Calculations on face and vertex regular polyhedra and application to finite element analysis

Abstract

Polyhedron is a solid figure bounded by plane faces. Face and vertex regular polyhedra are the polyhedra whose faces are regular polygons and the arrangement of polygons around each vertex is identical. Here general equations to calculate the properties of the face and vertex regular polyhedra are developed. This includes equations for radius of the escribed sphere and internal solid angle of a vertex. Using these equations the radius of the escribed sphere of face and vertex regular polyhedrda are found including that of Snub Cube and Snub Dodecahedron. It is also shown that sphere is a limiting case of a polyhedron. As application to finite element analysis, approximating the boundary by the sides of the finite elements is proposed. Also a method of defining the Lagrange interpolating polynomial is proposed. 2D tessellations are filling of infinite plane using polygons and 3D tessellations are filling of infinite space using polyhedra. With the piecewise polynomial selected in the above manner it is shown that the only possible regular tessellations that can be used in finite elements are Equilateral Triangle and Square in 2D and Triangular Regular Prism and Cube in 3D. It is shown in general that "any polygon having two axis of symmetry with nodes are selected at vertices cannot be used as a finite element i f its Lagrange polynomial contains the complete polynomial of degree two" and "any polyhedron having a polygonal face with two axis of symmetry and having six or more number of vertices with the nodes are selected at vertices cannot be used as a finite element i f its Lagrange polynomial contains a two variable complete polynomial of degree two".