### Nauka Znanost

Shared publicly -**Coxeter–Conway friezes**

The grid of numbers in the picture is an example of a

**Coxeter–Conway frieze**. The frieze takes the form of an infinitely wide checkerboard of positive integers, satisfying the following two conditions.

1. There are finitely many rows, and all of the entries in the top and the bottom row must be equal to 1.

2. If a, b, c and d are four numbers surrounding a particular empty square to the square's left, top, bottom and right respectively, then the numbers must satisfy the

*unimodular law,*meaning that ad–bc = 1.

Coxeter–Conway friezes are named after

**H.S.M. Coxeter**(1907–2003) and

**John H. Conway**(1937–), who studied them in the 1970s. Conway and Coxeter found a remarkable connection between their friezes and triangulations of convex polygons, the details of which are as follows.

A

*triangulation*of a convex n-gon is a way to divide the n-gon into non-overlapping triangles in such a way that the corners of each triangle are three of the corners of the original n-gon. For example, the picture shows a triangulation of a heptagon (or 7-gon) which, inevitably, consists of 7–2 = 5 triangles.

The

*width*of a Coxeter–Conway frieze is the number of rows in the frieze excluding the top and bottom rows of 1s; for example, the frieze in the picture has width 4. Conway and Coxeter proved that, for a fixed n, there is a one to one correspondence between Coxeter–Conway friezes of width n–3 on the one hand, and triangulations of a convex n-gon on the other hand.

To see how this works, let us look at the seven corners of the heptagon, working clockwise from the lowest corner shown. The number of triangles meeting at each of the seven corners is 2, 2, 1, 4, 2, 1, 3. These are exactly the entries of the second to top row of the frieze! The same idea can be used to match every frieze of width 4 with a triangulation of a heptagon, and vice versa. It follows from this result that the entries in any given row of a frieze of width n–3 must form a periodic cycle of length 7, and indeed this is the case for the frieze in the picture.

The friezes and their corresponding triangulations are closely related to triangulations of n-gons in the

**Farey graph**; this is closely related to the Farey–Ford tessellation, which I posted about a few weeks ago (https://plus.google.com/101584889282878921052/posts/GrZP3ajnRGU). Friezes also have connections to other important areas of modern mathematical research, including the cluster algebras of Fomin and Zelevinsky.

**Relevant links**

Wikipedia entry on John H. Conway: http://en.wikipedia.org/wiki/John_Horton_Conway

Wikipedia entry on H.S.M. Coxeter: http://en.wikipedia.org/wiki/Coxeter

An arXiv paper by Sophie Morier-Genoud, Valentin Ovsienko and Serge Tabachnikov, which gives more details on some of the topics mentioned above: http://arxiv.org/abs/1402.5536

(Note: Usually, names of mathematicians are listed in alphabetical order, but for some reason, Coxeter is usually listed before Conway in the context of these friezes.)

#mathematics #sciencesunday #spnetwork arXiv:1402.5536

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