Platonic solids, Archimedean solids and semi-equivelar maps on the sphere

Datta, B and Maity, D (2022) Platonic solids, Archimedean solids and semi-equivelar maps on the sphere. In: Discrete Mathematics, 345 (1).

PDF dis_mathematics_345_2022 - Published Version Download (729kB)

Abstract

A map X on a surface is called vertex-transitive if the automorphism group of X acts transitively on the set of vertices of X. A map is called semi-equivelar if the cyclic arrangement of faces around each vertex is same. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar maps on the torus, the Klein bottle and other surfaces which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and anti-prisms are vertex-transitive maps on S2. Here we show that there is exactly one semi-equivelar map on S2 which is not vertex-transitive. As a consequence, we show that all the semi-equivelar maps on RP2 are vertex-transitive. Moreover, every semi-equivelar map on S2 can be geometrized, i.e., every semi-equivelar map on S2 is isomorphic to a semi-regular tiling of S2. In the course of the proof of our main result, we present a combinatorial characterisation in terms of an inequality of all the types of semi-equivelar maps on S2. Here we present combinatorial proofs of all the results. © 2021 Elsevier B.V.

Item Type:	Journal Article
Publication:	Discrete Mathematics
Publisher:	Elsevier B.V.
Additional Information:	The copyright for this article belongs to Elsevier B.V.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	28 Nov 2021 09:53
Last Modified:	28 Nov 2021 09:53
URI:	http://eprints.iisc.ac.in/id/eprint/70328

Actions (login required)

View Item