{
"cells": [
{
"cell_type": "markdown",
"id": "b74cda4c-5160-41f9-a549-22da2d79c2a1",
"metadata": {},
"source": [
"[Oregon Curriculum Network](http://4dsolutions.net/ocn/)
\n",
"[School of Tomorrow](School_of_Tomorrow.ipynb)\n",
"\n",
"\n",
"# Defining the U, V & W Modules\n",
"\n",
"These three quants (quantim modules) are not in Synergetics as published, but do extend the same ideas. David Koski is behind this naming schema, with the U an allusion to Tell Anderson's U.\n",
"\n",
"\n",
"\n",
"V6 made of (3U+2V+1W)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "a3bff75c-e85d-420f-b4ad-96bc24f170f7",
"metadata": {},
"outputs": [
{
"data": {
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"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from IPython.display import YouTubeVideo\n",
"YouTubeVideo(\"EbDKiOCcfBY\") "
]
},
{
"cell_type": "markdown",
"id": "912d4610-679a-4da3-b71f-25ed55a48a27",
"metadata": {},
"source": [
" \n",
"\n",
"Synergetics as published focused on the A, B, T (1/24), E and S modules.\n",
"\n",
"To review, A and B modules together build the regular tetrahedron and octahedron, the complementary space-fillers of the IVM. 24 As (12 left, 12 right) build the regular tetrahedron.\n",
"\n",
"\n",
"\n",
"S module: 24 of them (12 left, 12 right) nestle in the voids twixt the octa of volume 4, and a faces-flush icosahedron of volume approximately 2.91.\n",
"\n",
"\n",
"\n",
"The E module: 120 of them (60 left, 60 right) build the thirty diamond-faced rhombic triacontahedron (RT). The RT made of Es \"shrink wraps\" an IVM ball. \n",
"\n",
"The RT made of 120 Ts have a volume of precisely 5. The E and T share the same angles (have the same shape) and differ in size only minutely.\n",
"\n",
"\n",
"\n",
"\n",
"#### Additional Nomenclature\n",
"\n",
"U3 means all edges phi-up (φ scaled) so volume goes up as a factor of φ to the 3rd.\n",
"\n",
"u3 means all edges phi-down (1/φ scaled) so volume decreases as a factor of (1/φ) to the 3rd.\n",
"\n",
"Ditto for other letters (E3, S3, v3, w3 and so on)."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "a8b3964e-ce5d-49f6-a124-8c4df8584be3",
"metadata": {},
"outputs": [],
"source": [
"from math import sqrt as rt2"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "d379123d-b10c-4003-9b44-fd6402e0bcff",
"metadata": {},
"outputs": [],
"source": [
"φ = (rt2(5)+1)/2 # golden ratio\n",
"Syn3 = rt2(9/8) # not to be confused with Smod\n",
"\n",
"S = (φ**-5) / 2 # home base Smod\n",
"\n",
"Cubocta = 20\n",
"SuperRT = Syn3 * Cubocta\n",
"\n",
"E3 = (rt2(2) / 8)\n",
"E = E3 * (φ**-3) # home base Emod\n",
"e3 = E * (φ**-3)\n",
"e6 = e3 * (φ**-3)\n",
"\n",
"S_factor = S / E # 2*sqrt(7-3*sqrt(5))"
]
},
{
"cell_type": "markdown",
"id": "31cb57f2-546f-49af-b785-aba99d2d9e8e",
"metadata": {},
"source": [
"For computing tetrahedron volumes from six edges, we have a choice of formulae. I'm using Gerald de Jong's (GdJ). David works with Piero della Francesca's."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "7671426f-ac04-4245-8625-ffcaaa10d3d8",
"metadata": {},
"outputs": [],
"source": [
"from tetravolume import Tetrahedron as T"
]
},
{
"cell_type": "markdown",
"id": "4acb05f4-ec7e-4999-9850-d02960d57578",
"metadata": {},
"source": [
"# W Module\n",
"\n",
"Expected volume: E3 + E or $(\\sqrt{2}/4)\\phi^{-1}$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "92516564-6eea-4e11-879c-9ed242684839",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.21850801222441055"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"W = E3 + E\n",
"W"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "c2ce2c4a-1766-470a-b81a-756226ce9074",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.21850801222441052"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(rt2(2)/4)*φ**-1"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "6d015a98-5e73-4379-9b87-2dc76bd4f12c",
"metadata": {},
"outputs": [],
"source": [
"# in R = 1 units\n",
"a = rt2(φ**-2 + 1)\n",
"b = rt2(φ**2 + 1 )\n",
"c = rt2(3) \n",
"d = rt2(3)\n",
"e = rt2(3*φ**-2)\n",
"f = rt2(φ**-2 + 1)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "f7f8d273-8e28-4de9-9576-77eaaecaa721",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1.1755705045849463,\n",
" 1.902113032590307,\n",
" 1.7320508075688772,\n",
" 1.7320508075688772,\n",
" 1.07046626931927,\n",
" 1.1755705045849463)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a, b, c, d, e, f"
]
},
{
"cell_type": "markdown",
"id": "23015a52-8f50-4d4a-b01d-2e3a3df461f7",
"metadata": {},
"source": [
"The above dimensions assume R = 1, where R is the radius of any IVM ball. The volume-from-edges formula used below assumes D = 1, so the lengths get halved. Also, the order in which the edges get entered is: three from any apex, then the edges around the base in the same order.\n",
"\n",
""
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "866cd047-0a69-4e6e-b2af-1d7516327281",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.218508012224411$"
],
"text/plain": [
"0.218508012224411"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"T(a/2, c/2, e/2, b/2, f/2, d/2).ivm_volume() # D = 1"
]
},
{
"cell_type": "markdown",
"id": "c2db67a8-1823-4ef9-b740-411cb04bb6ff",
"metadata": {},
"source": [
"# U3 Module\n",
"\n",
"Expected volume: 2E3 or $(\\sqrt{2}/4)$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "cb5d1109-43a6-49b2-abdc-1db50a4134ed",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.3535533905932738"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U3 = 2 * E3\n",
