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Open AccessArticle

Critically-Finite Dynamics on the Icosahedron

Department of Mathematics and Statistics, California State University, Long Beach, CA 90840, USA
Symmetry 2020, 12(1), 177; https://doi.org/10.3390/sym12010177
Received: 11 December 2019 / Revised: 10 January 2020 / Accepted: 16 January 2020 / Published: 19 January 2020
(This article belongs to the Special Issue Symmetry and Dynamical Systems)
A recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how to build such “dynamical polyhedra” for other icosahedral maps. First, algebra associated with the icosahedron determines a special family of maps with 60 periodic critical points. The topological behavior of each map is then worked out and results in a geometric algorithm out of which emerges a system of edges—the dynamical polyhedron—in natural correspondence to a map’s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps’ geometric behavior fall into combinatorial classes the presentation of which concludes the paper. View Full-Text
Keywords: icosahedron; dynamics; equivariant icosahedron; dynamics; equivariant
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MDPI and ACS Style

Crass, S. Critically-Finite Dynamics on the Icosahedron. Symmetry 2020, 12, 177. https://doi.org/10.3390/sym12010177

AMA Style

Crass S. Critically-Finite Dynamics on the Icosahedron. Symmetry. 2020; 12(1):177. https://doi.org/10.3390/sym12010177

Chicago/Turabian Style

Crass, Scott. 2020. "Critically-Finite Dynamics on the Icosahedron" Symmetry 12, no. 1: 177. https://doi.org/10.3390/sym12010177

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