Skip to main content
SpringerLink
Log in

Regular polyhedra of index two, I

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Brückner, M.: Vielecke und Vielflache. Leipzig (1900)

  • Conder, M.: http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt (2011)

  • Conder M., Dobesanyi P.: Determination of all regular maps of small genus. J. Comb. Theory Ser. B 81, 224–242 (2001)

    Article  MATH  Google Scholar 

  • Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York, 1973. cf. 2nd edn. Macmillan, London (1973)

  • 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 

  • Coxeter H.S.M., Moser W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1980)

    Google Scholar 

  • 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)

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Cutler, A.M.: Regular Polyhedra of Index 2. PhD Thesis, Northeastern University, Boston (2009). http://iris.lib.neu.edu/math_diss/2

  • Cutler, A.M.: Regular polyhedra of index two, II. Beitr. Algebra Geom. 52 (2011, to appear)

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Dress A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, II: complete enumeration. Aequationes Math. 29, 222–243 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Gordan P.: Über die Auflösung der Gleichungen vom fünften Grade. Math. Ann. 13, 375–404 (1878)

    Article  MathSciNet  Google Scholar 

  • Grove L.C., Benson C.T.: Finite Reflection Groups, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1985)

    Google Scholar 

  • Grünbaum B.: Regular polyhedra—old and new. Aequationes Math. 16, 1–20 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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 

  • 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 

  • McMullen P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355–387 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • McMullen P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1–35 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • McMullen P.: Regular apeirotopes of dimension and rank 4. Discrete Comput. Geom. 42, 224–260 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • McMullen P., Schulte E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449–478 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • McMullen P., Schulte E.: Abstract Regular Polytopes Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  • 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 

  • Monson B.R., Weiss A.I.: Realizations of regular toroidal maps. Can. J. Math. 51(6), 1240–1257 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Pellicer D., Schulte E.: Regular polygonal complexes in space, I. Trans. Am. Math. Soc. 362(12), 6679–6714 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Richter, D.A.: The Regular Polyhedra of Index Two. http://homepages.wmich.edu/~drichter/regularpolyhedra.htm (2010)

  • Schulte E.: Chiral polyhedra in ordinary space, I. Discrete Comput. Geom. 32, 55–99 (2004)

    MATH  MathSciNet  Google Scholar 

  • Schulte E.: Chiral polyhedra in ordinary space, II. Discrete Comput. Geom. 34, 181–229 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

  • Wills J.M.: Combinatorially regular polyhedra of index 2. Aequationes Math. 34, 206–220 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Northeastern University, Boston, MA, 02115, USA

    Anthony M. Cutler & Egon Schulte

Authors
  1. Anthony M. Cutler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Egon Schulte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Egon Schulte.

Additional information

E. Schulte was supported by NSA-grant H98230-07-1-0005 and NSF-grant DMS-0856675.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cutler, A.M., Schulte, E. Regular polyhedra of index two, I. Beitr Algebra Geom 52, 133–161 (2011). https://doi.org/10.1007/s13366-011-0015-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-011-0015-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation