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

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Open AccessArticle
Topological Invariance under Line Graph Transformations
Symmetry 2012, 4(2), 329-335; https://doi.org/10.3390/sym4020329 - 08 Jun 2012
Viewed by 3276
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 L [...] 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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Open AccessArticle
Knots in Art
Symmetry 2012, 4(2), 302-328; https://doi.org/10.3390/sym4020302 - 05 Jun 2012
Cited by 1 | Viewed by 3502
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 Tamil [...] 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)
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Open AccessArticle
Diagrammatics in Art and Mathematics
Symmetry 2012, 4(2), 285-301; https://doi.org/10.3390/sym4020285 - 22 May 2012
Viewed by 4375
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 mathematical [...] 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)
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Open AccessArticle
Following Knots down Their Energy Gradients
Symmetry 2012, 4(2), 276-284; https://doi.org/10.3390/sym4020276 - 27 Apr 2012
Cited by 3 | Viewed by 2553
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)
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Open AccessArticle
Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators
Symmetry 2012, 4(2), 265-275; https://doi.org/10.3390/sym4020265 - 16 Apr 2012
Cited by 10 | Viewed by 3078
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) Printed Edition available
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