(Check out this web-site: Simeon's Triangle Puzzle )

I have tried to figure out how this could be, because everything I know about it tells me it can't be. However, I can't seem to make it NOT work.

Two right angle triangles. They are each cut up into four identical pieces. In the first, all the pieces fit together so that there are NO empty spaces; in the second, presumed to be identical in size to the first, the pieces are slightly rearranged, and now, there IS a space in the triangle.