This is a close up of the sketch in my moleskine (click on the picture to enlarge) which shows a Menger sponge. A Menger sponge is a fractal shape, and so an accurate rendition is not possible. I've gone to 'level 4', with 'level 1' as a cube.

To make a Menger sponge, start with a cube, and make a square tunnel through each side. Each face is 8/9th the area it started with. This can be thought of as eight squares in a ring. In the centre of each of these squares, remove another square tunnel. Wash, rinse, repeat.

When taken to infinity, we end up with a very holey solid. It has a fractional dimension, it's a fractal.

It actually has a dimension of 2.72683. What's this mean? Well, imagine a line, double it in size. It gets twice as big. I.e. you need two original lines to make the new one. That's a change of 2^{1}. I.e. this has one dimension.

Take a square, double it in size, it's area increases four times. I.e. you need four original squares to make the new one. That's a change of 2^{2}. I.e. this has two dimensions.

Take a square, double it in size, it's volume increases eight times. I.e. you need eight original cubes to make the new one. That's a change of That's 2^{3}. I.e. this has three dimensions.

Now, to make a larger menger sponge, we need to increase it in linear size three times. That's not a problem, with cubes we'd need 3^{3} cubes (27 cubes), the dimension is still 3.

With menger sponges we'd need 8 for the top and bottom layer, and 4 for the inner layer, so that's 20 smaller spongers to make one larger sponge.

This means that 20=3^{dimension}, so ln(20)=dimension*ln(3)

In turn this means that the dimension of the sponge is ln(20)/ln(3) or approximately 2.72683. It's more solid that a flat surface, e.g. paper, but less solid than a solid, e.g. a cube.

For more information on this topic, I can highly recommend 'Flatterland' by Ian Stewart. The classic prequel is out of copyright and available online as well as a bound edition.