### 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".