5.4 The Perfect Symmetric Structure in Proximity Matrix and the Crystallographic Structure

For a proximity matrix D with some perfect symmetry pattern. The convergence is to a matrix with a simpler structure that can be used to describe the symmetry pattern in D.

We usually do not encounter proximity matrices with a perfect symmetric structure in a medical study of the sort looked at here. One can search for a proximity matrix with a symmetric structure in physics, chemistry or molecular biology.

One icosahedron, Figure 9a, is formed with an inversion center together with the twelve vertices. The proximity matrix of these thirteen points is calculated as the =78 pairs of Euclidean distance. It takes the sequence of correlation matrices fifteen iterations to converge to the matrix in Figure 9b. has a simple and clear pattern with only five different numbers in it, 0, 1, -1, a, and -a (a =). Ones exist only on the main diagonal, which means no two points are grouped together. The coefficients for to the twelve vertices are all 0 since is the inversion center and all the twelve vertices surround it in a three-dimensional symmetric manner.

Figure 9. Example for Perfect Symmetry: The Icosahedron.(a). The Structure of Icosahedron with 12 Vertices and One Inversion Center;(b). The Converged Matrix for the Icosahedron in (a), .

For any of the twelve vertices, there are five layers of relationship. Let us use to illustrate these five layers of structure. The first layer consists of itself and the coefficient is 1; the second layer contains vertices () on the same pentagon, the coefficients from to all these five vertices are equal to a = ; the inversion center, , alone forms the third layer with coefficient 0 for exact symmetry; the fourth layer of vertices () is similar to that of the second layer except the signs of the coefficients are negative; the fifth layer of is exactly opposite to relative to the inversion center, the coefficient is -1 for and .

There are three leading eigenvalues for the three-dimensional structure and an extra eigenvalue =1 characterizing the inversion center for . The first three eigenvectors are identical to the original three coordinates with a scale change while the fourth eigenvector has an one for and zeros for the twelve vertices, =(1,0,0,…,0).