Tiling can be fascinating and beautiful, but is limited to two dimensions. My idea is to make tiling 3D by using transparent and opaque blocks. The blocks fit together in any of the "convex uniform honeycomb" arrangements (see link). My preferred tilings would be a cubic or quarter cubic honeycomb (see
link).

The area to be tiled must be recessed (e.g. at least 20cm) to allow for the 3D tiles to be stacked.

The blocks are made out of a hard material; the transparent blocks made of glass or crystal, the opaque blocks made of ceramic. The blocks are of any suitable tiling size (at least 5cm). The transparent blocks are preferrably larger than the opaque blocks so you can see through the tiling matrix.

The 'grout' between tiles must be transparent and preferrably of the same refractive index as the transparent blocks (e.g. silicone or epoxy).

Because of precision required in installing the blocks a 'grout' layer could be pre-attached onto each surface of the blocks. A sealing film is then removed from the pre-attached grout layer when installing (sort of like the film on a sliced cheese).

illustrationhttp://i.imgur.com/Y7Tym.jpg Here's a section of the 3D tiling (quarter cubic honeycomb). First image is a cross section (to show its flat on the top surface). [xaviergisz, Jul 19 2011]

[+] because this made me think and made me
start looking through Wikipedia which led to
geometry and all kinds of fun stuff.

(I got distracted by the problem of self-
intersecting cubes. It led to a question: you take
a cube, rotate it through 45° around any one of its
three axes, then superimpose it on the original to
give two intersecting cubes. You then do the
same again, each time duplicating the last cube,
picking an axis, and rotating it 45°. How many
different orientations of the cube can you do
before returning to the original position? I
guessed there were only 8 possible orientations
(rotated by 0 or by 45° in any of three axes), but I
don't think this is the case.)

I'm not sure I am visualising the idea correctly. Will the floor still be flat with only the illusion of 3d?

//How many different orientations of the cube can you do before returning to the original position?//

Are you looking for the fewest possible turns without undoing any twists? Otherwise the corners of the cube will create an infinite amount of points and basically make a sphere looking shape first I would think.

I went iterative as far as 28 cubes without repetition (for some reason 3^3 seemed like a likely possibility). It won't get truly spherical, if the total number is high enough it will appear close.

[poc] it was very easy in Cinema4D. Beyond 16384
cubes, it got too complex and wouldn't render. But
it shows no signs of reaching maximum complexity.
My guess is that any series of 45° rotations leaves
you displaced from the original by an angle (or
angles) which are irrational or something.

[MaxwellBuchanan]; I suspect (I should at least try and do
the maths but I can't be bothered...) that the ONlY way to
get the cube back to it's initial orientation is to precisely
(reversely) follow the movements that were taken.
It would make a good (probably impossible...) game: here is
a cube (or some shape) in a particular orientation; put it
back "straight" in the minimum number of moves.