# How do Shapes Fill Space?

### Factsheets and Posters

All posters (22.8 MB) All factsheets (9.9 MB)

#### Where do we start?

Patterns and tilings appear everywhere and are one of the oldest parts of mathematics.

#### Factsheet 1

A discussion of proof and the classification of the regular and semi-regular tilings.

#### What are the secrets of the Islamic master craftsmen?

Tilings are an essential tool is creating beautifully intricate Islamic patterns.

#### Factsheet 2

Learn how to produce Islamic patterns from simple tilings.

#### How do triangles meet?

What happens when we put 3,4,5,6 or even more triangles round a vertex?

#### Factsheet 3

How we can move from triangles round a point to three dimensional shapes and curious geometries.

#### How do 3d shapes meet?

Can we use our understanding of the step from 2d to 3d to get to 4d? We jolly well can!

#### Factsheet 4

Showing that there are only 6 regular shapes in 4d and only 3 for every higher dimension!

#### What do you see?

We begin the second story with a curious tiling, discovered by Roger Penrose.

#### Factsheet 5

The two shapes of the Penrose tiling do tile the plane, but never periodically. There is never a finite region that can simply be translated round.

#### What shapes fill the plane?

The amazing discovery that not even a computer search can give the answer for any set of shapes!

#### Factsheet 6

A history of aperiodic shapes that can tile the plane but not periodically, and the projection method that can be used to find many examples.

#### New Maths, New Science!

The discovery of new mathematical ideas and language can lead to new scientific advances!

#### Factsheet 7

The substitution rule, a second powerful tool to find new aperiodic tilings, many of which produce beautiful images.
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