Polytope contractions within icosahedral symmetry

Abstract

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 coordinates of the dominant point -which specifies the orbit in an appropriate basis -toward smaller positive values. Exact icosahedral symmetry is maintained at any stage of the contraction. All icosahedral orbits or polytopes can be built by successive contractions. This definition of contraction is general and can be applied to orbits of any finite reflection group. PACS Nos.: 61.05.-a, 61.18.-j, 61.48.-c.Résumé : La symétrie icosaédrique est omniprésente dans la nature et la compréhension des déformations possibles des structures est critique pour déterminer les propriétés fondamentales. Nous présentons ici un cadre de travail pour générer et représenter les déformations de ce type de structures, tout en préservant la symétrie icosaédrique. Ceci est fait en prenant des points d'une orbite du groupe icosaèdre comme des sommets d'un polytope icosaédrique. La contraction de l'orbite est définie par des variations continues des coordonnées du point dominant -ceci positionne l'orbite dans une base appropriée -vers de plus petites valeurs positives. Nous maintenons la symétrie icosaédrique à toutes les étapes de la contraction. Toutes les orbites/polytopes icosaédriques peuvent être construites par contractions successives. Cette définition de la contraction est générale et peut être appliquée aux orbites de n'importe quel groupe de réflexion fini. [Traduit par la Rédaction]

Polytope Contractions within Weyl Group Symmetries

“…A motivation for the polytope contractions can be found in [1,2,9]. The analogy with the definition of contractions of Lie algebras established some 60 years ago [7], is in itself a motivation for our present undertaking.…”

“…Although the method is quite general, in terms of the dimension of the space and the reflection group symmetry, it has been used specifically for polytopes with icosahedral symmetry only [2]. More precisely, a polytope is being deformed into another polytope by the contraction process, while the icosahedral symmetry is preserved at all stages of the process.…”

“…Contractions of reflection generated polytopes in 3D real Euclidean space R 3 were introduced in [2]. Although the method is quite general, in terms of the dimension of the space and the reflection group symmetry, it has been used specifically for polytopes with icosahedral symmetry only [2].…”

Polytopes vibrations within Coxeter group symmetries

“…We will use Hooke's Law approximations for isotropic materials, which holds at least for small amplitudes so that the atomic bond forces can be described spring forces. Using the polytope structures described in [4,5], we discuss below three reasons to analyze the dynamical properties of polytopes.…”

“…The description of the transition between some polytopes was shown in [4,5] using several parameters. This enables us to find the vibration modes connected to these parameters.…”