Polyhedra in Physics, Chemistry and Geometry

  title={Polyhedra in Physics, Chemistry and Geometry
  author={Michael Francis Atiyah and Paul Sutcliffe},
  journal={Milan Journal of Mathematics},
In this article we review some problems in physics, chemistry and mathematics that lead naturally to a class of polyhedra which include the Platonic solids. Examples include the study of electrons on a sphere, cages of carbon atoms, central configurations of gravitating point particles, rare gas microclusters, soliton models of nuclei, magnetic monopole scattering and geometrical problems concerning point particles. 
Geometrical Problems Related to Crystals, Fullerenes, and Nanoparticle Structure
This paper focuses on three groups of geometrical problems, closely related to material sciences in general and particularly to crystal/quasicrystal structures along with their formations andExpand
Two Groups of Geometrical Problems Related to Study of Fullerenes & Crystals
The paper focuses on two groups of geometrical problems closely related to formations of crystal/quasi-crystal structures, and fullerenes. In section one we discuss a minimum radius of local identityExpand
The Relativistic Geometry and Dynamics of Electrons
Atiyah and Sutcliffe (Proc R Soc Lond Ser A 458:1089–1115, 2002) made a number of conjectures about configurations of N distinct points in hyperbolic 3-space, arising from ideas of Berry and RobbinsExpand
Colloidal spheres confined by liquid droplets: Geometry, physics, and physical chemistry
I discuss how colloidal particles organize when they are confined by emulsion droplets. In these systems, the interplay between surface tension and interparticle repulsion drives the formation ofExpand
A chemical synthetic route towards "colloidal molecules".
Current understanding of various physical phenomena and capability to fabricate new functional materials have been considerably enriched by the development of synthetic strategies that are capable of generating copious quantities of colloidal entities of good size uniformity. Expand
Dipole-dipole minimum energy configuration for Platonic, Archimedean and Catalan solid structures
Abstract Several magnetic materials consisting of dipoles owe their properties to the specific nature of the dipole-dipole interaction. In the present work, we study systems of dipoles where theExpand
Anyons in geometric models of matter
A bstractWe show that the “geometric models of matter” approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, usingExpand
Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations
It is well-known that Klein's lectures on the icosahedron and the solution of equations of fifth degree is one of the most important and influential books of 19th-century mathematics. In the presentExpand
Polyhedra obtained from Coxeter groups and quaternions
We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtainedExpand
Platonic Solids and High Genus Covers of Lattice Surfaces
We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associatedExpand


An Atlas of Fullerenes
Introduction. 1: Fullerene cages. 2: Electronic structure. 3: Steric strain. 4: Symmetry and spectroscopy. 5: Fullerene isomerisation. 6: Carbon gain and loss. Appendix: The spiral computer program.Expand
Electrons on the Sphere