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Symmetry
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Rigidity and Symmetry
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Rigidity and Symmetry

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 March 2015) | Viewed by 46766

Special Issue Editor

Dr. Bernd Schulze

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, UK
Interests: discrete geometry; rigidity theory; symmetry; triangulations; polytope theory; graph theory; matroids; combinatorics; algebraic methods; representation theory; computational geometry; interdisciplinarity; rigidity and flexibility of symmetric framework systems; “equal-area-triangulation problems”

Special Issue Information

Dear Colleagues,

The mathematical theory of “rigidity” investigates the rigidity and flexibility of structures which are defined by geometric constraints (fixed lengths, fixed areas, fixed directions, etc.) on a set of rigid objects (points, lines, polygons, etc.). This theory has both combinatorial and geometric aspects and draws on techniques from a wide range of mathematical areas, including graph theory, matroid theory, positive semidefinite programming, representation theory of symmetry groups, algebraic and projective geometry, and even operator theory.
Since the rigidity and flexibility properties of a structure—either man-made, such as a building, bridge or mechanical linkage, or found in nature, such as a biomolecule, protein or crystal—are critical to its form, behavior and functioning, rigidity theory has many practical applications in fields such as engineering, robotics, computer-aided design, materials science, medicine and biochemistry.
Since symmetry appears widely in both natural and man-made structures, the study of structures with additional symmetry—both finite structures with point group symmetry and infinite structures with periodic or crystallographic symmetry—is an important area of research which has seen a dramatic increase in interest over the last few years, both in mathematics and in the applied sciences.
The aim of this special issue of “Symmetry” is to present a broad spectrum of the latest developments and current research concerning the rigidity and flexibility of symmetric geometric constraint systems, and to foster the exchange of ideas among workers who focus on different aspects and applications of the field.

Dr. Bernd Schulze
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • rigidity of frameworks
  • geometric constraint systems
  • flexibility
  • mechanisms
  • linkages
  • symmetric frameworks
  • point group symmetry
  • periodic frameworks
  • crystallographic symmetry
  • molecular structures
  • pebble game algorithms
  • gain graphs
  • packings

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Published Papers (7 papers)

