Pentagonal maps on the torus and the plane

Maity, D (2019) Pentagonal maps on the torus and the plane. In: Beitrage zur Algebra und Geometrie, 60 (1). pp. 17-37.

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Abstract

We know pentagonal tilings of the plane where vertex degree sequence (vds) of faces are same. There are three such tilings, namely, Prismatic pentagonal tiling (by type 1 pentagons of vds 3 3 4 2 ), Cairo pentagonal tiling (by type 2 pentagons of vds 3 2 4 1 3 1 4 1 ), and Floret pentagonal tiling (by type 3 pentagons of vds 3 4 6 1 ) (https://en.wikipedia.org/wiki/Pentagonaltiling). In this article, we are interested on the class of pentagonal tilings of the plane whose faces are pentagons of types 1, 2 and 3. There are infinitely many tilings of the plane by pentagons of types 1 and 2. We show that there are exactly six pentagonal tilings on the plane, including above three, by the faces of types 1, 2 and 3, where the face cycles of the degree four vertices are of same type. We also consider maps on the torus which are quotient of above type of tilings on the plane by latices. We have found bounds of the number of orbits of faces under the automorphism group of the maps. We show that our bounds on the number of orbits are sharp. © 2018, The Managing Editors.

Item Type:	Journal Article
Publication:	Beitrage zur Algebra und Geometrie
Publisher:	Springer Verlag
Additional Information:	The copyright for this article belongs to Springer Verlag
Keywords:	Face-transitive map; Maps on torus; Pentagonal tiling; Polyhedral map
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	13 Dec 2022 04:17
Last Modified:	13 Dec 2022 04:17
URI:	https://eprints.iisc.ac.in/id/eprint/78298

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