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Thompson B, Hyde ST. A Theoretical Schema for Building Weavings of Nets via Colored Tilings of Two-Dimensional Spaces and Some Simple Polyhedral, Planar and Three-Periodic Examples. Isr J Chem 2018. [DOI: 10.1002/ijch.201800121] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Affiliation(s)
- Benjamin Thompson
- Dept of Applied Mathematics, Research School of Physical Sciences; Australian National University; Canberra Australia
| | - Stephen T. Hyde
- Dept of Applied Mathematics, Research School of Physical Sciences; Australian National University; Canberra Australia
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Brinkmann G, Goetschalckx P, Schein S. Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations. Proc Math Phys Eng Sci 2017. [DOI: 10.1098/rspa.2017.0267] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The use of operations on polyhedra possibly dates back to the ancient Greeks, who were the first to describe the Archimedean solids, which can be constructed from the Platonic solids by local symmetry-preserving operations (e.g. truncation) on the solid. By contrast, the results of decorations of polyhedra, e.g. by Islamic artists and by Escher, have been interpreted as decorated polyhedra—and not as new and different polyhedra. Only by interpreting decorations as combinatorial operations does it become clear how closely these two approaches are connected. In this article, we first sketch and compare the operations of Goldberg, Fuller, Caspar & Klug and Coxeter to construct polyhedra with icosahedral symmetry, where all faces are pentagons or hexagons and all vertices have three neighbours. We point out and correct an error in Goldberg’s construction. In addition, we transform the term
symmetry-preserving
into an exact requirement. This goal, symmetry-preserving, could also be obtained by taking global properties into account, e.g. the symmetry group itself, so we make precise the terms
local
and
operation
. As a result, we can generalize Goldberg’s approach to a systematic one that uses
chamber operations
to encompass all local symmetry-preserving operations on polyhedra.
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Affiliation(s)
- Gunnar Brinkmann
- Applied Mathematics, Computer Science and Statistics, Ghent University Krijgslaan 281-S9, 9000 Ghent, Belgium
| | - Pieter Goetschalckx
- Applied Mathematics, Computer Science and Statistics, Ghent University Krijgslaan 281-S9, 9000 Ghent, Belgium
| | - Stan Schein
- California Nanosystems Institute and Department of Psychology, MC 951563, University of California, Los Angeles, CA 90095, USA
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Kovács F. Twofold orthogonal weavings on cuboids. Proc Math Phys Eng Sci 2016; 472:20150576. [DOI: 10.1098/rspa.2015.0576] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Some closed polyhedral surfaces can be completely covered by two-way, twofold (rectangular) weaving of strands of constant width. In this paper, a construction for producing all possible geometries for such weavable cuboids is proposed: a theorem on spherical octahedra is proven first that all further theory is based on. The construction method of weavable cuboids itself relies on successive truncations of an initial tetrahedron and is also extended for cases of degenerate (unbounded) polyhedra. Arguments are mainly based on the plane geometry of the development of the respective polyhedra, in connection with some of three-dimensional projective properties of the same.
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