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Tilings

A set of five laminated posters with background notes, from The Beauty of Mathematics poster collection.

The Tilings poster set

The posters illustrate different forms of tilings:

• Archimedean Tilings
• Tessellating Pentagons
• Rep-tiles
• Penrose Rhombs
• The Hyperbolic Plane

Each poster describes the tilings and also gives brief mathematical or historical information, such as the discovery of the tilings, or where a tiling has been used in art and architecture.

See below for more information about the background notes as well as samples from the notes. Larger images of the posters may be seen by going to the Poster Gallery.

• Poster Gallery
• Order
• Other poster sets:
• Curves from Circles
• Polyhedra
• Curves from Lines
• Fractals

“Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.”—Hermann Weyl

The Tilings background notes

A descriptive booklet accompanies the posters, explaining the mathematics behind the images in the posters and related topics.

The four pages of background notes give more information about the tilings on the five posters and about tilings in general. The topics covered are:

• Types of Tilings
• Symmetry
• Regular tilings
• Duality
• Inflation
• Description and properties of the tilings on each poster
• Glossary
• References

The notes help interested students and teachers explore further, and provide ideas for activities connected to the theme of the posters.

Extracts from the background notes

Types of Tilings

A tiling is a collection of tiles which cover the plane without gaps or overlaps, like that shown alongside. The study of all types of tilings is a huge topic and in the remainder of these notes tiles with curved, fractal or infinite boundaries are not considered. In particular, all the tiles are polygons. Each tiling also uses only a small number of polygons and in many cases only one or two.

The tilings on The Beauty of Mathematics posters are coloured only for illustrative purposes; in general the colouring is ignored when considering the properties.

In a tiling of the plane by polygons, two tiles may meet exactly in a single side of each tile and so fit together ‘edge to edge’; the tiling itself is said to be edge-to-edge if this is the case for all pairs of adjacent tiles. Most of the tilings considered here are edge-to-edge, though not all are.

Archimedean Tilings

It is possible to prove by elementary means that there are only eleven edge-to-edge tilings by regular polygons, such that the arrangement of polygons around each vertex is the same. By calculating the interior angles of regular polygons, a list of all possible arrangements around a point may be made (there are 21). For example, a square, pentagon and 20-gon may be placed together at a vertex, as shown alongside. Consideration of each arrangement in turn either verifies that a tiling is possible, or shows that the arrangement cannot be extended to more vertices. In the example shown, the arrangement cannot be repeated for all the vertices around the pentagon.

Many tilings exist with more than one type of vertex arrangement. There are 20 edge-to-edge tilings by regular polygons which have two different vertex arrangements.

Tessellating Pentagons

At the time of writing (2004) the only convex pentagons known to tessellate are the fourteen types which are shown on The Beauty of Mathematics poster Tessellating Pentagons, but it is not known whether this list is complete. The tessellation problem for other convex polygons is completely solved: all triangles, all quadrilaterals and exactly three types of hexagons tessellate; no convex polygon with more than six sides tessellates.

© A K Jobbings 2004-2009