KaleidoTile app for iPhone and iPad

Explore polyhedra and tilings while creating colorful kinematic art. Polyhedron mode lets you create traditional polyhedra, while Tiling mode smooths your polyhedra into tilings of the sphere, Euclidean plane or hyperbolic plane. Decorate your work with images from your photo library, fresh images from the camera, or colors. Within each symmetry group, any polyhedron or tiling morphs to any other; for example, you may morph a dodecahedron to an icosahedron, confirming that they share the same symmetries.

KaleidoTile comes with a guide to Tilings and Symmetry.