Abstract
An abstract regular polytope \(\mathcal{P}\) of rank n can only be realized faithfully in Euclidean space \(\mathbb{E}^{d}\) of dimension d if d≥n when \(\mathcal{P}\) is finite, or d≥n−1 when \(\mathcal{P}\) is infinite (that is, \(\mathcal{P}\) is an apeirotope). In case of equality, the realization P of \(\mathcal{P}\) is said to be of full rank. If there is a faithful realization P of \(\mathcal{P}\) of dimension d=n+1 or d=n (as \(\mathcal {P}\) is finite or not), then P is said to be of nearly full rank. In previous papers, all the at most four-dimensional regular polytopes and apeirotopes of nearly full rank have been classified. This paper classifies the regular polytopes and apeirotopes of nearly full rank in all higher dimensions.
Article PDF
Similar content being viewed by others
References
Arocha, J.L., Bracho, J., Montejano, L.: Regular projective polyhedra with planar faces, I. Aequ. Math. 59, 55–73 (2000)
Bracho, J.: Regular projective polyhedra with planar faces, II. Aequ. Math. 59, 160–176 (2000)
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, pp. 76–105.)
Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)
Coxeter, H.S.M.: The evolution of Coxeter–Dynkin diagrams. In: Bisztriczky, T., McMullen, P., Schneider, R., Ivić Weiss, A. (eds.) Polytopes: Abstract, Convex and Computational. NATO ASI Series C, vol. 440, pp. 21–42. Kluwer, Dordrecht (1994)
Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 4th edn. Springer, Berlin (1980)
Debrunner, H.E.: Dissecting orthoschemes into orthoschemes. Geom. Dedic. 33, 123–152 (1990)
Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, I: Grünbaum’s new regular polyhedra and their automorphism group. Aequ. Math. 23, 252–265 (1981)
Dress, A.W.M.: A combinatorial theory of Grünbaum’s new regular polyhedra, II: complete enumeration. Aequ. Math. 29, 222–243 (1985)
Du Val, P.: Homographies, Quaternions and Rotations. Oxford University Press, Oxford (1964)
Elte, E.L.: The Semiregular Polytopes of the Hyperspaces. Hoitsema, Groningen (1912)
Gosset, T.: On the regular and semi-regular figures in space of n dimensions. Messenger Math. 29, 43–48 (1900)
Grünbaum, B.: Regular polyhedra—old and new. Aequ. Math. 16, 1–20 (1977)
McMullen, P.: Realizations of regular polytopes. Aequ. Math. 37, 38–56 (1989)
McMullen, P.: Locally projective regular polytopes. J. Comb. Theory, Ser. A 65, 1–10 (1994)
McMullen, P.: Regular polytopes of full rank. Discrete Comput. Geom. 32, 1–35 (2004)
McMullen, P.: Four-dimensional regular polyhedra. Discrete Comput. Geom. 38, 355–387 (2007)
McMullen, P.: Regular apeirotopes of dimension and rank 4. Discrete Comput. Geom. 42, 224–260 (2009)
McMullen, P.: Rigidity of regular polytopes (in preparation)
McMullen, P., Schulte, E.: Constructions for regular polytopes. J. Comb. Theory, Ser. A 53, 1–28 (1990)
McMullen, P., Schulte, E.: Regular polytopes from twisted Coxeter groups and unitary reflexion groups. Adv. Math. 82, 35–87 (1990)
McMullen, P., Schulte, E.: Higher toroidal regular polytopes. Adv. Math. 117, 17–51 (1996)
McMullen, P., Schulte, E.: Regular polytopes in ordinary space. Discrete Comput. Geom. 17, 449–478 (1997)
McMullen, P., Schulte, E.: Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications, vol. 92. Cambridge University Press, Cambridge (2002)
Pellicer, D., Schulte, E.: Regular polygonal complexes in space, I. Trans. Am. Math. Soc. (to appear)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
McMullen, P. Regular Polytopes of Nearly Full Rank. Discrete Comput Geom 46, 660–703 (2011). https://doi.org/10.1007/s00454-011-9335-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-011-9335-y