Optimally spaced rotational grid points

Optimal rotational spacing was obtained for n = 2 to 12 orientations of an object in three-dimensional space. For n = 2, 3, and 4 the minimum spacing, χ(min), was 180°, with rotation space not being completely filled. For n = 5 and 6 all spacings are equal to 151.05 and 141.06° respectively. The n = 7 case has two spacings at 134.04 and 180°. The n = 8 case has two spacings at 130.18 and 153.56°. The n = 10 case has three spacings at 128.53, 141.05 and 164.8°. The n = 12 case has two spacings at 120 and 180°. The best arrangement found for n = 9 and 11 was to remove one grid point from n = 10 and 12 respectively. The coordination about each point and the orientations of the grid difference rotation axes are given. The axes for n = 5 are directed toward the vertices of a regular dodecahedron; the axes for n= 12 are directed toward the vertices and faces of a cube. Products of two rotations of equal magnitude to give a third rotation of the same magnitude were considered and classified into conrotatory and disrotatory types. For n > 12 the Lattman treatment was extended to include third-order terms. Examples of Lattmanian angle grids are given and the grid spacings are compared to theoretical estimates.