Regular polyhedra—old and new

Abstract

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.

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References

  1. M. Brückner 1900Vielecke und Vielflache. Teubner, Leipzig 1900.

    Google Scholar 

  2. M. Burt 1966Spatial arrangement and polyhedra with curved surfaces and their architectural applications. M.Sc. thesis, Technion-Israel Institute of Technology, Haifa 1966.

    Google Scholar 

  3. H. S. M. Coxeter 1937Regular skew polyhedra in three and four dimensions, and their topological analogues. Proc. London Math. Soc. (2) 43 (1937), 33–62. (Improved reprint in “Twelve Geometric Essays”, Southern Illinois University Press, Carbondale 1968.)

    Google Scholar 

  4. H. S. M. Coxeter 1973Regular Polytopes. (3rd edit.) Dover, New York 1973.

  5. H. S. M. Coxeter 1974Regular Complex Polytopes. Cambridge Univ. Press 1974.

  6. H. S. M. Coxeter andW. O. J. Moser 1972Generators and Relations for Discrete Groups. (3rd edit.) Springer, Berlin 1972.

    Google Scholar 

  7. H. S. M. Coxeter andG. C. Shephard 1977A regular 3-complex with toroidal cells. J. Combinat. Theory (to appear).

  8. P. Du Val 1964Homographies, Quaternions and Rotations. Clarendon Press, Oxford 1964.

    Google Scholar 

  9. V. A. Efremovič andYu. S. Ilyašenko 1962Regular polygons in E n. [In Russian] Vestnik Moskov. Univ. 1962, No. 5, pp. 18–24.

    Google Scholar 

  10. L. Fejes Tóth 1964Regular Figures. Pergamon, New York 1964.

    Google Scholar 

  11. J. R. Gott, III 1967Pseudopolyhedrons. Amer. Math. Monthly74 (1967), 497–504.

    Google Scholar 

  12. B. Grünbaum 1977Regularity of graphs, complexes and designs. Proc. Internat. Colloq. on Combinatorics and Graph Theory, Paris 1976. (To appear.)

  13. P. McMullen 1967Combinatorially regular polytopes. Mathematika14 (1967), 142–150.

    Google Scholar 

  14. 1968Affine and projectively regular polytopes. J. London Math. Soc.43 (1968), 755–757.

    Google Scholar 

  15. P. Pearce 1966Synestructures. A report to the Graham Foundation.

  16. L. Poinsot 1810Mémoire sur les polygones et les polyèdres. J. École Polytech.10 (1810), 16–48.

    Google Scholar 

  17. G. Ringel 1974Map Color Theorem. Springer, New York 1974.

    Google Scholar 

  18. A. H. Schoen 1968aInfinite regular warped polyhedra (IRWP) and infinite periodic minimal surfaces (IPMS). Abstract 658-30. Notices Amer. Math. Soc.15 (1968), 727.

    Google Scholar 

  19. 1968bRegular saddle polyhedra (RSP). Abstract 68T-D6. Notices Amer. Math. Soc.15 (1968), 929–930.

    Google Scholar 

  20. G. C. Shephard 1952Regular complex polytopes. Proc. London Math. Soc. (3)2 (1952), 82–97.

    Google Scholar 

  21. A. Wachman, M. Burt andM. Kleinmann 1974Infinite Polyhedra. Technion-Israel Institute of Technology, Haifa 1974.

    Google Scholar 

  22. A. F. Wells 1954aThe geometrical basis of crystal chemistry. Part 1. Acta Cryst.7 (1954), 535–544.

    Google Scholar 

  23. 1954bThe geometrical basis of crystal chemistry. Part 2. Acta Cryst.7 (1954) 545–554.

    Google Scholar 

  24. 1969The geometrical basis of crystal chemistry. X. Further study of three-dimensional polyhedra. Acta Cryst. B25 (1969), 1711–1719.

    Google Scholar 

  25. R. Williams 1972Natural Structure. Eudaemon Press, Moorpark, California 1972.

    Google Scholar 

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This work was supported in part by National Science Foundation Grant MPS74-07547 A01.

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Grünbaum, B. Regular polyhedra—old and new. Aeq. Math. 16, 1–20 (1977). https://doi.org/10.1007/BF01836414

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AMS (1970) subject classification

  • Primary 50B30