Wednesday, July 29, 2009

A geometry problem

Godel, Escher, Bach cover
Look at the cover of Godel, Escher, Bach. The blocks are carved such that each of the directions casts the shadow of a different letter. Now, it is clearly not possible to do this with any three shapes. What conditions are necessary for it to be possible?

I suspect the answer to this is if each shape contains a point somewhere on every horizontal and vertical line, that guarantees that you can build the block. If this is not the case, as in the letter "I", restrictions are created on the other symbols: "I" "I" and "O" could be done, but "I" "O" and "O" couldn't.

This assumes that the block can be in pieces, though. If any of the shapes is disconnected, like an "O" with a dot in the center, the resulting block will be disconnected, too. Is there a set of shapes, though, where each shape is connected, but the resulting block must be disconnected? I can't think of any, but I'm not sure.