{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "[Oregon Curriculum Network](http://www.4dsolutions.net/ocn)
\n", "[Discovering Math with Python](Introduction.ipynb)\n", "\n", "\n", "# FOCUSING ON THE S FACTOR\n", "#### $\\phi$ DOODLES Using $\\LaTeX$\n", "\n", "\"Eyeballing\n", "\n", "First, some identity checks (not proofs), using Decimal objects:\n", "\n", "$\\sqrt{2}-(\\sqrt{2}(\\phi^{-3}))= 2\\sqrt{2}(\\phi^{-2})$\n", "\n", "$(\\phi^{-2})+(\\phi^{-3})+(\\phi^{2}) = 1$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from math import sqrt as rt2\n", "from decimal import Decimal, getcontext\n", "context = getcontext()\n", "context.prec = 50" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "one = Decimal(1) # 28 digits of precision by default, more on tap\n", "two = Decimal(2)\n", "three = Decimal(3)\n", "five = Decimal(5)\n", "nine = Decimal(9)\n", "eight = Decimal(8)\n", "sqrt2 = two.sqrt()\n", "sqrt5 = five.sqrt()\n", "Ø = (one + sqrt5)/two\n", "S3 = (nine/eight).sqrt() # Got Synergetics?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In showing off the Decimal type, I'm advertising high precision, but not \"infinite precision\". Please be tolerant of our epsilons (tiny abberations)." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('0.99999999999999999999999999999999999999999999999995')" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(Ø**-2) + (Ø**-3) + ( Ø**-2)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('1.0803630269509058144061726281963757019894604868056')" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sqrt2 - sqrt2 * Ø**-3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## JITTERBUG TRANSFORMATION\n", "\n", "We call this the S Factor by the way. VE:Icosa :: S:E is what to remember. VE is the 12-around-1 nuclear sphere based agglomeration whereas Icosa is dervied from Jitterbugging, a mathematical transformation with a more technical name if you're a math snob (I can be).\n", "\n", "\"Jitterbug" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('1.0803630269509058144061726281963757019894604868055')" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "two * sqrt2 * (Ø**-2)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('18.512295868219161196009899292654531923571426913640')" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "icosa = five * sqrt2 * Ø ** 2\n", "icosa" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true }, "outputs": [], "source": [ "ve = Decimal(20)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "s_factor = ve / icosa" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('1.0803630269509058144061726281963757019894604868056')" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s_factor # see?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## \"SMALLGUY\"\n", "\n", "Above is an expression for the volume of said Icosa in tetravolumes. \n", "\n", "We may think of it as \"two applications of the S-Factor bigger\" than a smaller cubocta, with edges, get this, equal in magnitude to the *volume* of the edge 2 icosa. \n", "\n", "David Koski and I got to calling this cubocta \"SmallGuy\" (feel free to substitute your own moniker). \n", "\n", "The Concentric Hierarchy has a *Sesame Street* flavor (kids' TV show) in some walkx of life, lending to our penchant for colloquialisms." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('15.860645438769607979506732934761348026253292147309')" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SmallGuy = icosa * one/s_factor * one/s_factor\n", "SmallGuy" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Another way to reach the SmallGuy is to start with the volume 20 cubocta and shrink its edges by the S Factor, which means volume shrinks by a factor of the reciprocal of said S Factor to the 3rd power or $1/s\\_factor ^{3}$" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('15.860645438769607979506732934761348026253292147310')" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ve * (one/s_factor)**3" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('1.8512295868219161196009899292654531923571426913641')" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SmallGuy_edge = two * (one/s_factor) # effect on edges\n", "SmallGuy_edge" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## RHOMBIC TRIACONTAHEDRON (RT)\n", "\n", "\"Rhombic" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('21.213203435596425732025330863145471178545078130654')" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "superRT = ve * S3\n", "superRT" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "S3 is our conversion constant for going between XYZ cube volumes and [IVM](https://github.com/4dsolutions/Python5/blob/master/Generating%20the%20FCC.ipynb) tetra volumes. The two mensuration systems each have their own unit volume, by convention a .5 radius edge cube versus a 1.0 diametered edged tetrahedron, or use edges 1 and 2 if preferred, their ratio will be the same, with the cube a bit bigger.