Oregon Curriculum Network

Discovering Math with Python

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 [3]:
```(Ø**-2) + (Ø**-3) + ( Ø**-2)

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Out[3]:
```

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In [4]:
```sqrt2 - sqrt2 * Ø**-3

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Out[4]:
```

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In [5]:
```two * sqrt2 * (Ø**-2)

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Out[5]:
```

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

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Out[6]:
```

```
In [7]:
```ve = Decimal(20)

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In [8]:
```s_factor = ve / icosa

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In [9]:
```s_factor # see?

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Out[9]:
```

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

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Out[10]:
```

```
In [11]:
```ve * (one/s_factor)**3

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Out[11]:
```

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

```
Out[12]:
```

```
In [13]:
```superRT = ve * S3
superRT

```
Out[13]:
```

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

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Out[14]:
```

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

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In [16]:
``````
emod
```

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Out[16]:
```

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In [17]:
```smod = emod * s_factor
smod

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Out[17]:
```

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In [18]:
```smod/emod

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Out[18]:
```

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

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Out[19]:
```

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In [20]:
```small_ve = ve / Decimal(8)

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In [21]:
```skew_icosa = small_ve * s_factor * s_factor

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In [22]:
``````
skew_icosa
```

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Out[22]:
```

```
In [23]:
```skew_icosa + (24 * smod)

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Out[23]:
```

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

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Out[24]:
```

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In [25]:
```Decimal(20) * Ø**-4

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Out[25]:
```

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!