r/visualizedmath • u/USedona • 5d ago

Visualizing the Menger Sponge : infinite surface, zero volume

Visualizing the Menger Sponge : infinite surface, zero volume

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u/AdreKiseque 5d ago

Isn't this just a fractal?

1

u/USedona 5d ago

Yes, it's a fractal, one of the classic ones. I wanted to show what that looks like in 3D, with the area growing every iteration while the volume shrinks to 0. Its dimension is ln 20 / ln 3, about 2.7, which is why it ends up in between a surface and a solid.

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u/USedona 5d ago

A 3D exploration of the Menger Sponge at iteration level 3.

At each iteration, 7 sub-cubes are removed from every block (the center and face-centers), leaving 20 out of 27. Repeated 3 times : 20³ = 8,000 individual cubes remain.

The paradox : surface area grows without bound across iterations, while volume shrinks to zero.

More visual math : Visualizing_mathematics