3-Isogonal Planar Tilings are not 3-Isogonal on the Torus
DOI:
https://doi.org/10.18311/jims/2023/29802Keywords:
Covering Maps, Isogonal Maps, Symmetric Group.Abstract
A 3-isogonal tiling is an edge-to-edge tiling by regular polygons having 3 distinct transitivity classes of vertices. We know that there are sixty-one distinct 3-isogonal tilings on the plane. In this article, we discuss and determine the bounds of the vertex orbits of the plane’s 3-isogonal lattices on the torus and will show that these bounds are sharp.
Downloads
Metrics
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Marbarisha M. Kharkongor, Debashis Bhowmik, Dipendu Maity
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2022-07-28
Published 2023-07-12
References
D. Bhowmik and A. K. Upadhyay, Some semi-equivelar maps of Euler characteristics−2, Nat. Acad. Sci. Lett., 44 (2021), 433-436.
D. Bhowmik and A. K. Upadhyay, A classification of semi-equivelar maps on the surface of Euler characteristic -1, Indian J. Pure Appl. Math., 52 (2021), 289-296.
B. Datta, Vertex-transitive covers of semi-equivelar toroidal maps, https://arxiv.org/abs/2004.09953.
B. Datta and D. Maity, Semi-equivelar and vertex-transitive maps on the torus, Beitr¨age Algebra Geom., 58 (2017), 617-634.
B. Datta and D. Maity, Semi-equivelar maps on the torus and the Klein bottle are Archimedean, Discrete Math., 341 (12) (2018), 3296-3309.
B. Gr¨unbaum and G. C. Shephard, Tilings by regular polygons: Patterns in the plane from Kepler to the present, including recent results and unsolved problems, Math. Mag., 50 (1977), 227-247.
B. Gr¨unbaum and G. C. Shephard, The geometry of planar graphs. Combinatorics (Swansea), London Math. Soc. LNS, Cambridge Univ. Press, Cambridge, 52 (1981), 124-150.
M. M. Kharkongor, D. Bhowmik and D. Maity, Quotient maps of 2,3-uniform tilings of the plane on the torus, https://arxiv.org/abs/2101.04373.
O. Kr¨otenheerdt, Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene. I, II, III, Wiss. Z. Martin-LUther-Univ. Halle-Wittenberg Math.-Natur. Reihe, 18,19 (1969, 1970), 18:273-290, 19:19-38, 97-241.
E. H. Spanier, Algebraic Topology, Springer-Verlag, New York, 1966.