Polytopes vibrations within Coxeter group symmetries

Abstract

Abstract.We are considering polytopes with exact reflection symmetry group G in the real 3-dimensional Euclidean space R 3 . By changing one simple element of the polytope (position of one vertex or length of an edge), one can retain the exact symmetry of the polytope by simultaneously changing other corresponding elements of the polytope. A simple method of using the symmetry of polytopes in order to determine several resonant frequencies is presented. Knowledge of these frequencies, or at least their ratios can be used for control of some principal changes of the polytopes.

Harmonic Oscillator with a Step and/or a Ramp

“…Remark. In this paper, we restrict ourselves to a > 0 in the potential (12). Note that unlike the case of a harmonic oscillator with a step (section 2), the constraint on a breaks the generality.…”

Geometrical structures of nested polyhedra

“…The polytopes which differ by a value of the positive coordinates of dominant points but have the same symmetry group can be obtained from one another by various deformations, for example contraction, vibrations etc. (see for example [1,3,14]). The number of vertices and faces of such polytopes does not change during the deformation, which preserve the symmetry group of the polytope, although the faces of dimension greater than 1 can have different size.…”

“…Polyhedra have been known since antiquity. They are ubiquitous in architecture, biology, chemistry, mathematics or physics [3,6,15].…”

Polytope Contractions within Weyl Group Symmetries

“…Exact coordinates of the families of polytopes with dominant point (a, b, c) of the groups A 3 , B 3 For the A 3 vertices, we have…”

“…During that process the overall symmetry of the polytope does not change. If we let the coordinates of one vertex vibrate, then the coordinates of the rest of the vertices have to vibrate in opposite phase to preserve the radius of the polytope [3].…”