"U3"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "d83b9793-fd61-463a-9042-78a3e0387e2a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.3535533905932738"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rt2(2)/4"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "9c2ad896-1c80-43a0-95c7-c4b915fe6bd9",
"metadata": {},
"outputs": [],
"source": [
"# in R = 1 units\n",
"a = rt2(φ**-2 + 1)\n",
"b = rt2(φ**2 + 1)\n",
"c = rt2(3)\n",
"d = rt2(3) \n",
"e = rt2(φ**2 + 1)\n",
"f = rt2(φ**-2 + 1)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "dee7000d-98b0-4f90-af06-1dd3ad798c15",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1.1755705045849463,\n",
" 1.902113032590307,\n",
" 1.7320508075688772,\n",
" 1.7320508075688772,\n",
" 1.902113032590307,\n",
" 1.1755705045849463)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a, b, c, d, e, f"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "a94da294-c83d-4c2b-ac9d-0582c0dd06f7",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.353553390593274$"
],
"text/plain": [
"0.353553390593274"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"T(a/2, c/2, e/2, b/2, f/2, d/2).ivm_volume() # D = 1"
]
},
{
"cell_type": "markdown",
"id": "ee66662b-7321-44e4-8e67-9d8c9ab10034",
"metadata": {},
"source": [
"# V3 Module\n",
"\n",
"Expected volume: V3 = 3E3 + E or or $(\\sqrt{2}/4)\\phi$"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "9bbb3353-13c3-483f-82fe-a1dac279c837",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.5720614028176844"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"V3 = 3 * E3 + E\n",
"V3"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "c5cfa255-eb8d-4356-9c1f-87caaa846d7e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.5720614028176844"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(rt2(2)/4)*φ"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "9d44cb61-c5b7-4593-821b-51223695767a",
"metadata": {},
"outputs": [],
"source": [
"a = rt2(φ**-2 + 1)\n",
"b = rt2(φ**2 + 1)\n",
"c = rt2(3)\n",
"d = rt2(3)\n",
"e = rt2(φ**2 + 1)\n",
"f = rt2(3*φ**2)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "0e8f3bfc-9013-46b4-9c58-2218bf4cd8d3",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1.1755705045849463,\n",
" 1.902113032590307,\n",
" 1.7320508075688772,\n",
" 1.7320508075688772,\n",
" 1.902113032590307,\n",
" 2.8025170768881473)"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a, b, c, d, e, f"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "45641217-41b2-4e8a-81fa-bc20aa8f138e",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.572061402817684$"
],
"text/plain": [
"0.572061402817684"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"V3 = T(a/2, c/2, e/2, b/2, f/2, d/2).ivm_volume() # D = 1\n",
"V3"
]
},
{
"cell_type": "markdown",
"id": "cf6f99c0-8bf9-4ce7-aa55-444a474b7fca",
"metadata": {},
"source": [
"# Relationships Between U, V, W"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "95785d4b-76bd-412f-b13b-0b881c575484",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.61803398874989$"
],
"text/plain": [
"1.61803398874989"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"V3/U3"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "6f441219-bdcc-4861-9b31-f77eff699f34",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.618033988749895"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U3/W"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "e2d33124-525f-4942-a553-d660c3fe2f0f",
"metadata": {},
"outputs": [],
"source": [
"U = U3 * (φ**-3) \n",
"V = V3 * (φ**-3)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "76f9dd47-c528-4ae2-a8e1-3d468e327e9a",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.21850801222441$"
],
"text/plain": [
"0.218508012224410"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U + V"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "bd096b6b-d036-494b-b8f4-b1466d054879",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.21850801222441055"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"W"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "7412d3c8-f770-41eb-b740-eae00bcd3f87",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.618033988749895$"
],
"text/plain": [
"0.618033988749895"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U/V"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "9dbe5181-7a67-4bd3-a77d-4dafa291a003",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.618033988749895$"
],
"text/plain": [
"0.618033988749895"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"V/W"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "54f651f6-7a43-49d3-bc52-811a1ef4d3b4",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 0.353553390593274$"
],
"text/plain": [
"0.353553390593274"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"V + W"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "71ff2807-8bf8-4ea4-9d7f-1074b5c58939",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.3535533905932738"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"U3"
]
},
{
"cell_type": "markdown",
"id": "75868ded-86d3-485b-ae4a-58746395307d",
"metadata": {},
"source": [
"\n",
"\n",
"From David:\n",
"\n",
"More on the u v w sequence:\n",
"This part of the helix, starts with a V and the largest is a V9
\n",
"V, V3, V6, V9
\n",
"4 V tetrahedra 3U & 3W scaled of course
"
]
}
],
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