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The 17 plane symmetry groups

Published online by Cambridge University Press:  03 November 2016

R. L. E. Schwarzenberger
R. L. E. Schwarzenberger*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL
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Extract

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.

Type
Research Article
Information
The Mathematical Gazette , Volume 58 , Issue 404 , June 1974 , pp. 123 - 131
Copyright
Copyright © Mathematical Association 1974

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References

Burckhardt, J. J., Die Bewegungsgruppen der Kristallographie. Birkhäuser (Basel, 1966).CrossRefGoogle Scholar
Bülow, R., Neubüser, J. and Wondratschek, H., Acta crystallogr. 27A, 517–535 (1971).Google Scholar
Weyl, H., Symmetry. Princeton University Press (1952).CrossRefGoogle Scholar
Buerger, M. J., Elementary crystallography. Wiley (New York, 1956).Google Scholar