Thread: Reptile puzzle

Here's a puzzle I've been thinking about, but one that I have no answer for.

Is it possible to take a polygon that can tile itself (a rep-tile) and place one or more marks on its edges so that when placed in the tiling all the marks line up and leave a larger tile that can be used the same way? Is is possible to do this so that the larger repeated tile looks exactly like the original (same number of marks in the same places)?

So far, I'd bet on it being impossible with a single tile. I'm not sure about sets of tiles that can be used to tile larger copies of each other. It's easy to do with three differently marked squares (L, T and + tiles), but might not be possible with just two. I don't know about other types of polyominoes.

Also, if you can do it with exactly two marks per tile, by drawing a line between them, you'd get a space-filliing curve. (With even numbers of marks, you could get something similar.)