{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Random IVM Tetrahedron Volumes\n",
"\n",
"\n",
"\n",
"
![\"Quadray](\"https://live.staticflickr.com/1213/4723083407_1e315f2877_o.gif\")
\n",
"
\n",
"The four quadrays (in blue)\n",
"
\n",
"
\n",
"\n",
"According to a proof by Dr. Robert Gray, at one time archived on a Synergetics listserv, any tetrahedron of IVM vertexes (assuming non-coplanar) will have a whole number volume. \n",
"\n",
"In this Notebook, we randomly generate IVM tetrahedrons and compute their volumes. Within the error of floating point, we demonstrate this result."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from tetravolume import Qvector"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"from itertools import permutations"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(0, 1, 1, 2),\n",
" (1, 2, 0, 1),\n",
" (0, 2, 1, 1),\n",
" (0, 1, 2, 1),\n",
" (2, 1, 0, 1),\n",
" (1, 1, 2, 0),\n",
" (1, 2, 1, 0),\n",
" (2, 1, 1, 0),\n",
" (1, 0, 2, 1),\n",
" (2, 0, 1, 1),\n",
" (1, 0, 1, 2),\n",
" (1, 1, 0, 2)]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"combos = set()\n",
"for combo in permutations((0,1,1,2)):\n",
" combos.add(combo)\n",
"combos = list(combos)\n",
"combos"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One frequency:\n",
"\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"ivm_vector(a=365, b=159, c=336, d=0)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import random\n",
"\n",
"def any_ball():\n",
" # random coefficients for arbitrary choice of vectors...\n",
" coeffs = [random.randint(0, 300) for _ in range(12)]\n",
" vectors = ([coeffs[i] * Qvector(combos[i]) for i in range(12)])\n",
" vector_sum = Qvector((0,0,0,0))\n",
" for v in vectors:\n",
" vector_sum = vector_sum + v\n",
" return vector_sum\n",
"\n",
"any_ball()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"A,B,C,D = any_ball(), any_ball(), any_ball(), any_ball()"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"lengths = [(\"AB\", (A-B).length()),\n",
" (\"AC\", (A-C).length()),\n",
" (\"AD\", (A-D).length()),\n",
" (\"BC\", (B-C).length()),\n",
" (\"CD\", (C-D).length()),\n",
" (\"DB\", (D-B).length())]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[('AB', 373.0321701944753),\n",
" ('AC', 333.39616074574104),\n",
" ('AD', 476.14808620848197),\n",
" ('BC', 534.9635501601955),\n",
" ('CD', 544.9155897935019),\n",
" ('DB', 280.3194606159194)]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lengths"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"from tetravolume import Tetrahedron"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"t = Tetrahedron(*[lengths[i][1] for i in range(6)])"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"44190451.00000004"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t.ivm_volume()"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"14170140.999999965"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def demo():\n",
" A,B,C,D = any_ball(), any_ball(), any_ball(), any_ball()\n",
" lengths = [(\"AB\", (A-B).length()),\n",
" (\"AC\", (A-C).length()),\n",
" (\"AD\", (A-D).length()),\n",
" (\"BC\", (B-C).length()),\n",
" (\"CD\", (C-D).length()),\n",
" (\"DB\", (D-B).length())]\n",
" t = Tetrahedron(*[lengths[i][1] for i in range(6)])\n",
" return t.ivm_volume()\n",
"\n",
"demo()"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"41663156.754389174"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t.xyz_volume()"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.display import YouTubeVideo\n",
"YouTubeVideo(\"v8vjw3gCBgs\") # https://youtu.be/v8vjw3gCBgs"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.3"
}
},
"nbformat": 4,
"nbformat_minor": 4
}