A three dimensional fractal, derived from a Sierpinski Carpet, which is in turn a 4 sided version of the Sierpinski Triangle. Investigated by Karl Menger in 1926.

The level 0 carpet is simply a square. Level 1 is formed by taking the square and removing a square one ninth of the area from the middle, thus:

_________________
| |
| |
| _____ |
| | | |
| | | | level 1 Sierpinski Carpet
| |_____| |
| |
| |
|_________________|

Subsequent levels are formed recursively by replacing each smaller square section with a level 1 square, thus:

_________________
| _ _ _ |
| |_| |_| |_| |
| _____ |
| _ | | _ |
| |_| | | |_| | level 2 Sierpinski Carpet
| |_____| |
| _ _ _ |
| |_| |_| |_| |
|_________________|

Now the fun begins with the 3D version, where you do the same thing, but with cubes. Removing the central square on each side and through the middle produces a level 1 Menger Sponge. They are called fractal sponges because each solid component is still a part of the whole (the opposite kind is called dust).

Of course the most interesting thing about a Menger Sponge is that a perfect one (infinitely many iterations) will have an **infinite surface area but zero volume**. Weird, but also quite cool. In fact, it's so cool that the MIT origami club (amongst others) have undertaken to build a level 3 cube out of 66048 business cards!

Crazy paper-folding mathmeticians...