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Simulation of High Dimensional Spheres

We can generate the spheres both by numerical simulation and analytically. The two methods are illustrated together in Fig. 6. For a numerical simulation, the motions of a molecular machine would be determined by the technique of molecular dynamics [Karplus & McCammon, 1986]. If the ``pins'' have been identified--which probably requires understanding how the machine works--then we could obtain a set of y_j. When the machine parts are at equilibrium, these should have independent Gaussian distributions (Assumption 2, Assumption 3). So instead of doing molecular dynamics, we can use any set of real numbers having a Gaussian distribution. These are easily created by adding together many pseudo-random numbers that have a flat distribution [Miller, 1981]. The central limit theorem assures us that the resulting sum approaches a Gaussian distribution [Breiman, 1969]. A set of D such numbers forms the point in Y space.

  

Figure 6: Simulation of a fourth-dimensional sphere.

A four-dimensional Y space was projected onto the two-dimensional space represented by the page. This is equivalent to a plane cross section through the space [Shannon, 1949]. The continuous grey-tone distribution represents the analytic probability density, f_D(r)(equation (48) in Appendix 22 and Fig. 4) for D = 4and $\sigma = 1$. Each small open circle ($\bigcirc$) represents a numerical simulation produced from four normally distributed values with mean 0 and standard deviation 1 according to equation (29). Each normally distributed value was the sum of 100 pseudo-random numbers with a flat distribution.

To see what the distribution of these points looks like, we can map the sphere onto a plane. The method is equivalent to moving cities from their particular latitudes and longitudes on a globe to their longitudes at the equator. In Fig. 6 we have mapped 4-dimensional Y space points onto the page by this method. From equation (24) the radial distance from the origin to a point in Y space is given by:

$$r_y = \sqrt{\sum_{j=1}^{D} {y_j}^{2}} .$$ (29)

When $\sigma = 1$, the distribution has a maximum at $r_{max} = \sqrt{D-1}$ (Appendix 22), so the radius was normalized by dividing by $\sqrt{D-1}$. The direction (angle) of each point was arbitrarily taken from two of the y_j coordinates.

To graph the corresponding smooth analytic function, we chose points on the page and determined their distance from the origin to obtain r_y. We then find the probability density directly from the f_D(r) function given by equation (48) of Appendix 22.

Fig. 7 shows the correspondence between the simulated points and the smooth analytic function. As the dimensionality increases, the spheres become sharper. The concentration of points at a particular radius is a consequence of the enormously increasing volume as the radius increases in higher dimensions. So, although the density is highest in the center according to equation (24), the majority of points are found far from the center.

  

Figure 7: Increase of Sphere Sharpness with Increasing Dimensionality.

A series of sphere projections are shown for $D = 4, 8, \ldots, 1024$ dimensions. The first one is the same as Fig. 6, but reduced in size. Only 10,000 Gaussian values were precalculated, so the number of simulated points that could be calculated decreased as the dimensionality increased.