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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bokowski, J., Ewald, G., Kleinschmidt, P.: On combinatorial and affine automorphisms of polytopes. Isr. J. Math. 47(2–3), 123–130 (1984)
Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)
Grünbaum, B.: Convex polytopes, 2nd ed. (Kaibel, V., Klee, V., Ziegler, G.M., eds.), Graduate Texts in Mathematics, vol. 221. Springer, New York (2003)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman & Company, New York (1987)
Hubard, I.: Two-orbit polyhedra from groups. Eur. J. Comb. 31(3), 943–960 (2010)
Hubard, I., Orbanić, A., Weiss, A.I.: Monodromy groups and self-invariance. Can. J. Math. 61(6), 1300–1324 (2009)
Matteo, N.: Two-orbit convex polytopes and tilings. Discrete. Comput. Geom (2015). doi:10.1007/s00454-015-9754-2
McMullen, P.: Combinatorially regular polytopes. Mathematika 14(02), 142–150 (1967)
McMullen, P.: On the combinatorial structure of convex polytopes. PhD Thesis, University of Birmingham (June 1968)
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Cambridge University Press, Cambridge (2002)
Monson, B., Schulte, E.: Semiregular polytopes and amalgamated C-groups. Adv. Math. 229(5), 2767–2791 (2012)
Schattschneider, D., Senechal, M.: Tilings. Handbook of Discrete and Computational Geometry, pp. 43–62. CRC Press, Boca Raton (1997)
Schulte, E.: The existence of non-tiles and non-facets in three dimensions. J. Comb. Theory Ser. A 38(1), 75–81 (1985)
Schulte, E.: Combinatorial space tiling. Symmetry Cult. Sci. 22(3–4), 477–491 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge János Pach
Rights and permissions
About this article
Cite this article
Matteo, N. Combinatorially Two-Orbit Convex Polytopes. Discrete Comput Geom 55, 662–680 (2016). https://doi.org/10.1007/s00454-015-9752-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-015-9752-4