The Menger Sponge -- Prototype Borg Cube?
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.
Almost everyone played with wooden building blocks at some point as a child. That activity might have seemed pointless at the time, but it will now help you understand a particular fractal. Isn't childhood development weird?
Think of gluing toy building blocks together to make a large cube. The cube is three by three, meaning that it has twenty-seven blocks in total, and each face has nine blocks each (although some of those blocks show up on multiple faces, of course). Now take the cube, and remove the center block of each face, as well as the center block of the entire cube. What you'll have left is a hollow set of 'lines,' each made up of three blocks, defining the edges of the cube.
Now look closer. Imagine that each of those three blocks, which define each edge of the cube, is made of smaller building blocks. These blocks are miniature versions of the original cube, with smaller building blocks all glued together, three by three. Do the same thing to each of these mini-cubes that you did to the larger one. Remove the center block of each face and the center block of the cube. Now each of the blocks that make up the original hollow cube is also made up of a hollow cube, and the surface will begin to look pitted.
Now picture that each of those hollowed out cubes is also made up of three by three building blocks, and - well, you get the picture.
Because the edges of each cube, no matter how tiny, are left intact, the structure maintains its shape even as it gets more and more tiny hollows drilled into it. At last, it's nothing more than, well, a Menger Sponge. As the number of divisions reaches infinity, the whole thing becomes a kind of lattice with no volume inside, just surfaces of infinitely pitted and thinned walls.
Each of those pitted walls is called a Sierpinski Carpet. We are taught to think of things as having one, two, or three dimensions (and sometimes four, when we're feeling frisky), but the Sierpinski Carpet is supposed to straddle the division between a one-dimensional line and a two dimensional plane. Clearly it occupies an area, but the surface is so pitted that it technically doesn't fill the area, so much as scribble a bunch of lines over it. As crazy as it sounds, 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.