Karl Menger · 1926
Start with a solid cube. Cut it into 27, take 7 away, then do the same to every cube that is left.
Iteration n
Iteration 5 needs 3.2 million cubes, more than a browser can turn smoothly. The true sponge is what remains after infinitely many rounds.
Divide. Cut every cube into a 3 × 3 × 3 grid of 27 smaller cubes, like a Rubik's Cube.
Remove. Take out the cube at the centre and the one in the middle of each face. 20 remain.
Repeat. Do both steps again on each of the 20, and keep going.
Measurements at this iteration
Carried on forever, the volume falls to zero while the surface grows without bound. The limit has dimension log 20 / log 3 ≈ 2.727.
Material
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Iteration 0 · 1 cube
Drag to orbit. Scroll or pinch to zoom, all the way inside.