Polyvolumes


Python for STEM Teachers
Oregon Curriculum Network

BUILDING A PANDAS DATAFRAME

Welcome everyone.

In this very simple demo, I start with a listing of Polyhedrons with volumes in increasing order.


In [1]:
import pandas as pd
import numpy as np

shapes = ['Tetrahedron', 'Cube', 'Octahedron', 'Rhombic Dodecahedron', 'Cuboctahedron']
table = pd.DataFrame([1,3,4,6,20], index = shapes)

In [2]:
table


Out[2]:
0
Tetrahedron 1
Cube 3
Octahedron 4
Rhombic Dodecahedron 6
Cuboctahedron 20

Unlike a numpy array, which is of uniform data type (example: all floating point), a pandas Dataframe is expected to have labels for both columns and rows. The data columns themselves by be of varying type. A Dataframe consists as an arrangement, left to right, of instances of the pandas Series type, i.e. the columns.


In [3]:
table.columns = ["Volume",] # relabel the one column

In [4]:
table


Out[4]:
Volume
Tetrahedron 1
Cube 3
Octahedron 4
Rhombic Dodecahedron 6
Cuboctahedron 20

$\phi = (1 + \sqrt{5})/2$


In [5]:
phi = (1 + np.sqrt(5))/2 # a constant aka "golden ratio"

Now comes our common need to add another row to our table.

However the Icosahedron's volume is smaller than the Cuboctahedron's.

How shall we maintain Volume order?


In [6]:
table.loc["Icosahedron","Volume"] = 5 * np.sqrt(2) * phi ** 2
shapes.append("Icosahedron")

In [7]:
table


Out[7]:
Volume
Tetrahedron 1.000000
Cube 3.000000
Octahedron 4.000000
Rhombic Dodecahedron 6.000000
Cuboctahedron 20.000000
Icosahedron 18.512296

Lets add one of a few interesting RTs (RT = rhombic triacontahedron). The original Icosahedron, just added, has a dual with crossing edges that we'll also be adding. Together, the Icosahedron and its intersecting Dual, the Pentagonal Dodecahedron, define an RT of tetravolume


In [8]:
table.loc["RT5","Volume"] = 5
shapes.append("RT5")

In [9]:
byvolume = table.sort_values(by="Volume")

In [10]:
byvolume


Out[10]:
Volume
Tetrahedron 1.000000
Cube 3.000000
Octahedron 4.000000
RT5 5.000000
Rhombic Dodecahedron 6.000000
Icosahedron 18.512296
Cuboctahedron 20.000000

The concentric hierarchy icosahedron, obtained by jitterbugging the Cuboctahedron of volume 20, has a volume of:

$$ 5 \sqrt{2}\phi^{2} $$

In [11]:
byvolume["Avols"] = byvolume["Volume"] * 24  # A = B = T = 1/24

In [12]:
byvolume


Out[12]:
Volume Avols
Tetrahedron 1.000000 24.000000
Cube 3.000000 72.000000
Octahedron 4.000000 96.000000
RT5 5.000000 120.000000
Rhombic Dodecahedron 6.000000 144.000000
Icosahedron 18.512296 444.295101
Cuboctahedron 20.000000 480.000000

In [13]:
penta_dodeca = 3 * np.sqrt(2) * (phi**2 + 1)
shapes.append("Pentagonal Dodeca")
penta_dodeca


Out[13]:
15.350018208050784

In [14]:
byvolume.loc["Pentagonal Dodeca","Volume"] = penta_dodeca
byvolume.loc["Pentagonal Dodeca","Avols"] = penta_dodeca * 24

In [15]:
SuperRT = 20 * np.sqrt(9/8) # S3

In [16]:
SuperRT


Out[16]:
21.213203435596423

In [17]:
byvolume.loc["SuperRT", "Volume"] = SuperRT
byvolume.loc["SuperRT", "Avols"] = SuperRT *24
shapes.append("SuperRT")

In [18]:
byvolume


Out[18]:
Volume Avols
Tetrahedron 1.000000 24.000000
Cube 3.000000 72.000000
Octahedron 4.000000 96.000000
RT5 5.000000 120.000000
Rhombic Dodecahedron 6.000000 144.000000
Icosahedron 18.512296 444.295101
Cuboctahedron 20.000000 480.000000
Pentagonal Dodeca 15.350018 368.400437
SuperRT 21.213203 509.116882

What other RTs might we want to add? The RT of exactly 5 tetravolumes is not the SuperRT scaled down by phi in its linear dimensions, (1/phi) to the 3rd power by volume. The latter has a volume of 5+ and a radius of 1 exactly, whereas the RT of 5 has a radius of 0.9994... that of the RT of 5+.


