Behaviour upon Iteration

Since the edges become shorter and shorter with each iteration, all three basic bodies Tetrahedron, Octahedron and Icosahedron will finally approximate the sphere. But Tetrahedron and Octahedron produce grids with relatively large differences of triangle size and thus an unevenly coverage of the sphere with nodes.

Four Iterations of the three basic bodies.

The Icosahedron grid is therefore best suited to generate a uniform distribution of nodes over the sphere. The number of nodes and the number of triangles roughly increase by a factor of 4 per iteration:

Number of Nodes and Triangles in Icosahedron Grids
Iteration Nodes Triangles
0 12 20
1 42 100
2 162 420
3 642 1700
4 2562 6820
5 10242 27300
6 40962 109220

It might be expected that the number of nodes is three times the number of triangles. But since each node is used by up to six triangles, the asymptotic average number of nodes per triangle is only 3/8=0.375:

Average Number of Nodes per Triangle
Iteration Nodes/Triangle
0 0.6
1 0.42
2 0.3857
3 0.3776
4 0.3757
5 0.3752
6 0.375041

In a plane, the described mechanism of subdividing the triangle edges reduces the triangle size by a factor of 2 with each iteration. On the sphere, the subdividsion produces triangles of different sizes, as shown above. To estimate the number of iterations necessary to produce a grid of a given triangle size, it is useful to look at the largest circumcircle radii of each grid: all triangles in the grid have circumcircles of at most this size, therefore it is a measure of the spatial resolution that can be obtained with the grid.

Decrease of the largest circumcircle radius as function of iteration number for the three basic bodies.

The plot shows that the maximum circumcircle radius decreases by a factor which asymptotically reaches 2.

eof.