Polyhedra with Hollow Faces

  • Branko Grünbaum
Part of the NATO ASI Series book series (ASIC, volume 440)

Abstract

The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term “regular polyhedra” was, and is, contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call “polyhedra”, with those special ones that deserve to be called “regular”. But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,…—the writers failed to define what are the “polyhedra” among which they are finding the “regular” ones. True, we now know what are the convex polyhedra, which we think are the polyhedra Euclid had in mind; hence there is no stigma attached to the use of a term like “regular convex polyhedron”. But where in the literature do we find acceptable definitions of polyhedra that could be specialized to give the “regular Kepler-Poinsot polyhedra” ? For these, a better expression would be to say that they are “regularpolyhedra”—a distinct kind of objects, constructed according to more or less explicit procedures, and without any connection to what the separate parts of that ungainly word may mean.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bachmann F. and Schmidt E., n-gons. Translated from German by C. W. L. Garner. Mathematical Expositions No. 18, Toronto Univ. Press, 1975Google Scholar
  2. [2]
    Barnette D.W., Gritzmann P. and Höhne R., On valences of polyhedra. J. Combinat. Theory A 58(1991), 279–300CrossRefMATHGoogle Scholar
  3. [3]
    Berlekamp E.R., Gilbert E. N. and Sinden F. W., A polygon problem. Amer. Math. Monthly 72(1965), pp. 233–241MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Brückner M., Vielecke und Vielflache. Theorie und Geschichte. Teubner, Leipzig 1900MATHGoogle Scholar
  5. [5]
    Brückner M., Über die diskontinuierlichen and nicht-konvexen gleicheckiggleich-flächigen Polyeder. Verh. des dritten Internat. Math.-Kongresses Heidelberg 1904 Teubner, Leipzig 1905, pp. 707–713Google Scholar
  6. [6]
    Brückner M., Über die gleicheckig-gleichflächigen, diskontinuirlichen und nichtkonvexen Polyeder. Nova Acta Leop. 86(1906), No. 1, pp. 1–348 + 29 platesGoogle Scholar
  7. [7]
    Brückner M., Zur Geschichte der Theorie der gleicheckig-gleichfläch;igen Polyedfer. Unterrichtsblätter für Mathematik und Naturwissenschaften, 13(1907), 104–110, 121-127 +plateGoogle Scholar
  8. [8]
    Coxeter H. S. M. and Moser W. O. J., Generators and Relations for Discrete Groups. 4th ed. Springer, Berlin 1980Google Scholar
  9. [9]
    Douglas J., Geometry of polygons in the complex plane. J. Math. Phys. 19(1940), pp. 93–130MATHGoogle Scholar
  10. [10]
    Dress A. W. M., A combinatorial theory of Grünbaum’s new regular polyhedra, Part I: Grünbaum’s new regular polyhedra and their automorphism group. Aequationes Math. 23(1981), 252–265MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Dress A. W. M., A combinatorial theory of Grünbaum’s new regular polyhedra, Part II: Complete enumeration. Aequationes Math. 29(1985), 222–243MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Farris S. L., Completely classifying all vertex-transitive and edge-transitive polyhedra, Part I: necessary class conditions. Geometriae Dedicata 26(1988), 111–124MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Farris S. L., Completely classifying all vertex-transitive and edge-transitive polyhedra, Part II: finite, fully-transitive polyhedra. J. of Geometry (to appear)Google Scholar
  14. [14]
    Grünbaum B., Regular polyhedra-old and new. Aequationes Math. 16(1977), 1–20MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Grünbaum B., Regular polyhedra. Companion Encyclopaedia of the History and Philosophy of the Mathematical Sciences, I. Grattan-Guinness, ed. Routledge, London 1993 (to appear)Google Scholar
  16. [16]
    Grünbaum B., Metamorphoses of polygons. In: “The Lighter Side of Mathematics”, Proc. Strens Conference, R. K. Guy et al. eds. Math. Assoc. of America (to appear)Google Scholar
  17. [17]
    Grünbaum B. and Shephard G. C., Polyhedra with transitivity properties. C. R. Math. Rep. Acad. Sci. Canada, 6(1984), 61–66MathSciNetMATHGoogle Scholar
  18. [18]
    Grünbaum B. and Shephard G. C., Duality of polyhedra. In: “Shaping Space: A Polyhedral Approach”, Proc. “Shaping Space” Conference, Smith College, April 1984. M. Senechal and G. Fleck, eds. Birkhäuser, Boston 1988, pp. 205–211Google Scholar
  19. [19]
    Grünbaum B. and Shephard G. C., Rotation and winding numbers for planar polygons and curves. Trans. Amer. Math. Soc. 322(1990), 169–187MathSciNetMATHGoogle Scholar
  20. [20]
    Grünbaum B. and Shephard G. C., Isohedra with non-convex faces. J. of Geometry (to appear)Google Scholar
  21. [21]
    Grünbaum B. and Shephard G. C., A new look at Euler’s theorem for polyhedra. Amer. Math. Monthly (to appear)Google Scholar
  22. [22]
    Günther S., Vermischte Untersuchungen zur Geschichte der mathematischen Wissenschaften. Teubner, Leipzig 1876Google Scholar
  23. [23]
    Hess E., Über gleicheckige und gleichkantige Polygone. Schriften der Gesell-schaft zur Beförderung der gesammten Naturwissenschaften zu Marburg, Band 10, Abhandlung 12, pp. 611–743, 29 figures. Th. Kay, Cassel 1874.Google Scholar
  24. [24]
    Hess E., Ueber zwei Erweiterungen des Begriffs der regelmässigen Körper. Sitzungsberichte der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg 1875, pp. 1–20Google Scholar
  25. [25]
    Hess E., Ueber die zugleich gleicheckigen und gleichflächigen Polyeder. Schriften der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg, Band 11, Abhandlung 1, pp. 1–97, 11 figures. Th. Kay, Cassel 1876Google Scholar
  26. [26]
    Hess E., Ueber einige merkwürdige nichtkonvexe Polyeder. Sitzungsberichte der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg 1877, pp. 1–13Google Scholar
  27. [27]
    Meister A. L. F., Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus. Novi Comm. Soc. Reg Scient. Gotting. 1(1769/70), pp. 144–180 + platesGoogle Scholar
  28. [28]
    Neumann B. H., Some remarks on polygons. J. London Math. Soc. 16(1941), pp. 230–245MathSciNetCrossRefGoogle Scholar
  29. [29]
    Robertson S. A., Polytopes and Symmetry. London Math. Soc. Lecture Note Series No. 90. Cambridge Univ. Press 1984Google Scholar
  30. [30]
    Stewart B. M., Adventures Among The Toroids. 2nd ed. Okemos MI, 1980MATHGoogle Scholar
  31. [31]
    Szillasi L., Regular toroids. Structural topology 13(1986), 69–80Google Scholar
  32. [32]
    Wilson S. E., New techniques for the construction of regular maps. Ph. D. thesis, University of Washington, Seattle 1976Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • Branko Grünbaum
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations