Ingeometry, thesphinx tiling is atessellation of the plane using the "sphinx", apentagonalhexiamond formed by gluing sixequilateral triangles together. The resultant shape is named for its reminiscence to theGreat Sphinx atGiza. A sphinx can bedissected into any square number of copies of itself,[1] some of themmirror images, and repeating this process leads to anon-periodic tiling of the plane. The sphinx is therefore arep-tile (aself-replicating tessellation).[2] It is one of few knownpentagonal rep-tiles and is the only known pentagonal rep-tile whose sub-copies are equal in size.[3]
An outer boundary ("frame") in the shape of a sphinx can also be tiled in a non-recursive way for all orders. We define the order of a sphinx frame on a triangular lattice by the number of triangles at the "tail" end. An order-2 frame can be tiled by four sphinxes in exactly one way (as shown in the figure), an order-3 frame can be tiled by 9 sphinxes in 4 ways, etc. The number of tilings grows exponentially as with the order of the frame, where[4]
