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Two-Orbit Convex Polytopes and Tilings

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Abstract

We classify the convex polytopes whose symmetry groups have two orbits on the flags. These exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, and their duals. The combinatorially regular two-orbit convex polytopes are certain 2n-gons for each \(n \ge 2\). We also classify the face-to-face tilings of Euclidean space by convex polytopes whose symmetry groups have two flag orbits. There are finitely many families, tiling one, two, or three dimensions. The only such tilings which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral–octahedral honeycomb, and the rhombic dodecahedral honeycomb.

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Acknowledgments

The author would like to thank his advisor Egon Schulte for guidance, assistance, and providing the original problem, Peter McMullen for suggested improvements, and the reviewers for very thoughtful, thorough, and helpful suggestions.

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  1. Department of Mathematics, 567 Lake Hall, Northeastern University, Boston, MA, 02115, USA

    Nicholas Matteo

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  1. Nicholas Matteo
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Matteo, N. Two-Orbit Convex Polytopes and Tilings. Discrete Comput Geom 55, 296–313 (2016). https://doi.org/10.1007/s00454-015-9754-2

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