# DVFILL

(Direction Vector FILLing)

This routine constructs a homogeneous set of points on a unit sphere (direction vectors). It starts with 12 face-centers on a regular dodecahedron, i.e. the 12 corners of a regular icosahedron. These form 20 regular triangles. If the centers of these 20 triangles are added to the 12 initial directions, a new set of triangles is created, which, by a Wigner-Seitz construction on the unit sphere corresponds to 32 faces, 12 pentagons, and 20 hexagons. Thus we have constructed a soccerball. By iterative addition of the triangle centers, any number N of homogeneously distributed points can be generated, which can be written in the form N = 3i x 10 + 2 , with i being an integer (N = 12, 32, 92, 272, ... ).

But there is a second way of generating a finer mesh of regular triangles from a cruder one. Instead of additional centers, the midpoints of the edges can be added. This roughly corresponds to an increase in the number of directions by a factor of four, instead of three. By combining these two procedures, a number N = 3i x j2 x 10 + 2  of directions can be created. These are the allowed values of NSPA in the DVFILL routine ("magic numbers"). The smallest are 12 (i=0, j=1), 32 (i=1, j=1), 42 (i=0, j=2), 92 (i=2, j=1 and i=0, j=3), 122 (i=1, j=2), 162 (i=0, j=4), .... The default for the construction of the fine grid is 1082 = 33 x 4 x 10 + 2 .