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Convex Polyhedra with Regular Faces

  • Norman W. Johnson (a1)
Extract

An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.

That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.

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References
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1. Aškinuze, V. G., O čisle polupravil'nyh mnogogrannikov, Mat. Prosvesc, 1 (1957), 107118.
2. Ball, W. W. R., Mathematical recreations and essays, 11th ed., revised by H. S. M. Coxeter (London, 1939).
3. Coxeter, H. S. M., Regular and semi-regular polytopes, Math. Z., 46 (1940), 380407.
4. Coxeter, H. S. M., Regular polytopes, 2nd ed. (New York, 1963).
5. Coxeter, H. S. M., Longuet-Higgins, M. S., and Miller, J. C. P., Uniform polyhedra, Philos. Trans. Roy. Soc. London, Ser. A, 246 (1954), 401450.
6. Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 14), 2nd ed. (Berlin, 1963).
7. Cundy, H. M. and Rollett, A. P., Mathematical models, 2nd printing (London, 1954).
8. Freudenthal, H. and van der Waerden, B. L., Over een bewering van Euclides, Simon Stevin, 25 (1947), 115121.
9. Grünbaum, B. and Johnson, N. W., The faces of a regular-faced polyhedron, J. London Math. Soc, 40 (1965), 577586.
10. Heath, T. L., The thirteen books of Euclid's elements, vol. 3 (London, 1908; New York, 1956).
11. Johnson, N. W., Convex polyhedra with regular faces (preliminary report), Abstract 576-157, Notices Amer. Math. Soc., 7 (1960), 952.
12. Kepler, J., Harmonice Mundi, Opera Omnia, vol. 5 (Frankfurt, 1864), 75334.
13. Zalgaller, V. A., Pravil'nogrannye mnogogranniki, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astron., 18 (1963), No. 7, 58.
14. Zalgaller, V. A. and others, O pravil'nogrannyh mnogogrannikah, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astron., 20 (1965), No. 1, 150152.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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