Abstract
From a given abstract n-polytope P and a given integer k we derive two abstract polytopes Clk(P) and \({\widetilde {Cl}_k}\left( P \right)\) of ranks n and n−1, respectively. These constructions generalise the truncation of convex polyhedra and the dual of a geometric construction yielding Petrie’s polyhedron {4,6|4}. We determine sufficient and necessary conditions to guarantee that Clk(P) and \({\widetilde {Cl}_k}\left( P \right)\) are regular.
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References
L. W. Berman, B. Monson, D. Oliveros and G. I. Williams: The monodromy group of a truncated simplex, J. Algebraic Combin. 42 (2015), 745–761.
G. Birkhoff: Lattice theory, volume 25 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, R. I., third edition, 1979.
J. Conway, H. Burgiel and Ch. Goodman-Strauss: The symmetries of things, A K Peters, Wellesley, 2008.
H. S. M. Coxeter: Regular polytopes, Dover Publications, Inc., New York, third edition, 1973.
H. S. M. Coxeter: Regular skew polyhedra in three and four dimensions, and their topological analogues, Proc. London Math. Soc. 43 (1937), 33–62.
L. Danzer: Regular incidence-complexes and dimensionally unbounded sequences of such. I, in: Convexity and graph theory (Jerusalem, 1981), volume 87 of NorthHolland Math. Stud., 115–127. North-Holland, Amsterdam, 1984.
M. del R. Francos: Chamfering operation on k-orbit maps, Ars Math. Contemp. 7 (2014), 507–524.
I. Gleason and I. Hubard: The antiprism of an abstract polytope, in preparation.
I. Gleason and I. Hubard: Products of abstract polytopes, https://arxiv.org/abs/1603.03585 (2016).
B. Grünbaum: Uniform tilings of 3-space, Geombinatorics 4 (1994), 49–56.
B. Grünbaum and G. C. Sheppard: Tilings and Patterns, Freeman, New York, 1987.
I. Helfand: Constructions of k-orbit Abstract Polytopes, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Northeastern University.
G. A. Jones and J. S. Thornton: Operations on maps, and outer automorphisms, J. Combin. Theory Ser. B 35 (1983), 93–103.
P. McMullen and E. Schulte: Self-dual regular 4-polytopes and their PetrieCoxeter-polyhedra, Results Math. 12 (1987), 366–375.
P. McMullen and E. Schulte: Abstract Regular Polytopes, Cambridge University Press, 2002.
B. Monson, D. Pellicer and G. Williams: Mixing and monodromy of abstract polytopes, Trans. Amer. Math. Soc. 366 (2014), 2651–2681.
A. Orbanić, D. Pellicer and A. I. Weiss: Map operations and k-orbit maps, J. Combin. Theory Ser. A 117 (2010), 411–429.
D. Pellicer: Extensions of regular polytopes with preassigned Schläffli symbol, J. Combin. Theory Ser. A 116 (2009), 303–313.
D. Pellicer and A. I. Weiss: Uniform maps on surfaces of non-negative euler characteristic, Symmetry: Cult. and Sci. 22 (2011), 159–196.
E. Schulte: On arranging regular incidence-complexes as faces of higher-dimensional ones, European J. Combin. 4 (1983), 375–384.
E. Schulte: Chiral polyhedra in ordinary space. I, Discrete Comput. Geom. 32 (2004), 55–99.
E. Schulte and A. I. Weiss: Chiral polytopes, in: Applied geometry and discrete mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 493–516. Amer. Math. Soc., Providence, RI, 1991.
S. E. Wilson: Operators over regular maps, Paci.c J. Math. 81 (1979), 559–568.
S. E. Wilson: Parallel products in groups and maps, J. Algebra 167 (1994), 539–546.
