Academic literature on the topic 'Snub dodecahedron'

The article is devoted to modeling and visualization of the formation of flat-nosed (snub-nosed) dodecahedron (snub dodecahedron). The purpose of the research is to model the snub dodecahedron (flat-nosed dodecahedron) and visualize the process of its formation. The formation of the faces of the flat-nosed dodecahedron consists in the truncation of the edges and vertices of the Platonic dodecahedron with the subsequent rotation of the new faces around their centers. The values of the truncation of the dodecahedron edges, the angle of rotation of the faces and the length of the edge of the flat-nosed dodecahedron are the parameters of three equations composed as the distances between the vertices of triangles located between the faces of the snub dodecahedron. The solution of these equations was carried out by the method of successive approximations. The results of the calculations were used to create an electronic model of the flat-nosed dodecahedron and visualize its formation. The task was generally achieved in the AutoCAD system using programs in the AutoLISP language. Software has been created for calculating the parameters of modeling a snub dodecahedron and visualizing its formation.

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.

This paper describes a new application of polyhedral theory to the growth of the outer sheath of certain viruses. Such structures are often modular, consisting of one or two types of units arranged in a symmetric pattern. In particular, the polyoma virus has a structure apparently related to the snub dodecahedron. Here, we consider the problem of how such patterns might grow in time, starting from a given number N of randomly placed circles on the surface of a sphere. The circles are first jostled by random perturbations, then their radii are enlarged, then they are jostled again, and so on. This ‘yin–yang’ method of growth can result in some very close packings. When N =12, the closest packing corresponds to the snub tetrahedron, and when N =24 the closest packing corresponds to the snub cube. However, when N =60 the closest packing does not correspond to the snub dodecahedron but to a less-symmetric arrangement. Special attention is given to the structure of the human polyoma virus, for which N =72. It is shown that the yin–yang procedure successfully assembles the observed structure provided that the 72 circles are pre-assembled in clusters of six. Each cluster consists of five circles arranged symmetrically around a sixth at the centre, as in a flower with five petals. This has implications for the assembly of the capsomeres in a polyoma virus.

Closed-form solution to the Snub Dodecahedron by Mark S. Adams and other solutions from H. S. M. Coxeter, Eric W. Weisstein, and Harish C. Rajpoot. Jupyter Notebook format available. Python script calculates the volume of the Snub Dodecahedron using five different methods.

The control over the self-assembly of complex structures is a long-standing challenge of material science, especially at the colloidal scale, as the desired assembly pathway is often kinetically derailed by the formation of amorphous aggregates. Here, we investigate in detail the problem of the self-assembly of the three Archimedean shells with five contact points per vertex, i.e., the icosahedron, the snub cube, and the snub dodecahedron. We use patchy particles with five interaction sites (or patches) as model for the building blocks and recast the assembly problem as a Boolean satisfiability problem (SAT) for the patch–patch interactions. This allows us to find effective designs for all targets and to selectively suppress unwanted structures. By tuning the geometrical arrangement and the specific interactions of the patches, we demonstrate that lowering the symmetry of the building blocks reduces the number of competing structures, which in turn can considerably increase the yield of the target structure. These results cement SAT-assembly as an invaluable tool to solve inverse design problems.