Cube fractals

Robert Fathauer’s fractal

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"]