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Discrete & Computational Geometry

, Volume 55, Issue 3, pp 662–680 | Cite as

Combinatorially Two-Orbit Convex Polytopes

  • Nicholas MatteoEmail author
Article

Abstract

Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group coincide). Hence, a combinatorially two-orbit convex polytope is isomorphic to one of a known finite list, all of which are 3-dimensional: the cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic triacontahedron. The same is true of combinatorially two-orbit normal face-to-face tilings by convex polytopes.

Keywords

Two-orbit Convex polytopes Tilings Half-regular Quasiregular 

Mathematics Subject Classification

52B15 51M20 51F15 52C22 

References

  1. 1.
    Bokowski, J., Ewald, G., Kleinschmidt, P.: On combinatorial and affine automorphisms of polytopes. Isr. J. Math. 47(2–3), 123–130 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)zbMATHGoogle Scholar
  3. 3.
    Grünbaum, B.: Convex polytopes, 2nd ed. (Kaibel, V., Klee, V., Ziegler, G.M., eds.), Graduate Texts in Mathematics, vol. 221. Springer, New York (2003)Google Scholar
  4. 4.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman & Company, New York (1987)zbMATHGoogle Scholar
  5. 5.
    Hubard, I.: Two-orbit polyhedra from groups. Eur. J. Comb. 31(3), 943–960 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hubard, I., Orbanić, A., Weiss, A.I.: Monodromy groups and self-invariance. Can. J. Math. 61(6), 1300–1324 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Matteo, N.: Two-orbit convex polytopes and tilings. Discrete. Comput. Geom (2015). doi: 10.1007/s00454-015-9754-2
  8. 8.
    McMullen, P.: Combinatorially regular polytopes. Mathematika 14(02), 142–150 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    McMullen, P.: On the combinatorial structure of convex polytopes. PhD Thesis, University of Birmingham (June 1968)Google Scholar
  10. 10.
    McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Monson, B., Schulte, E.: Semiregular polytopes and amalgamated C-groups. Adv. Math. 229(5), 2767–2791 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schattschneider, D., Senechal, M.: Tilings. Handbook of Discrete and Computational Geometry, pp. 43–62. CRC Press, Boca Raton (1997)Google Scholar
  13. 13.
    Schulte, E.: The existence of non-tiles and non-facets in three dimensions. J. Comb. Theory Ser. A 38(1), 75–81 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schulte, E.: Combinatorial space tiling. Symmetry Cult. Sci. 22(3–4), 477–491 (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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