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Non-discrete Frieze Groups

Published online by Cambridge University Press:  20 November 2018

Alan F. Beardon
Alan F. Beardon*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK e-mail: afb@dpmms.cam.ac.uk
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Abstract

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The classification of Euclidean frieze groups into seven conjugacy classes is well known, and many articles on recreational mathematics contain frieze patterns that illustrate these classes. However, it is only possible to draw these patterns because the subgroup of translations that leave the pattern invariant is (by definition) cyclic, and hence discrete. In this paper we classify the conjugacy classes of frieze groups that contain a non-discrete subgroup of translations, and clearly these groups cannot be represented pictorially in any practicalway. In addition, this discussion sheds light onwhy there are only seven conjugacy classes in the classical case.

Keywords

51M0451N3020E45frieze groupsisometry groups

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 234 - 243
Copyright
Copyright © Canadian Mathematical Society 2016

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