The Complement of the Menger Sponge

The complement of an evolving Menger sponge is all of the material that has been removed so far.


Level zero

I described the Menger sponge fractal in this blog three weeks ago. At level zero, you begin with a solid cube of, say, gold.

Level one

Trisect the level zero cube into 3x3x3 = 27 smaller cubes, like a Rubik's cube without the labels. Remove the center cube from each of the six faces, as well as the cube at the very center. This leaves these 27 - 7 = 20 smaller cubes at level one.

Level one complement

The level one complement is all the material from level zero that was removed to obtain level one. I have made it a brighter shade of gold to provide contrast.

The union

The level one cube and its complement fit together nicely to recreate level zero.

Until next year

That's enough for now. I wanted to get to level one so I could use it my annual graphics-only holiday card.

Published with MATLAB® R2021a

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