Introduction to Tiling Theory

In this mini-lecture, we explore tilings found in everyday life and give the mathematical definition of a tiling. In particular, we think about: (i) traditional Islamic tilings; (ii) floor, wallpaper, pavement, and architectural tilings; (iii) the three regular tilings using either equilateral triangles, squares, or regular hexagons; (iv) a variety of tilings using strange and complicated tiles; (v) the Voderberg tile; (vi) properties we do not want in a tiling: disconnected tiles, holes, cut points, whiskers, tiles connected by whiskers, or an infinite tile; (vi) properties we do want in a tiling: deformable (topologically equivalent) to a disk, and covers the entire plane without the interiors of tiles intersecting (called a ‘packing’); (vii) the definition of a protoset and the definition of a tiling using a protoset; (viii) monohedral tilings, which are tilings with just one prototile; (ix) trihedral tilings, which are tilings with three prototiles; and (x) vertices and edges in tiling.