Tessellations of the Plane

A 'star' tessellation with the vertex figure 12/5, 12/5, 3/2

There are traditionally held to be eleven tessellations of the plane.  Three regular tessellations and eight semi-regular. 

This page however, concentrates on other uniform tessellations of the plane.  These involve infinite, retrograde and star polygons. Pages on each category of tessellation are reached by pressing the page headings below. Clicking on individual vertex diagrams will lead to enlarged pictures of said vertex within its tessellation.

Kepler described many of the tessellations involving normal polygons, including those with crossed vertex figures in his 1619 "Opera Omnia".  Tessellations involving star polygons were first described in a paper "Memoir  sur les Figures Isoscèles" by A.Badoureau published in the Journal de l'Ecole Polytechnique (1881).  This is the same paper in which Badoureau also described 37 of the non-convex uniform polyhedra.  A number of these tessellations are displayed on a panel at the Science Museum in London.  The first complete list of the uniform tessellations of the plane was given by H.S.M.Coxeter, M.S.Longuet-Higgins and J.C.P.Miller in "Uniform Polyhedra", published in Philosophical Transactions of the Royal Society of London, Series A Volume 246 pp 401-450 (1954).  Unfortunately, Coxeter et al only give a table of such tessellations without an explanation as to how the list was developed.

For each tessellation, the notation: K:V, B:65, C: 2 4 | 4 = t{4,4} gives the symbols assigned by Kepler, the figure number in Badoureau's paper and the notation given by Coxeter.  A dash ('-') signifies the paper did not include the tessellation.  In order to show Coxeter's notation where one figure is shown above the other, the notation a : b is used for a above b, 'oo' is used for infinity.

All of the tessellations and edge networks on these pages (including the hyperbolics) were generated using Melinda Green and Don Hatch's excellent "Tyler" applet. It is linked here.

The 'traditional' eleven tessellations

Infinite and semi-infinite tessellations

Seven further tessellations involving infinite polygons.

'Star' and 'Retrograde' tessellations:

Another 22 tessellations, bringing the total number of uniform plane tessellations I have generated to 40. This covers every tessellation listed by Coxeter et al. which the authors "believe to be complete".

In a number of instances, a number of tessellations can be formed which share the same network of edges and vertices (a "network").  I have grouped tessellations below by network. It can though be shown that for networks with three edges per vertex then only one tessellation can exist, and for networks with four edges per vertex only three tessellations can exist.  

Clicking on the 'Type' heading leads to a short page focusing on each group of tessellations.

Type 1

Retrograde alternatives to the familiar 6,4,3,4

Type 2

A faceting of the 8,8,4

Type 3

A faceting of the 12,12,3

Type 4

A faceting of the 6,4,3,4

Type 5

A second faceting of the 8,8,4

Type 6

A second faceting of the 12,12,3

Type 7

A third faceting of the 8,8,4

Type 8

A faceting of the 12,6,4

Type 9

A third faceting of the 12,12,3

Type 10

A stellation of the 8,8,4

Type 11

A stellation of the 12,12,3

Type 12

A variant of the 4,4,3,3,3

Type 13

A variant of the 4,3,4,3,3

Type 14  NEW

A chiral tessellation

Hyperbolic Star Tessellations

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