Mod_Fun


Oregon Curriculum Network
Discovering Math with Python

FOCUSING ON THE S FACTOR

$\phi$ DOODLES Using $\LaTeX$

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)


Out[3]:
Decimal('0.99999999999999999999999999999999999999999999999995')

In [4]:
sqrt2 - sqrt2 * Ø**-3


Out[4]:
Decimal('1.0803630269509058144061726281963757019894604868056')

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).


In [5]:
two * sqrt2 * (Ø**-2)


Out[5]:
Decimal('1.0803630269509058144061726281963757019894604868055')

In [6]:
icosa = five * sqrt2 * Ø ** 2
icosa


Out[6]:
Decimal('18.512295868219161196009899292654531923571426913640')

In [7]:
ve = Decimal(20)

In [8]:
s_factor = ve / icosa

In [9]:
s_factor  # see?


Out[9]:
Decimal('1.0803630269509058144061726281963757019894604868056')

"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


Out[10]:
Decimal('15.860645438769607979506732934761348026253292147309')

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


Out[11]:
Decimal('15.860645438769607979506732934761348026253292147310')

In [12]:
SmallGuy_edge = two * (one/s_factor) # effect on edges
SmallGuy_edge


Out[12]:
Decimal('1.8512295868219161196009899292654531923571426913641')

RHOMBIC TRIACONTAHEDRON (RT)


In [13]:
superRT = ve * S3
superRT


Out[13]:
Decimal('21.213203435596425732025330863145471178545078130654')

S3 is our conversion constant for going between XYZ cube volumes and IVM 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.

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 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


Out[14]:
Decimal('21.213203435596425732025330863145471178545078130654')

E MODULE


In [15]:
emod = (superRT * Ø**-3)/Decimal(120)

In [16]:
emod


Out[16]:
Decimal('0.041731316927773654299439512001665297072526423571412')

In [17]:
smod = emod * s_factor
smod


Out[17]:
Decimal('0.045084971874737120511467085914095294300772949514403')

In [18]:
smod/emod


Out[18]:
Decimal('1.0803630269509058144061726281963757019894604868056')

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


Out[19]:
Decimal('1.0803630269509058144061726281963757019894604868056')

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)

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


Out[22]:
Decimal('2.9179606750063091077247899380617129367814492116541')

In [23]:
skew_icosa + (24 * smod)


Out[23]:
Decimal('3.9999999999999999999999999999999999999999999999998')

The figure below is an S-Factor radius, meaning from the center to each diamond face center on the surface.

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


Out[24]:
Decimal('2.9179606750063091077247899380617129367814492116540')

In [25]:
Decimal(20) * Ø**-4


Out[25]:
Decimal('2.9179606750063091077247899380617129367814492116540')

A MODULE

The A and B modules have the same volume (1/24), as does the T modules. We review these in other Notebooks.

3D Print Me!