In [19]:
byvolume.loc["RTe", "Volume"] = byvolume.loc["SuperRT","Volume"] * phi**-3

In [20]:
byvolume.loc["RTe", "Avols"] = byvolume.loc["RTe", "Volume"] * 24

In [21]:
shapes.append("RTe") # RT of 120 E modules

In [22]:
byvolume.sort_values(by="Volume", inplace=True)

3D Print Me!

In [23]:
byvolume


Out[23]:
Volume Avols
Tetrahedron 1.000000 24.000000
Cube 3.000000 72.000000
Octahedron 4.000000 96.000000
RT5 5.000000 120.000000
RTe 5.007758 120.186193
Rhombic Dodecahedron 6.000000 144.000000
Pentagonal Dodeca 15.350018 368.400437
Icosahedron 18.512296 444.295101
Cuboctahedron 20.000000 480.000000
SuperRT 21.213203 509.116882

In [24]:
comments = pd.Series(index=shapes, dtype=str)

In [25]:
comments.loc["Tetrahedron"] = "unit of volume, self dual"
comments.loc["Cube"] = "intersected tetrahedron duals"
comments.loc["Octahedron"] = "cube dual"
comments.loc["Rhombic Dodecahedron"] = "space-filler, sphere container"
comments.loc["Cuboctahedron"] = "RD's dual"
comments.loc["Icosahedron"] = "Platonic, pent. dodeca's dual"
comments.loc["RT5"] = "120 T modules, radius 0.9994"
comments.loc["Pentagonal Dodeca"] = "Platonic, Icosa's dual"
comments.loc["RTe"] = "120 E modules, radius 1.0000"
comments.loc["SuperRT"] = "RTe scaled up to Phi radius"

In [26]:
comments.shape


Out[26]:
(10,)

In [27]:
comments


Out[27]:
Tetrahedron                  unit of volume, self dual
Cube                     intersected tetrahedron duals
Octahedron                                   cube dual
Rhombic Dodecahedron    space-filler, sphere container
Cuboctahedron                                RD's dual
Icosahedron              Platonic, pent. dodeca's dual
RT5                       120 T modules, radius 0.9994
Pentagonal Dodeca               Platonic, Icosa's dual
SuperRT                    RTe scaled up to Phi radius
RTe                       120 E modules, radius 1.0000
dtype: object

In [28]:
byvolume["Comments"] = comments

In [29]:
byvolume


Out[29]:
Volume Avols Comments
Tetrahedron 1.000000 24.000000 unit of volume, self dual
Cube 3.000000 72.000000 intersected tetrahedron duals
Octahedron 4.000000 96.000000 cube dual
RT5 5.000000 120.000000 120 T modules, radius 0.9994
RTe 5.007758 120.186193 120 E modules, radius 1.0000
Rhombic Dodecahedron 6.000000 144.000000 space-filler, sphere container
Pentagonal Dodeca 15.350018 368.400437 Platonic, Icosa's dual
Icosahedron 18.512296 444.295101 Platonic, pent. dodeca's dual
Cuboctahedron 20.000000 480.000000 RD's dual
SuperRT 21.213203 509.116882 RTe scaled up to Phi radius

In [30]:
from IPython.display import YouTubeVideo
YouTubeVideo("RGH8m0LdaTM")


Out[30]:

My video narrative could be a little more clear on how we're inserting rows in the same way we insert columns, but with .loc for adding the new labeling. In general, consider embedding videos in your Notebooks if you think these could be helpful to those making use of it down the road.

Want to see the embedded Youtube?

Here's the NBVIEWER view.