\n", "\n", "\"Regular\n", "\n", "SuperRT is the RT (rhombic triacontahedron) formed by the Icosa and its dual, the Pentagonal Dodecahedron, the two five-fold symmetric shapes in [the Platonic set](https://youtu.be/vk-cpknOz9E) of five polys. The Icosa we're talking about is the one above, derived from the VE of volume 20, through Jitterbugging.\n", "\n", "If we shrink SuperRT down by $\\phi^{-3}$ volume-wise (all edges are now $\\phi^{-1}$ their initial length), and carve it into 120 modules (60 left, 60 right), then lo and behold, we have the E modules.\n", "\n", "Another expression for SuperRT volume is $15\\sqrt{2}$." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('21.213203435596425732025330863145471178545078130654')" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Decimal('15') * sqrt2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## E MODULE\n", "\n", "\"module_studies\"" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": true }, "outputs": [], "source": [ "emod = (superRT * Ø**-3)/Decimal(120)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('0.041731316927773654299439512001665297072526423571412')" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "emod" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('0.045084971874737120511467085914095294300772949514403')" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "smod = emod * s_factor\n", "smod" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('1.0803630269509058144061726281963757019894604868056')" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "smod/emod" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The S factor again, yes? \n", "\n", "$\\sqrt{2}-(\\sqrt{2}(\\phi^{-3}))= 2\\sqrt{2}(\\phi^{-2})$ = S Factor.\n", "\n", "Another expression for the S Factor is $24E + 8e3$ where E means emod, and $e3$ means $E * \\phi^{-3}$." ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('1.0803630269509058144061726281963757019894604868056')" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Decimal(24) * emod + Decimal(8) * emod * Ø**-3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## S MODULE\n", "\n", "Now lets shrink the 20 volumed VE by halving all edges, reducing volume by a factor of 8, to 2.5\n" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": true }, "outputs": [], "source": [ "small_ve = ve / Decimal(8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\"S\n", "\n", "As every grade schooler knows, if at all aware of their heritage, said VE inscribes inside the octahedron of volume 4, as does an Icosahedron with flush faces. We do a kind of jitterbugging that makes the VE larger instead of smaller. Two applications of the S Factor does the trick." ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": true }, "outputs": [], "source": [ "skew_icosa = small_ve * s_factor * s_factor" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('2.9179606750063091077247899380617129367814492116541')" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "skew_icosa" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('3.9999999999999999999999999999999999999999999999998')" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "skew_icosa + (24 * smod)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The figure below is an S-Factor radius, meaning from the center to each diamond face center on the surface.\n", "\n", "\"S-Factor\n", "
by D.B. Koski using vZome
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "David Koski writes (on Facebook): \n", "\n", "The volume 4, edge 2 octahedron, has a volume of 4 tetrahedral units or 84S + 20s3 modules\n", "\n", "S = $(\\phi^{-5})/2$ = .045084\n", "\n", "s3 = $(\\phi^{-8})/2$ = .010643\n", "\n", "The icosahedron inside of this octahedron has a volume of 84S+20s3 - 24S = 60S+20s3 = 2.917960 = $20(\\phi^{-4})$.\n", "Surprisingly, this icosahedron has an edge of 1.08036 or the Sfactor!" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('2.9179606750063091077247899380617129367814492116540')" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Decimal(60) * smod + Decimal(20) * smod * Ø**-3" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Decimal('2.9179606750063091077247899380617129367814492116540')" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Decimal(20) * Ø**-4" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A MODULE\n", "\n", "The A and B modules have the same volume (1/24), as does the [T modules](http://www.grunch.net/synergetics/tmod.html). We review these in other Notebooks.\n", "\n", "\"Plane\n", "\n", "
3D Print Me!
" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 2 }