Python for STEM Teachers
Oregon Curriculum Network
Welcome everyone.
In this very simple demo, I start with a listing of Polyhedrons with volumes in increasing order.
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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)
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table
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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.
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table.columns = ["Volume",] # relabel the one column
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table
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$\phi = (1 + \sqrt{5})/2$
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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?
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table.loc["Icosahedron","Volume"] = 5 * np.sqrt(2) * phi ** 2
shapes.append("Icosahedron")
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table
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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
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table.loc["RT5","Volume"] = 5
shapes.append("RT5")
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byvolume = table.sort_values(by="Volume")
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byvolume
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The concentric hierarchy icosahedron, obtained by jitterbugging the Cuboctahedron of volume 20, has a volume of:
$$ 5 \sqrt{2}\phi^{2} $$
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byvolume["Avols"] = byvolume["Volume"] * 24 # A = B = T = 1/24
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byvolume
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penta_dodeca = 3 * np.sqrt(2) * (phi**2 + 1)
shapes.append("Pentagonal Dodeca")
penta_dodeca
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In [14]:
byvolume.loc["Pentagonal Dodeca","Volume"] = penta_dodeca
byvolume.loc["Pentagonal Dodeca","Avols"] = penta_dodeca * 24
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SuperRT = 20 * np.sqrt(9/8) # S3
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SuperRT
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byvolume.loc["SuperRT", "Volume"] = SuperRT
byvolume.loc["SuperRT", "Avols"] = SuperRT *24
shapes.append("SuperRT")
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byvolume
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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+.
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byvolume.loc["RTe", "Volume"] = byvolume.loc["SuperRT","Volume"] * phi**-3
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byvolume.loc["RTe", "Avols"] = byvolume.loc["RTe", "Volume"] * 24
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shapes.append("RTe") # RT of 120 E modules
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byvolume.sort_values(by="Volume", inplace=True)
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byvolume
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comments = pd.Series(index=shapes, dtype=str)
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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"
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comments.shape
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comments
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byvolume["Comments"] = comments
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byvolume
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In [30]:
from IPython.display import YouTubeVideo
YouTubeVideo("RGH8m0LdaTM")
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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.