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Symmetry, Volume 4, Issue 2 (June 2012), Pages 265-335

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Research

Open AccessArticle Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators
Symmetry 2012, 4(2), 265-275; doi:10.3390/sym4020265
Received: 29 January 2012 / Revised: 26 March 2012 / Accepted: 5 April 2012 / Published: 16 April 2012
Cited by 4 | PDF Full-text (262 KB)
Abstract This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group. Full article
(This article belongs to the Special Issue Polyhedra)
Open AccessArticle Following Knots down Their Energy Gradients
Symmetry 2012, 4(2), 276-284; doi:10.3390/sym4020276
Received: 3 February 2012 / Revised: 19 March 2012 / Accepted: 18 April 2012 / Published: 27 April 2012
Cited by 2 | PDF Full-text (325 KB) | HTML Full-text | XML Full-text
Abstract This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Diagrammatics in Art and Mathematics
Symmetry 2012, 4(2), 285-301; doi:10.3390/sym4020285
Received: 8 March 2012 / Revised: 25 April 2012 / Accepted: 28 April 2012 / Published: 22 May 2012
PDF Full-text (30730 KB)
Abstract
This paper explores two-way relations between visualizations in mathematics and mathematical art, as well as art in general. A collection of vignettes illustrates connection points, including visualizing higher dimensions, tessellations, knots and links, plotting zeros of polynomials, and new and rapidly developing [...] Read more.
This paper explores two-way relations between visualizations in mathematics and mathematical art, as well as art in general. A collection of vignettes illustrates connection points, including visualizing higher dimensions, tessellations, knots and links, plotting zeros of polynomials, and new and rapidly developing mathematical discipline, diagrammatic categorification. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Figures

Open AccessArticle Knots in Art
Symmetry 2012, 4(2), 302-328; doi:10.3390/sym4020302
Received: 9 May 2012 / Revised: 15 May 2012 / Accepted: 15 May 2012 / Published: 5 June 2012
Cited by 1 | PDF Full-text (5009 KB)
Abstract
We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or [...] Read more.
We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Topological Invariance under Line Graph Transformations
Symmetry 2012, 4(2), 329-335; doi:10.3390/sym4020329
Received: 6 April 2012 / Revised: 1 June 2012 / Accepted: 4 June 2012 / Published: 8 June 2012
PDF Full-text (177 KB) | HTML Full-text | XML Full-text
Abstract
It is shown that the line graph transformation G L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of [...] Read more.
It is shown that the line graph transformation G L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of L(G). As a consequence, the homology of K is isomorphic to that of G. This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of G as a line graph transformation invariant. Full article

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