Getting a good look

We describe a convenient 2 dimensional projection scheme and also some ways for choosing orientations that highlight the band structure. We have used a dot to indicate the vertices which have degree 6 and a triangle to indicate the vertices with lesser degree. Rotating a configuration to certain orientations can highlight the structure of the configuration so that, for example, points fall on a few horizontal bands with the same (or nearly the same) z coordinate.

PROJECTION OF THE SPHERE TO A PLANAR REGION

The basic projection projects points outward to the circumscribing cylinder with vertical axis then cuts the cylinder vertically and unfolds it to a rectangle. Cylindrical coordinates (r, q , z) are mapped to Cartesian coordinates (q , z). The map from rectangular coordinates uses the function Arg(x,y) which is found in Mathematica. The inverse map from the rectangle to the sphere in rectangular coordinates is given by

X( th , z ) = { Sqrt[ 1-z2 ] Cos[th] , Sqrt[ 1-z2 ] Sin[th] , z }.

This map is area preserving-- for points near the North Pole, the East-West distance is stretched and distance in the North direction is compressed, but the product is constant. The reader may be intuitively persuaded by recalling that the surface area of the "annulus" of sphere between z=a and z=b depends only on b-a. This level outward projection is different from the Mercator projection since the image of that projection has constant distance between evenly spaced latitude lines.

This area preserving property allows us to easily generate random uniformly distributed points on the surface of the sphere. First we generate points in the rectangle and then we map them back to the sphere.

We have normalized the horizontal values to range from –1 to 1 so that evenly spaced q values can be more easily recognized.

ORIENTING THE SPHERE

Orienting the sphere is achieved by finding an orthonormal basis and computing the coordinates of points with respect to these coordinates. We have found four particular orientations or bases to be fruitful—one generated by ranking the points according to their partial Coulomb energy and three which arise from different orders of the eigenvectors of the variance-covariance matrix of the points.

COULOMB ARRANGEMENT: Each point Pk of a configuration experiences a sum of effects from the other particles called the partial Coulomb potential,

Ek = Sum for j not equal to k [ 1 / | Pk – Pj | ] .

An orthonormal basis {v1, v2, v3} is constructed so that the point with largest partial potential is the direction v3, and the point with the second largest potential has theta=0.

 

VARIANCE-COVARIANCE EIGENVECTOR BASES: For the configuration P represented as a matrix with ith row given by the coordinates of the ith point, the variance covariance (VC) matrix, VC (P)= Transpose(P).P , a 3 x 3 matrix.For eigenvalues a1 a2<=a3 of VC(P) let v1,v2,v3be the corresponding eighenvectors. The three cyclic permutations of these eigenvalues represent our bases and resulting orientation.

Our calculations reveal the variance covariance matrices of energy minimizing configurations are approximately equal to n/3 times the 3 ´ 3 identity matrix. Here is an intuitive argument for why this occurs.

The off diagonal elements are usually close to zero. We consider a typical entry, the 1,2 entry, which is Sum[ xj yj ]. If points are fairly evenly spread out on the surface of the sphere, the projection of points to their first 2 coordinates will be roughly circularly symmetric on a unit disk, and the covariance will be close to 0.

Projecting further onto any coordinate axis will give a set of points with variance the same as any other such projection. The trace of VC(P) is the sum of xj2 + yj2 + zj2 which is the sum of n 1’s or n. So each of the three diagonal entries is about n/3.

In general points on the sphere cannot be perfectly evenly spread out, apparently, and so these entries are approximate rather than exact. The perfectly symmetric cases, like the dodecahedron, yield the identity matrix times n/3 (and the search for eigenvectors is ill conditioned and unnecessary).

Why should the eigenvectors provide a good base or orientation? The VC matrix expressed with respect to the eigenbasis will have off diagonal entries zero. "Thus" the points will be perfectly balanced about the eigenvector directions.

Sometimes there is more than one good way to view the points. Each view seems to have instances where it reveals structure and other instances where it does not.