The Disappearing Hyperbolic Squares
The disk is the Poincaré model of the hyperbolic plane. Each red tile is a hyperbolic square. If either of the two largest red tiles is deleted, the resulting set of red tiles is congruent to the original set. This Demonstration shows the two congruences: the last image on the left is the original set minus the tile on top, while the last image on the right is the original set minus the tile on the bottom. Such a set that is invariant under two deletions does not exist in the Euclidean plane.