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Finite groups, wallpaper patterns and non-euclidean geometries

Published online by Cambridge University Press:  22 September 2016

A. F. Beardon
A. F. Beardon*
Affiliation:
Department of Pure Mathematics, University of Cambridge, Mill Lane, Cambridge CB2 1SB
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Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by

The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.

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Type
Research Article
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The Mathematical Gazette , Volume 62 , Issue 422 , December 1978, pp. 267 - 278
Copyright
Copyright © Mathematical Association 1978

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References

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