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Research

26 pages, 913 KiB  
Article
Symmetry in Sphere-Based Assembly Configuration Spaces
by Meera Sitharam, Andrew Vince, Menghan Wang and Miklós Bóna
Symmetry 2016, 8(1), 5; https://doi.org/10.3390/sym8010005 - 21 Jan 2016
Cited by 4 | Viewed by 5682
Abstract
Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring in nature can be modeled geometrically, starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and [...] Read more.
Many remarkably robust, rapid and spontaneous self-assembly phenomena occurring in nature can be modeled geometrically, starting from a collection of rigid bunches of spheres. This paper highlights the role of symmetry in sphere-based assembly processes. Since spheres within bunches could be identical and bunches could be identical, as well, the underlying symmetry groups could be of large order that grows with the number of participating spheres and bunches. Thus, understanding symmetries and associated isomorphism classes of microstates that correspond to various types of macrostates can significantly increase efficiency and accuracy, i.e., reduce the notorious complexity of computing entropy and free energy, as well as paths and kinetics, in high dimensional configuration spaces. In addition, a precise understanding of symmetries is crucial for giving provable guarantees of algorithmic accuracy and efficiency, as well as accuracy vs. efficiency trade-offs in such computations. In particular, this may aid in predicting crucial assembly-driving interactions. This is a primarily expository paper that develops a novel, original framework for dealing with symmetries in configuration spaces of assembling spheres, with the following goals. (1) We give new, formal definitions of various concepts relevant to the sphere-based assembly setting that occur in previous work and, in turn, formal definitions of their relevant symmetry groups leading to the main theorem concerning their symmetries. These previously-developed concepts include, for example: (i) assembly configuration spaces; (ii) stratification of assembly configuration space into configurational regions defined by active constraint graphs; (iii) paths through the configurational regions; and (iv) coarse assembly pathways. (2) We then demonstrate the new symmetry concepts to compute the sizes and numbers of orbits in two example settings appearing in previous work. (3) Finally, we give formal statements of a variety of open problems and challenges using the new conceptual definitions. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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14 pages, 1686 KiB  
Article
Characterizations of Network Structures Using Eigenmode Analysis
by Youngho Park and Sangil Hyun
Symmetry 2015, 7(2), 962-975; https://doi.org/10.3390/sym7020962 - 3 Jun 2015
Cited by 3 | Viewed by 5631
Abstract
We introduced an analysis to identify structural characterization of two-dimensional regular and amorphous networks. The analysis was shown to be reliable to determine the global network rigidity and can also identify local floppy regions in the mixture of rigid and floppy regions. The [...] Read more.
We introduced an analysis to identify structural characterization of two-dimensional regular and amorphous networks. The analysis was shown to be reliable to determine the global network rigidity and can also identify local floppy regions in the mixture of rigid and floppy regions. The eigenmode analysis explores the structural properties of various networks determined by eigenvalue spectra. It is useful to determine the general structural stability of networks that the traditional Maxwell counting scheme based on the statistics of nodes (degrees of freedom) and bonds (constraints) does not provide. A visual characterization scheme was introduced to examine the local structure characterization of the networks. The eigenmode analysis is under development for various practical applications on more general network structures characterized by coordination numbers and nodal connectivity such as graphenes and proteins. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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14 pages, 2606 KiB  
Article
Flexible Polyhedral Surfaces with Two Flat Poses
by Hellmuth Stachel
Symmetry 2015, 7(2), 774-787; https://doi.org/10.3390/sym7020774 - 27 May 2015
Cited by 5 | Viewed by 6703
Abstract
We present three types of polyhedral surfaces, which are continuously flexible and have not only an initial pose, where all faces are coplanar, but pass during their self-motion through another pose with coplanar faces (“flat pose”). These surfaces are examples of so-called rigid [...] Read more.
We present three types of polyhedral surfaces, which are continuously flexible and have not only an initial pose, where all faces are coplanar, but pass during their self-motion through another pose with coplanar faces (“flat pose”). These surfaces are examples of so-called rigid origami, since we only admit exact flexions, i.e., each face remains rigid during the motion; only the dihedral angles vary. We analyze the geometry behind Miura-ori and address Kokotsakis’ example of a flexible tessellation with the particular case of a cyclic quadrangle. Finally, we recall Bricard’s octahedra of Type 3 and their relation to strophoids. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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21 pages, 460 KiB  
Article
On the Self-Mobility of Point-Symmetric Hexapods
by Georg Nawratil
Symmetry 2014, 6(4), 954-974; https://doi.org/10.3390/sym6040954 - 18 Nov 2014
Cited by 3 | Viewed by 6128
Abstract
In this article, we study necessary and sufficient conditions for the self-mobility of point symmetric hexapods (PSHs). Specifically, we investigate orthogonal PSHs and equiform PSHs. For the latter ones, we can show that they can have non-translational self-motions only if they are architecturally [...] Read more.
In this article, we study necessary and sufficient conditions for the self-mobility of point symmetric hexapods (PSHs). Specifically, we investigate orthogonal PSHs and equiform PSHs. For the latter ones, we can show that they can have non-translational self-motions only if they are architecturally singular or congruent. In the case of congruency, we are even able to classify all types of existing self-motions. Finally, we determine a new set of PSHs, which have so-called generalized Dietmaier self-motions. We close the paper with some comments on the self-mobility of hexapods with global/local symmetries. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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35 pages, 1266 KiB  
Article
Symmetry Adapted Assur Decompositions
by Anthony Nixon, Bernd Schulze, Adnan Sljoka and Walter Whiteley
Symmetry 2014, 6(3), 516-550; https://doi.org/10.3390/sym6030516 - 27 Jun 2014
Cited by 5 | Viewed by 7525
Abstract
Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, [...] Read more.
Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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15 pages, 290 KiB  
Article
Symmetry Perspectives on Some Auxetic Body-Bar Frameworks
by Patrick W. Fowler, Simon D. Guest and Tibor Tarnai
Symmetry 2014, 6(2), 368-382; https://doi.org/10.3390/sym6020368 - 15 May 2014
Cited by 8 | Viewed by 6930
Abstract
Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied [...] Read more.
Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied to two long known but apparently little studied hinged-hexagon frameworks, and is shown to detect auxetic behaviour in both. In contrast, for double-link frameworks based on triangular and square tessellations, other affine deformations can mix with the isotropic expansion mode. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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21 pages, 617 KiB  
Article
The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks
by Ghada Badri, Derek Kitson and Stephen C. Power
Symmetry 2014, 6(2), 308-328; https://doi.org/10.3390/sym6020308 - 24 Apr 2014
Cited by 13 | Viewed by 6765
Abstract
A crystallographic bar-joint framework, C in Rd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω (C), is a singleton. Moreover, the [...] Read more.
A crystallographic bar-joint framework, C in Rd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω (C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, Td, determined by a full rank translation symmetry group and an associated motif of joints and bars. Full article
(This article belongs to the Special Issue Rigidity and Symmetry)
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