**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 were at one time 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.

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

A faceting of the 8,8,4

A faceting of the 12,12,3

A faceting of the 6,4,3,4

A second faceting of the 8,8,4

A second faceting of the 12,12,3

A third faceting of the 8,8,4

A faceting of the 12,6,4

A third faceting of the 12,12,3

A stellation of the 8,8,4

A stellation of the 12,12,3

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

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

A chiral tessellation