Regular polyhedra of index two, I

Original Paper

Abstract

A polyhedron in Euclidean 3-space \({{\mathbb{E}^3}}\) is called a regular polyhedron of index 2 if it is combinatorially regular but “fails geometric regularity by a factor of 2”; its combinatorial automorphism group is flag-transitive but its geometric symmetry group has two flag orbits. The present paper, and its successor by the first author, describe a complete classification of regular polyhedra of index 2 in \({{\mathbb{E}^3}}\). In particular, the present paper enumerates the regular polyhedra of index 2 with vertices on two orbits under the symmetry group. The subsequent paper will enumerate the regular polyhedra of index 2 with vertices on one orbit under the symmetry group.

Keywords

Regular polyhedra Kepler–Poinsot polyhedra Archimedean polyhedra Face-transitivity Regular maps on surfaces Abstract polytopes 

Mathematics Subject Classification (2000)

Primary 51M20 Secondary 52B15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brückner, M.: Vielecke und Vielflache. Leipzig (1900)Google Scholar
  2. Conder M., Dobesanyi P.: Determination of all regular maps of small genus. J. Comb. Theory Ser. B 81, 224–242 (2001)CrossRefMATHGoogle Scholar
  3. Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York, 1973. cf. 2nd edn. Macmillan, London (1973)Google Scholar
  4. Coxeter, H.S.M.: Regular skew polyhedra in 3 and 4 dimensions and their topological analogues. Proc. Lond. Math. Soc. (2) 43, 33–62 (1937) (Reprinted with amendments in Twelve Geometric Essays, Southern Illinois University Press (Carbondale, 1968), 76–105.)Google Scholar
  5. Coxeter H.S.M., Moser W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1980)Google Scholar
  6. Coxeter, H.S.M., DuVal, P., Flather, H.T., Petrie, J.F.: The 59 Icosahedra. University of Toronto Studies, Mathematical Series no. 6, University of Toronto, Toronto (1938)Google Scholar
  7. Coxeter H.S.M., Longuet-Higgins M.S., Miller J.C.P.: Uniform polyhedra. Philos. Trans. R. Soc. Lond. Ser. A 246, 401–450 (1954)CrossRefMATHMathSciNetGoogle Scholar
  8. Cutler, A.M.: Regular Polyhedra of Index 2. PhD Thesis, Northeastern University, Boston (2009). http://iris.lib.neu.edu/math_diss/2
  9. Cutler, A.M.: Regular polyhedra of index two, II. Beitr. Algebra Geom. 52 (2011, to appear)Google Scholar
  10. Dress A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, I: Grünbaum’s new regular polyhedra and their automorphism group. Aequationes Math. 23, 252–265 (1981)CrossRefMATHMathSciNetGoogle Scholar
  11. Dress A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, II: complete enumeration. Aequationes Math. 29, 222–243 (1985)CrossRefMATHMathSciNetGoogle Scholar
  12. Gordan P.: Über die Auflösung der Gleichungen vom fünften Grade. Math. Ann. 13, 375–404 (1878)CrossRefMathSciNetGoogle Scholar
  13. Grove L.C., Benson C.T.: Finite Reflection Groups, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1985)Google Scholar
  14. Grünbaum B.: Regular polyhedra—old and new. Aequationes Math. 16, 1–20 (1977)CrossRefMATHMathSciNetGoogle Scholar
  15. Grünbaum B.: Polyhedra with hollow faces. In: Bisztriczky, T., McMullen, P., Schneider, R., Ivić Weiss, A. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series C, vol. 440, pp. 43–70. Kluwer, Dordrecht (1994)Google Scholar
  16. Grünbaum B.: Are your polyhedra the same as my polyhedra?. In: Aronov, B. (eds) Discrete and Computational Geometry. Algorithms Comb., vol. 25, pp. 461–488. Springer, Berlin (2003)Google Scholar
  17. Grünbaum B., Shephard G.C.: Duality of polyhedra. In: Fleck, G., Senechal, M. (eds) Shaping Space—a Polyhedral Approach, pp. 205–211. Birkhäuser, Boston (1988)Google Scholar
  18. Martini H.: Hierarchical classification of euclidean polytopes with regularity properties. In: Bisztriczky, T., McMullen, P., Schneider, R., Ivić Weiss, A. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series C, vol. 440, pp. 71–96. Kluwer, Dordrecht (1994)Google Scholar
  19. Martini H.: Reguläre Polytope und Verallgemeinerungen. In: Giering, O., Hoschek, J. (eds) Geometrie und ihre Anwendungen, pp. 247–281. Verlag Carl Hanser, Munich, Vienna (1994)Google Scholar
  20. McMullen P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355–387 (2007)CrossRefMATHMathSciNetGoogle Scholar
  21. McMullen P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1–35 (2004)CrossRefMATHMathSciNetGoogle Scholar
  22. McMullen P.: Regular apeirotopes of dimension and rank 4. Discrete Comput. Geom. 42, 224–260 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. McMullen P., Schulte E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449–478 (1997)CrossRefMATHMathSciNetGoogle Scholar
  24. McMullen P., Schulte E.: Abstract Regular Polytopes Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002)Google Scholar
  25. McMullen P., Schulte E.: Regular and chiral polytopes in low dimensions. In: Davis, C., Ellers, E.W. (eds) The Coxeter Legacy—Reflections and Projections. Fields Institute Communications, vol. 48, pp.87–106. American Mathematical Society, Providence (2006)Google Scholar
  26. Monson B.R., Weiss A.I.: Realizations of regular toroidal maps. Can. J. Math. 51(6), 1240–1257 (1999)CrossRefMATHMathSciNetGoogle Scholar
  27. Pellicer D., Schulte E.: Regular polygonal complexes in space, I. Trans. Am. Math. Soc. 362(12), 6679–6714 (2010)CrossRefMATHMathSciNetGoogle Scholar
  28. Richter, D.A.: The Regular Polyhedra of Index Two. http://homepages.wmich.edu/~drichter/regularpolyhedra.htm (2010)
  29. Schulte E.: Chiral polyhedra in ordinary space, I. Discrete Comput. Geom. 32, 55–99 (2004)MATHMathSciNetGoogle Scholar
  30. Schulte E.: Chiral polyhedra in ordinary space, II. Discrete Comput. Geom. 34, 181–229 (2005)CrossRefMATHMathSciNetGoogle Scholar
  31. Schulte, E., Wills, J.M.: Combinatorial regular polyhedra in three-space. In: Hofmann, K.H., Wille, R. (eds.) Symmetry of Discrete Mathematical Structures and Their Symmetry Groups. Research and Exposition in Mathematics, vol. 15, pp. 49–88 (1991)Google Scholar
  32. Wills J.M.: Combinatorially regular polyhedra of index 2. Aequationes Math. 34, 206–220 (1987)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Managing Editors 2011

Authors and Affiliations

  1. 1.Northeastern UniversityBostonUSA

Personalised recommendations