{
"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",
"\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",
""
]
},
{
"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",
""
]
},
{
"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",
"\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",
""
]
},
{
"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": [
"\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",
"\n",
"