The mathematics of tiling

Introduction

The recently restored Leeds Tiled Hall cafe and the Central Library are stunning examples of Victorian architecture and tilings. The parquet floors, tiled walls, ceilings and staircases display amazing colourful tiling patterns made by using shapes like triangles, squares, hexagons, rectangles and octagons. But why were these particular shapes used to create the patterns? What is so special about them? How can you create your own tilings using these ones as stimuli?

Description

Images of tilings

When tiling it is important that the shape of the tile when repeated should cover the whole surface or plane without any gaps or overlaps. A repeating pattern is then formed and in mathematics we call a tiling like this a tessellation.

Let’s first consider a regular tessellation – this is one made up of regular polygons of the same size and shape. [Regular polygons have all their sides the same length and angles the same size.] Only three regular polygons tessellate: equilateral triangles, squares and hexagons.

But why only these three and not other regular polygons like regular pentagons, heptagons, octagons etc?

Well it’s all to do with angles – more precisely the internal angles of the regular polygons. Here is a table with the internal angles for regular polygons starting with an equilateral triangle.

Regular polygon	Internal angle
equilateral triangle	60°
square	90°
pentagon	108°
hexagon	120°
heptagon	102.6°
octagon	135°
more than eight sides	more than 135°

Let’s consider common points in the tilings where the shapes meet. At these points the sum of the angles must add up to 360°. This is the case for equilateral triangles, squares and hexagons.

But for regular pentagons, heptagons, octagons, this does not happen.

If we try and tessellate with these shapes and not allow overlaps there are always gaps when we try and fit two or more together. For example when we try to tessellate with regular pentagons (internal angle 108°) we see that we can place three of them at a common point but there is a gap. The internal angles add up to 324° (3 x 108°) and the gap is 36°. Similarly when we try to tessellate with heptagons and octagons there are gaps of 102.8° and 90°. So for regular polygons to tessellate and leave no gaps at each common point, the interior angles must divide exactly into 360°. This is the case for the equilateral triangle, square and hexagon but not the case for any other regular polygon as their internal angles do not divide exactly into 360°.

Semi-regular tessellations can be created by using two regular polygons repeatedly to tile the surface or plane, for example an octagon and a square.

And of course other shapes can be used apart from regular polygons to make tessellations and patterns as can be evidenced by the tilings in the Leeds Tiled Hall cafe and the Central Library.

I used these beautiful tilings as inspiration to create my own artwork. Here are some examples of my designs made by using polygons folded from coloured A size paper (click here for instructions to fold polygons or view the Youtube videos).

All the tilings and patterns in the Tiled Hall and Central Library (LS1 3AB) are freely accessible on a daily basis. Why not visit, be amazed at their beauty and then create your own tiling patterns using folded shapes or otherwise?

Tiled Hall, Leeds Library staircase and entrance to the Tiled Hall and art galle

Leeds Art Gallery and the Tiled Hall,

The Headrow, Leeds LS1 3AA, UK.

Viewpoint:

You need to go into the Tiled Hall Cafe via the entrance to the Leeds Art Gallery and Public Library

Author:

eduebm

Concepts:

Themes:

Last updated:

08/06/2011 - 10:35

Access information

Wheechair/buggy accessible:

Yes

Visiting times:

Mon-Wed 9am-8pm; Thu-Fri 9am-5pm; Sat 10am-5pm; Sun 1pm-5pm

Cost:

None

Access notes:

The Art Gallery/Central Library/Tiled Hall Café are all wheelchair/buggy accessible via ramped entrance at front, and lift.

There is lots of space around for small groups of people to gather and talk.

Add a Site

To tell us about the maths in your city, log in or register.

Comments

Submitted by Chen on Sat, 29/10/2011 - 19:16.

You are right that there are only 3 regular tessellations of the plane. Having considered regular tessellations, we can perhaps try to construct non-regular tesselations systematically.

By extending the method that you used (by considering the sum of angles at a point), we do obtain a small insight on the construction of non-regular tesselations. For example, since 135+135+60+60=/=360, we know that 2 regular octagons and two equilateral triangles cannot meet at a vertex and hence cannot be used to tessellate a plane. However, since 135+135+90=360, 2 regular octagons and a square can meet at a vertex and possibly tessellate a plane. Indeed, they can tessellate a plane and their tessellation can be seen at http://en.wikipedia.org/wiki/Truncated_square_tiling.

Have fun with tessellations!

This site requires a modern browser with javascript enabled for full functionality