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...