The Menger Sponge literally straddles the line between different dimensions

The Menger Sponge literally straddles the line between different dimensions: A particular fractal, called Menger's Sponge, is all about surface appearances. It's a purely theoretical shape that has infinite surface area and no volume whatsoever. And because of that, it doesn't occupy three dimensions. Or two. It manages to exist in fractional dimensions...

the Sierpinski Carpet is supposed to have a fractional dimension of 1.89, and the Menger Sponge, which has no real volume, has a fractional dimension of 2.73.