Constructing Semi-Regular Tilings

1. Introduction and Historical Background
2. Notation and Definitions
3. General Theorems
4. Hyperbolic Results
5. References

1. Introduction and Historical Background

The five familiar regular or Platonic polyhedra were well-known in antiquity. A toy regular dodecahedron was found in Padua in Etruscan ruins dating from 500 B.C. See [Mar,199]. Theatetus (419--369 B.C.) is given the credit for having developed the abstract notion of a regular polyhedra. Somewhat less familiar are the semi-regular polyhedra (whose faces are regular polygons of two or more different types arranged similarly about each vertex) that were also known to the Greeks. Because they were described in Archimedes' writings, they are often called Archimedean polyhedra. There are thirteen such polyhedra as well as two infinite classes which are often referred to as regular prisms and antiprisms.

A sphere that passes through all the vertices may be circumscribed about any of the regular or semi-regular polyhedra. If the polyhedron is then projected onto the sphere from its center, a tiling of the sphere by regular (spherical) polygons is obtained. Conversely, a tiling of the sphere by regular polygons (of three or more sides, three or more at a vertex) similarly arranged about each vertex can be used to produce one of the polyhedra mentioned above by connecting adjacent vertices with straight line segments. In this way we can think of the regular and semi-regular polyhedra as solving a very special tiling problem on the sphere. For example, the icosahedron is a regular tiling of the sphere using five triangles at avertex.

Though regular and semi-regular polyhedra were known in antiquity, this work seems to have been lost and the results were only gradually rediscovered during the Renaissance. Artists such as Albrect Durer and Pierro della Francesca made drawings of many of the semi-regular polyhedra. Johannes Kepler was the first person of this period to give a complete description of such figures in his Harmonices Mundi which appeared in 1619. The figure below gives examples of all of the classes of semi-regular tilings of the sphere.

In the Euclidean plane, tilings using only regular polygonal tiles were known and used in antiquity, but were not completely and systematically classified. It was Kepler, again, who generalized the idea of the Archimedean polyhedra and found all eleven so-called Archimedean tilings of the Euclidean plane. These tilings employ one or more types of regular polygon similarly arranged about each vertex. These eight tilings employing polygons of more than one type appear in the figure below.

Surprisingly, Kepler's work on Euclidean plane tilings seems to have been forgotten for the better part of 300 years. Grunbaum and Shephard claim that the first correct determination of the eleven Archimedean Euclidean tilings in modern times was by Robin in 1887, Sommerville in 1905, and then Andreini in 1907. Apparently they were aware neither of Kepler's work nor each others'. In fact, as late as 1931, certain authors appeared to be unaware of such work. Grunbaum and Shephard further claim that many other authors of this period gave defective treatments of this problem failing to list one or another of the types of Archimedean tilings. [G-S 1987 or G-S 1989, Chapter 2.10]

The nineteenth century creation of hyperbolic geometry provides another arena for creating tilings. The hyperbolic plane is amenable to an infinite number of different tilings by regular polygons. It is well-known that regular hyperbolic p-gons appearing q at a vertex will tile the hyperbolic plane whenever 1/ p + 1/q < 1/2. Such tilings are important because when p = q = 4n they are the universal covering spaces of orientable compact Riemann surfaces ([C-M, Chapter 5], [H-CV, Section 49]) and because of their intimate relation to modular groups and automorphic functions ([K-F], [L]). Drawings of such tilings, especially those employing the Poincare model, are aesthetically appealing. Pictures of such tilings, for example septagons (divided into fourteen alternating black-and-white right triangles) meeting three at a vertex appear in the mathematical literature as early as 1890 [K-F, Figure 33]. A similar illustration of triangulated regular hyperbolic hexagons meeting four at vertex (see the cover of [G]) was the inspiration for M. C. Escher's series of four woodcuts "Circle Limit I--IV" [Co, Figure 1], [D1, Figure 1]. The figure below shows a tiling by septagons three at a vertex, divided into alternating red and white triangles.

The general problem of tiling the hyperbolic plane with two or more different regular polygons similarly arranged about each vertex seems to have attracted less attention. In fact, the author was able to locate only a single such illustration [R, Figure 11] which shows a tiling of the hyperbolic plane using two regular octagons and a single hexagon at each vertex.

This figure shows a tiling of the hyperbolic plane using three regular poylgons at a vertex: a quadrilateral, an hexagon, and a 14-gon.

For an introduction to the terminology and defintions used to describe regular and semi-regular tilings, see 2. Notation and Definitions. Sections 3. General Theorems and 4. Hyperbolic Results describe some relatively simple geometric processes that allow one to construct several such tilings. For additional sources of material, see the References section.