#!/usr/bin/env python # coding: utf-8 # [Oregon Curriculum Network](http://www.4dsolutions.net/ocn)
# [Discovering Math with Python](Introduction.ipynb) # # # # FOCUSING ON THE S FACTOR # #### $\phi$ DOODLES Using $\LaTeX$ # # Eyeballing Code # # First, some identity checks (not proofs), using Decimal objects: # # $\sqrt{2}-(\sqrt{2}(\phi^{-3}))= 2\sqrt{2}(\phi^{-2})$ # # $(\phi^{-2})+(\phi^{-3})+(\phi^{2}) = 1$ # In[1]: from math import sqrt as rt2 from decimal import Decimal, getcontext context = getcontext() context.prec = 50 # In[2]: one = Decimal(1) # 28 digits of precision by default, more on tap two = Decimal(2) three = Decimal(3) five = Decimal(5) nine = Decimal(9) eight = Decimal(8) sqrt2 = two.sqrt() sqrt5 = five.sqrt() Ø = (one + sqrt5)/two S3 = (nine/eight).sqrt() # Got Synergetics? # In showing off the Decimal type, I'm advertising high precision, but not "infinite precision". Please be tolerant of our epsilons (tiny abberations). # In[3]: (Ø**-2) + (Ø**-3) + ( Ø**-2) # In[4]: sqrt2 - sqrt2 * Ø**-3 # ## JITTERBUG TRANSFORMATION # # 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). # # Jitterbug Transformation # In[5]: two * sqrt2 * (Ø**-2) # In[6]: icosa = five * sqrt2 * Ø ** 2 icosa # In[7]: ve = Decimal(20) # In[8]: s_factor = ve / icosa # In[9]: s_factor # see? # ## "SMALLGUY" # # Above is an expression for the volume of said Icosa in tetravolumes. # # 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. # # David Koski and I got to calling this cubocta "SmallGuy" (feel free to substitute your own moniker). # # The Concentric Hierarchy has a *Sesame Street* flavor (kids' TV show) in some walkx of life, lending to our penchant for colloquialisms. # In[10]: SmallGuy = icosa * one/s_factor * one/s_factor SmallGuy # 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 $1/s\_factor ^{3}$ # In[11]: ve * (one/s_factor)**3 # In[12]: SmallGuy_edge = two * (one/s_factor) # effect on edges SmallGuy_edge # ## RHOMBIC TRIACONTAHEDRON (RT) # # Rhombic Triacontahedron # In[13]: superRT = ve * S3 superRT # 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. # # Regular and Right Tetrahedrons Compared # # 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. # # 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. # # Another expression for SuperRT volume is $15\sqrt{2}$. # In[14]: Decimal('15') * sqrt2 # ## E MODULE # # module_studies # In[15]: emod = (superRT * Ø**-3)/Decimal(120) # In[16]: emod # In[17]: smod = emod * s_factor smod # In[18]: smod/emod # The S factor again, yes? # # $\sqrt{2}-(\sqrt{2}(\phi^{-3}))= 2\sqrt{2}(\phi^{-2})$ = S Factor. # # Another expression for the S Factor is $24E + 8e3$ where E means emod, and $e3$ means $E * \phi^{-3}$. # In[19]: Decimal(24) * emod + Decimal(8) * emod * Ø**-3 # ## S MODULE # # Now lets shrink the 20 volumed VE by halving all edges, reducing volume by a factor of 8, to 2.5 # # In[20]: small_ve = ve / Decimal(8) # S Module # # 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. # In[21]: skew_icosa = small_ve * s_factor * s_factor # In[22]: skew_icosa # In[23]: skew_icosa + (24 * smod) # The figure below is an S-Factor radius, meaning from the center to each diamond face center on the surface. # # S-Factor Radius #
by D.B. Koski using vZome
# David Koski writes (on Facebook): # # The volume 4, edge 2 octahedron, has a volume of 4 tetrahedral units or 84S + 20s3 modules # # S = $(\phi^{-5})/2$ = .045084 # # s3 = $(\phi^{-8})/2$ = .010643 # # The icosahedron inside of this octahedron has a volume of 84S+20s3 - 24S = 60S+20s3 = 2.917960 = $20(\phi^{-4})$. # Surprisingly, this icosahedron has an edge of 1.08036 or the Sfactor! # In[24]: Decimal(60) * smod + Decimal(20) * smod * Ø**-3 # In[25]: Decimal(20) * Ø**-4 # ## A MODULE # # 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. # # Plane Nets # #
3D Print Me!