Length", "[", + RowBox[{"Select", "[", + RowBox[{"valid", ",", + RowBox[{ + RowBox[{"With", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"components", "=", + RowBox[{"ConnectedComponents", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"graph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", + "\[IndentingNewLine]", + RowBox[{ + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"graph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "1", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], + ",", "3"}], "]"}], "&&", + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"graph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "2", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], + ",", "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}], "]"}]}], ";", + "\[IndentingNewLine]", + RowBox[{"dualGraph", "=", + RowBox[{"IGDualGraph", "[", "graph", "]"}]}], ";", + "\[IndentingNewLine]", + RowBox[{"cycles", "=", + RowBox[{ + RowBox[{"FindCycle", "[", + RowBox[{"dualGraph", ",", "Infinity", ",", "All"}], "]"}], "/.", + RowBox[{ + RowBox[{"(", + RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", + "x"}]}]}], ";", "\[IndentingNewLine]", + RowBox[{"valid", "=", + RowBox[{"Select", "[", + RowBox[{"cycles", ",", + RowBox[{ + RowBox[{"Not", "[", + RowBox[{"IGConnectedQ", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], + "]"}]}], ";", "\[IndentingNewLine]", + RowBox[{"vertexDecomposable", "=", + RowBox[{"Length", "[", + RowBox[{"Select", "[", + RowBox[{"valid", ",", + RowBox[{ + RowBox[{"With", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"components", "=", + RowBox[{"ConnectedComponents", "[", + RowBox[{"VertexDelete", "[", + RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", + "\[IndentingNewLine]", + RowBox[{ + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"dualGraph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "1", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], + ",", "3"}], "]"}], "&&", + RowBox[{"KVertexConnectedGraphQ", "[", + RowBox[{ + RowBox[{"Subgraph", "[", + RowBox[{"dualGraph", ",", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{ + "components", "\[LeftDoubleBracket]", "2", + "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], + ",", "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}], "]"}]}], ";", + "\[IndentingNewLine]", + RowBox[{"{", + RowBox[{"faceDecomposable", ",", "vertexDecomposable"}], "}"}]}]}], + "\[IndentingNewLine]", "]"}]}]], "Input", + CellChangeTimes->{{3.832340233432548*^9, 3.832340427782093*^9}, { + 3.8323404633434258`*^9, 3.832340465470111*^9}, {3.832341064134386*^9, + 3.832341105249135*^9}, {3.832341215107959*^9, 3.832341309459516*^9}, { + 3.83234151306717*^9, 3.8323415312540283`*^9}}, + CellLabel->"In[7]:=",ExpressionUUID->"58712807-a428-4b89-8b30-be4ce33480f8"], + +Cell["\<\ +Function to read data from a file and convert it into a list of graphs. The \ +easiest way to use this is to download a massive list from plantri (in graph6 \ +format if it is very large, to keep the file small), and give that filename \ +here. I highly recommend you put a semicolon at the end of that line to \ +suppress the output; it tends to be pretty massive.\ +\>", "Text", + CellChangeTimes->{{3.832358728187701*^9, 3.832358787855624*^9}, { + 3.832358819591075*^9, + 3.8323588520926237`*^9}},ExpressionUUID->"ab257888-085b-4fde-8617-\ +8abd299bbd66"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"readData", "[", "filename_", "]"}], ":=", + RowBox[{ + RowBox[{ + RowBox[{"graphFromPlantri", "[", + RowBox[{"#", ",", + RowBox[{"format", "\[Rule]", "graph6"}]}], "]"}], "&"}], "/@", + RowBox[{ + RowBox[{"Flatten", "[", + RowBox[{"StringSplit", "[", + RowBox[{ + RowBox[{"ReadList", "[", + RowBox[{ + RowBox[{"FileNameJoin", "[", + RowBox[{"{", + RowBox[{ + RowBox[{"NotebookDirectory", "[", "]"}], ",", "filename"}], "}"}], + "]"}], ",", "String"}], "]"}], ",", "\"\<: \>\""}], "]"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"2", ";;", + RowBox[{"-", "1"}], ";;", "2"}], "\[RightDoubleBracket]"}]}]}]], "Input",\ + + 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