How do Shapes Fill Space?
Factsheets and Posters
Where do we start?
Patterns and tilings appear everywhere and are one of the oldest parts of mathematics.
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.
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?
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!
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.
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!
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!
The substitution rule, a second powerful tool to find new aperiodic tilings, many of which produce beautiful images.
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Last modified: 19 July 2009