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On Laves' Graph Of Girth Ten

Published online by Cambridge University Press:  20 November 2018

H. S. M. Coxeter
H. S. M. Coxeter*
Affiliation:
University of Toronto
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1. Introduction. This note shows how a certain infinite graph of degree three, discovered by Laves in connection with crystal structure, can be inscribed (in sixteen ways, all alike) in an infinite regular skew polyhedron which has square faces, six at each vertex. One-eighth of the vertices of the polyhedron are vertices of the graph, and the three edges of the graph that meet at such a vertex are diagonals of alternate squares. Thus either diagonal of any face of the polyhedron can serve as an edge, and the whole graph can then be completed in a unique manner.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 7 , 1955 , pp. 18 - 23
Copyright
Copyright © Canadian Mathematical Society 1955

References

1. Coxeter, H. S. M., Regular skew polyhedra in three and four dimensions, and their topological analogues, Proc. London Math. Soc. (2), 43 (1937), 33–62.Google Scholar
2. Coxeter, H. S. M., The regular sponges, or skew polyhedra, Scripta Math., 6 (1939), 240–244.Google Scholar
3. Coxeter, H. S. M., Regular polytopes (London, 1948; New York, 1949).Google Scholar
4. Frucht, R., Remarks on finite groups defined by generating relations, Can. J. Math., 7 (1955), 8–17.Google Scholar
5. Heesch, H. and Laves, F., Über dünne Kugelpackungen, Z. Kristallogr., 85 (1933), 443–453.Google Scholar
6. Jensen, H., De seks uendelige regulaere polyedre. Mat. Tidsskr. (A), 1950, 53–60.Google Scholar
7. Laves, F., Zur Klassifikation der Silikate, Z. Kristallogr., 82 (1932), 1–14.Google Scholar
8. Melmore, S., The densest and the least dense packings of equal spheres, Mineralogical Magazine, 28 (1949), 479–485.Google Scholar
9. Tutte, W. T., A family of cubical graphs, Proc. Cambridge Philos. Soc, 43 (1947), 459–474.Google Scholar