Mapification of n-dimensional abstract polytopes and hypertopes

Antonio Breda d'Azevedo

Ars mathematica contemporanea, Volume 18 (2020) no. 1, pp. 73-86

Abstract

The n-dimensional abstract polytopes and hypertopes, particularly the regular ones, have gained great popularity over recent years. The main focus of research has been their symmetries and regularity. The planification of a polyhedron helps its spatial construction, yet it destroys symmetries. No “planification” of n-dimensional polytopes do exist, however it is possible to make a “mapification” of an n-dimensional polytope; in other words it is possible to construct a restrictedly-marked map representation of an abstract polytope on some surface that describes its combinatorial structures as well as all of its symmetries. There are infinitely many ways to do this, yet there is one that is more natural that describes reflections on the sides of (n − 1)-simplices (flags or n-flags) with reflections on the sides of n-gons. The restrictedly-marked map representation of an abstract polytope is a cellular embedding of the flag graph of a polytope. We illustrate this construction with the 4-cube, a regular 4-polytope with automorphism group of size 384. This paper pays a tribute to Lynne James’ last work on map representations.

Article information

DOI: 10.26493/1855-3974.1869.9f6

Keywords: Maps, hypermaps, representation, polytopes, hypertopes