Polytope contractions within icosahedral symmetry

  title={Polytope contractions within icosahedral symmetry},
  author={M. Bodner and J. Patera and M. Szajewska},
  journal={Canadian Journal of Physics},
Icosahedral symmetry is ubiquitous in nature, and understanding possible deformations of structures exhibiting it can be critical in determining fundamental properties. In this work we present a framework for generating and representing deformations of such structures while the icosahedral symmetry is preserved. This is done by viewing the points of an orbit of the icosahedral group as vertices of an icosahedral polytope. Contraction of the orbit is defined as a continuous variation of the… Expand
Polytopes vibrations within Coxeter group symmetries
Abstract We are considering polytopes with exact reflection symmetry group G in the real 3-dimensional Euclidean space R3. By changing one simple element of the polytope (position of one vertex orExpand
Polytope Contractions within Weyl Group Symmetries
A general scheme for constructing polytopes is implemented here specifically for the classes of the most important 3D polytopes, namely those whose vertices are labeled by integers relative to aExpand
Faces of root polytopes in all dimensions.
  • M. Szajewska
  • Mathematics, Medicine
  • Acta crystallographica. Section A, Foundations and advances
  • 2016
In this paper the root polytopes of all finite reflection groups W with a connected Coxeter-Dynkin diagram in {\bb R}^n are identified, their faces of dimensions 0 ≤ d ≤ n - 1 are counted, and theExpand
Reduction of orbits of finite Coxeter groups of non-crystallographic type
A reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices, which transform points of the orbit of any group into points of the orbitsExpand


The icosahedral point groups
Basic group-theoretical properties of the icosahedral point groups are derived. Here are given the permutations of the vertices of an icosahedron under the action of the elements of the icosahedralExpand
Mechanical deformation of spherical viruses with icosahedral symmetry.
The mechanical properties of crystalline shells of icosahedral symmetry on a substrate under a uniaxial applied force by computer simulations are studied to predict the elastic response for small deformations, and the buckling transitions at large deformations. Expand
The rings of n-dimensional polytopes
Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a realExpand
C70, C80, C90 and carbon nanotubes by breaking of the icosahedral symmetry of C60.
The icosahedral symmetry group H3 of order 120 and its dihedral subgroup H2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature and there is an uncountable number of different twisted fullerenes, all with precise icosahed symmetry. Expand
Self-assembly of regular hollow icosahedra in salt-free catanionic solutions
It is shown that in salt-free mixtures of anionic and cationic surfactants, such bilayers can self-assemble into hollow aggregates with a regular icosahedral shape, which are larger than any known icosahedra protein assembly or virus capsid. Expand
Mechanisms of plastic deformation of icosahedral quasicrystals
Abstract Icosahedral quasicrystals deform plastically by dislocation motion. The deformation behaviour can be modelled by simple equations of dislocation mobility, the damage of the quasicrystalExpand
On the Contraction of Groups and Their Representations.
  • E. Inonu, E. Wigner
  • Physics, Medicine
  • Proceedings of the National Academy of Sciences of the United States of America
  • 1953
The purpose of the present note is to investigate, in some generality, in which sense groups can be limiting cases of other groups, and how their representations can be obtained from the representations of the groups of which they appear as limits. Expand
Structural constraints on the three-dimensional geometry of simple viruses: case studies of a new predictive tool.
This new concept in virus biology provides for the first time predictive information on the structural constraints on coat protein and genome topography, and reveals a previously unrecognized structural interdependence of the shapes and sizes of different viral components. Expand
Discrete and continuous graded contractions of Lie algebras and superalgebras
Grading preserving contractions of Lie algebras and superalgebras of any type over the complex number field are defined and studied. Such contractions fall naturally into two classes: theExpand
Description of reflection-generated polytopes using decorated Coxeter diagrams
A new method of explicit description of n-dimensional polytopes generated by reflections of a single point (D-polytopes) or a face of maximal dimension (V-polytopes) is used to provide a comprehens...