64,839 research outputs found
The orbit rigidity matrix of a symmetric framework
A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks. The orbit matrix is a true
analog of the standard rigidity matrix for general frameworks, and its analysis
gives important insights into questions about the flexibility and rigidity of
classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the
power to predict symmetry-preserving finite mechanisms - giving a simplified
analysis which covers a wide range of the known mechanisms, and generalizes the
classes of known mechanisms. This initial exploration of the properties of the
orbit matrix also opens up a number of new questions and possible extensions of
the previous results, including transfer of symmetry based results from
Euclidean space to spherical, hyperbolic, and some other metrics with shared
symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure
Symmetry adapted Assur decompositions
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.Comment: 40 pages, 22 figure
Finite motions from periodic frameworks with added symmetry
Recent work from authors across disciplines has made substantial
contributions to counting rules (Maxwell type theorems) which predict when an
infinite periodic structure would be rigid or flexible while preserving the
periodic pattern, as an engineering type framework, or equivalently, as an
idealized molecular framework. Other work has shown that for finite frameworks,
introducing symmetry modifies the previous general counts, and under some
circumstances this symmetrized Maxwell type count can predict added finite
flexibility in the structure.
In this paper we combine these approaches to present new Maxwell type counts
for the columns and rows of a modified orbit matrix for structures that have
both a periodic structure and additional symmetry within the periodic cells. In
a number of cases, this count for the combined group of symmetry operations
demonstrates there is added finite flexibility in what would have been rigid
when realized without the symmetry. Given that many crystal structures have
these added symmetries, and that their flexibility may be key to their physical
and chemical properties, we present a summary of the results as a way to
generate further developments of both a practical and theoretic interest.Comment: 45 pages, 13 figure
Rigidity and flexibility of biological networks
The network approach became a widely used tool to understand the behaviour of
complex systems in the last decade. We start from a short description of
structural rigidity theory. A detailed account on the combinatorial rigidity
analysis of protein structures, as well as local flexibility measures of
proteins and their applications in explaining allostery and thermostability is
given. We also briefly discuss the network aspects of cytoskeletal tensegrity.
Finally, we show the importance of the balance between functional flexibility
and rigidity in protein-protein interaction, metabolic, gene regulatory and
neuronal networks. Our summary raises the possibility that the concepts of
flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl
Linking Rigid Bodies Symmetrically
The mathematical theory of rigidity of body-bar and body-hinge frameworks
provides a useful tool for analyzing the rigidity and flexibility of many
articulated structures appearing in engineering, robotics and biochemistry. In
this paper we develop a symmetric extension of this theory which permits a
rigidity analysis of body-bar and body-hinge structures with point group
symmetries. The infinitesimal rigidity of body-bar frameworks can naturally be
formulated in the language of the exterior (or Grassmann) algebra. Using this
algebraic formulation, we derive symmetry-adapted rigidity matrices to analyze
the infinitesimal rigidity of body-bar frameworks with Abelian point group
symmetries in an arbitrary dimension. In particular, from the patterns of these
new matrices, we derive combinatorial characterizations of infinitesimally
rigid body-bar frameworks which are generic with respect to a point group of
the form .
Our characterizations are given in terms of packings of bases of signed-graphic
matroids on quotient graphs. Finally, we also extend our methods and results to
body-hinge frameworks with Abelian point group symmetries in an arbitrary
dimension. As special cases of these results, we obtain combinatorial
characterizations of infinitesimally rigid body-hinge frameworks with
or symmetry - the most common symmetry groups
found in proteins.Comment: arXiv:1308.6380 version 1 was split into two papers. The version 2 of
arXiv:1308.6380 consists of Sections 1 - 6 of the version 1. This paper is
based on the second part of the version 1 (Sections 7 and 8
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
Frameworks, Symmetry and Rigidity
Symmetry equations are obtained for the rigidity matrix of a bar-joint
framework in R^d. These form the basis for a short proof of the Fowler-Guest
symmetry group generalisation of the Calladine-Maxwell counting rules. Similar
symmetry equations are obtained for the Jacobian of diverse framework systems,
including constrained point-line systems that appear in CAD, body-pin
frameworks, hybrid systems of distance constrained objects and infinite
bar-joint frameworks. This leads to generalised forms of the Fowler-Guest
character formula together with counting rules in terms of counts of
symmetry-fixed elements. Necessary conditions for isostaticity are obtained for
asymmetric frameworks, both when symmetries are present in subframeworks and
when symmetries occur in partition-derived frameworks.Comment: 5 Figures. Replaces Dec. 2008 version. To appear in IJCG
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