Classical Geometry: Euclidean, Transformational, Inversive, and Projective

CHAPTER 12

TESSELLATIONS

12.1 Tilings

A tiling or tessellation of the plane is a division of the plane into regions

called tiles, in such a manner that:

1. No region contains an interior point of another region.

2. Every point in the plane belongs to one of the regions.

In other words, the plane is completely covered by nonoverlapping tiles.

There is no requirement that the tiles be related in any way, but our interest is primarily in tilings where there are only a finite number of differently shaped tiles. A tessellation is of order-k, or k-hedral, if there is a finite set of k incongruent tiles S such that:

1. Every tile in the tessellation is congruent to some member of S.

2. Every member of S occurs at least once in the tessellation.

The members of S are called prototiles, and we say that S tiles the plane. The tiling in the figure below has a set of three prototiles, so it is an order-3 tiling.

12.2 Monohedral Tilings

The most natural question is “Which polygons are monohedral prototiles?” The words monohedral, dihedral, and trihedral are commonly used as synonyms for 1-hedral, 2-hedral, and 3-hedral, respectively.

It is not difficult to see that every parallelogram tiles the plane, as in (a) below.

As a consequence, every triangle tiles the plane because two copies ...