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I saw a post on Bluesky that proposed a problem that I had been thinking about for a long time!

I figured that with the help of dictionary and a Python script, I could find the desired examples.

Python code

Here’s my quick Python script to help find these. The dictionary has some words that I’m not familiar with (e.g. “cen,” “wir,” etc.) so I had to manually audit the results.

from wordfreq import top_n_list
import re
from collections import defaultdict

common_words = top_n_list('en', 50000)

def vowels(word):
    return re.findall(r'[aeiouy]', word, re.IGNORECASE)

def has_one_vowel(word):
    return len(vowels(word)) == 1 

def first_vowel(word):
    return vowels(word)[0]

def replace_vowels(word):
    return re.sub(r'[aeiouyAEIOUY]', '_', word)

one_vowel_words = filter(has_one_vowel, common_words)
dict = defaultdict(list)

for w in one_vowel_words:
    dict[replace_vowels(w)] = sorted(dict[replace_vowels(w)] + [first_vowel(w)])

vowel_subsets = [['a'], ['e'], ['i'], ['o'], ['u'], ['a', 'e'], ['a', 'i'], ['a', 'o'], ['a', 'u'], ['e', 'i'], ['e', 'o'], ['e', 'u'], ['i', 'o'], ['i', 'u'], ['o', 'u'], ['a', 'e', 'i'], ['a', 'e', 'o'], ['a', 'e', 'u'], ['a', 'i', 'o'], ['a', 'i', 'u'], ['a', 'o', 'u'], ['e', 'i', 'o'], ['e', 'i', 'u'], ['e', 'o', 'u'], ['i', 'o', 'u'], ['a', 'e', 'i', 'o'], ['a', 'e', 'i', 'u'], ['a', 'e', 'o', 'u'], ['a', 'i', 'o', 'u'], ['e', 'i', 'o', 'u'], ['a', 'e', 'i', 'o', 'u']]

for vs in vowel_subsets:
    print("".join(vs).upper() + ":", ", ".join([k for k, v in dict.items() if v == vs][0:10]))

Robert Fathauer’s fractal

On May 16th, 2024, Robert Fathauer tweeted a wonderful timelapse video of building a white cube fractal, which you can see below.

When I tried to learn more about this fractal, I found a Mathematica demo called “Shifting Cube Fractal”. I couldn’t get the demo to work quite properly, so I illustrated it myself by writing a Mathematica script to make this video:

Bluesky #MathArtMarch

In March 2025, @ayliean.bsky.social started #MathArtMarch with a list of prompts to use to make new art or post existing art. On Day 12, I saw @curved-ruler.bsky.social‘s cube fractal, and inspired it, Ayliean’s prompt, and Robert Fathauer’s timelapse, I modified the above video to show off some new perspectives.

This video has four frames where the sides of the cube are perfectly transparent, and these look very reminiscent of @curved-ruler.bsky.social‘s cube fractal.

Mathematica code

Here’s my quick and dirty Mathematica code used to make the first illustration. If you modify it, show me what you make on Bluesky! (@peterkagey.com)

nextGen[cube_, t_] := (
  s = Volume[cube]^(1/3)/2;
  c = RegionCentroid[cube];
  {
   Cube[c + s {1.5, -0.5 + t, -0.5 + t}, s],
   Cube[c + s {-1/2 + t, 1.5, -1/2 + t}, s],
   Cube[c + s {-0.5 + t, -0.5 + t, 1.5}, s]
   }
  )
frames = Table[
  cubes = {{Cube[]}};
  cubes = 
   Append[cubes, 
    Flatten[nextGen[#, (Sin[2 \[Pi] t + \[Pi]/4] + 1)/2] & /@ 
      cubes[[-1]]]];
  cubes = 
   Append[cubes, 
    Flatten[nextGen[#, (Sin[2 \[Pi] t + \[Pi]/4] + 1)/2] & /@ 
      cubes[[-1]]]];
  cubes = 
   Append[cubes, 
    Flatten[nextGen[#, (Sin[2 \[Pi] t + \[Pi]/4] + 1)/2] & /@ 
      cubes[[-1]]]];
  cubes = 
   Append[cubes, 
    Flatten[nextGen[#, (Sin[2 \[Pi] t + \[Pi]/4] + 1)/2] & /@ 
      cubes[[-1]]]];
  Graphics3D[
   Transpose[{Table[
      Hue[(3 n)/5 + (Sin[2 \[Pi]*t] + 1)/10], {n, 1, 5}], cubes}],
   ViewVector -> 
    100*{Sin[4 \[Pi]*t - \[Pi]/4], Sin[4 \[Pi]*t - \[Pi]/4], 
      Sqrt[2] Cos[4 \[Pi]*t - \[Pi]/4]},
   ViewVertical -> -{Sin[4 \[Pi]*t + \[Pi]/4], 
      Sin[4 \[Pi]*t + \[Pi]/4], Sqrt[2] Cos[4 \[Pi]*t + \[Pi]/4]},
   ViewAngle -> 0.025,
   ImageSize -> {720, 720}, Boxed -> False, Lighting -> "Neutral", 
   SphericalRegion -> True],
  {t, 1/200, 1, 1/200}
  ]
Export["frame_000.png", frames, "VideoFrames"]

As I discussed in Part 1, Part 2, and Part 3, I’m going through my old saved tweets and documenting them as I move to Bluesky (@peterkagey.com). Here are the last three of those tweets (all of which had video/GIF embeddings):

• Alison Martin packing (2024-03-08)
• Matt Henderson on Pringles (2022-02-20)
• Lenore with a video of a human polyhedron (2024-06-15)

Alison Martin packing

Can anyone tell me more about the geometry of this? Let me know on Bluesky!

Matt Henderson on Pringles

I watched this 12 minute YouTube video (ad?) “How Pringles Are Made In Factory” to see if Pringles would ever be extruded through a “letterbox,” but alas, they are just stamped out using molds.

Also check out this Pringles tesselation that I learned about on Bluesky from @mathgrrl.bsky.social via @johngolden.bsky.social.

These tessellations are by @theo.rooden.art.weaving on Instagram—go over there and like and share!

Lenore with a video of a human polyhedron

No, not a human pyramid—a human rhombic dodecahedron. (I reached out to the Los Angeles Public Library to see if their librarians could help me to figure out where this comes from!)

I imagine them saying to each other, “If we put our heads together, we can make a rhombic dodecahedron.”

I made a model of the polyhedron that rotates, and where the participants “put their heads together” to alternate between a 13-face solid and a rhombic dodecahedron.