{ "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", "\"uvw1\"\n", "\n", "V6 made of (3U+2V+1W)" ] }, { "cell_type": "code", "execution_count": 1, "id": "a3bff75c-e85d-420f-b4ad-96bc24f170f7", "metadata": {}, "outputs": [ { "data": { "image/jpeg": 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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": [ "\"Phi \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", "\"A\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", "\"S\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", "\"Rhombic\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", "\"reference_tet\"" ] }, { "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": [ "\"uvw_module_spiral2\"\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
" ] } ], "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": "3.9.12" } }, "nbformat": 4, "nbformat_minor": 5 }