Geodesic Grid - Parameters

Parameters

The simplest tessellation of the icosahedron is characterized by the following parameters.

Let r_u be the radius of the circumscribed sphere and a₁ be the edge length of the icosahedron. From a view point at the center of the sphere, each edge appears under an angle γ₁,

By splitting each edge into s line segments of length a₁/s, and by projection of the intermediate points back onto the sphere, each triangle is split into s2 smaller triangles, with associated viewing angles

and edge lengths a_s

(The out-projection lets a_s become larger than a₁/s.) The edges that are not generated by out-projection of one of the 30 edges of the icosahedron but from points inside the triangular mesh of any of the 20 surfaces have different lengths. That is, the triangles are generally not isosceles if s>1.

The solid contains 20s2 tetrahedra with apexes at the sphere center, with base surface areas A_s,t and orthogonal distance r_i,s,t to the center of the sphere, t=1,...,s2.

The total volume in all tetrahedra is

The volume filling factor

relative to the volume of the circumscribed sphere approaches 1 as s grows to infinity.

The following table collects numerical values of these parameters, assuming that the (s+1)(s+2)/2 points of a regular flat triangular grid on each of the 20 surfaces are projected radially onto the circum-sphere: