Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T19:46:10.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Frieze groups, cylinders, and quotient groups

Published online by Cambridge University Press:  23 January 2015

A. F. Beardon
A. F. Beardon*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 538 , March 2013 , pp. 95 - 100
Copyright
Copyright © The Mathematical Association 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Armstrong, M. A., Groups and symmetry, Springer-Verlag (1988).Google Scholar
2. Budden, F. J., The fascination of gropus, Cambridge University Press (1982).Google Scholar
3. Conway, J. H., Burgiel, H. and Goodman-Struass, C., The symmetries of things, A. K. Peters (2008).Google Scholar
4. Lyndon, R. C., Groups and geometry, London Math. Soc. Lecture Notes, vol. 101, Cambridge University Press (1985).Google Scholar
5. Martin, G. E., Transformation geometry, Springer-Verlag (1982).Google Scholar
6. Speiser, A., Theorie der Gruppen von endlicher Ordnung, Springer (1927).Google Scholar
7. Conway, J. H. and Coxeter, H. S. M., Triangulated polygons and frieze patterns, Math. Gaz. 57 (1973), pp. 87-89 and pp. 175–183.Google Scholar
8. Shephard, G. C., Additive frieze patterns and multiplication tables, Math. Gaz. 60 (1976) pp. 178–184.Google Scholar
9. Schwarzenberger, R., The 17 plane crystallographic groups, Math. Gaz. 58 (1974), pp. 123–131.Google Scholar