## ABSTRACT

This chapter explores several families of tilings that lend themselves to origami tessellations. There is a vast mathematical and artistic literature devoted to tilings and their properties. Polygonal tilings can be composed of a single type of polygon or several types, and individual tiles can be regular, semiregular, or completely irregular polygons. By using lattices and lattice patches, we can describe an entire tiling of unit regular polygons quite concisely and can construct the tiling fairly easily as well. The Archimedean tilings are called uniform because at each vertex, the same polygons occur in the same order around the vertex. A periodic tiling with k transitivity classes is said to be a k-uniform tiling. The k-uniform tilings are commonly labeled by the vertex figure for each transitivity class. The next level of the k-uniform tiling hierarchy beyond the uniform tilings consists of the 2-uniform tilings.