###### Example 8.2.7 Collecting correlated pairs

In a catalog of images, there are two types of pairs that might generate recurring distances. The cosmic crystallography method outlined previously detects what have been called Type II pairs in the literature: A Type II pair is a pair of points of the form \(\{p,T(p)\}\text{,}\) where \(T\) is a transformation from the group of isometries used to generate the manifold. If \(T\) transforms each point the same distance (i.e., if \(T\) is a Clifford translation), then this common distance will appear in the PSH as a spike.

The other type of recurring distance can arise from what's been called a Type I pair of points in the catalog. A Type I pair consists of any pair \(\{p,q\}\) of points. If we can see images of these points in a copy of the fundamental domain, say \({T(p),T(q)}\text{,}\) then since transformations preserve distance, \(d(p,q) = d(T(p),T(q))\text{,}\) and this common distance will have occurred at least twice in the PSH. The figure below shows a portion of the torus tiling of the plane depicted in Figure 8.2.2. The two types of pairs of points are visible: Type I pairs are joined by dashed segments, and Type II pairs are joined by solid segments.

Type I pairs will not produce discernible spikes in the PSH. Even in simulations for which several images of a pair of points are present, the spike generated by this set of pairs having the same distance is not statistically significant.

The collecting correlated pairs (CCP) method, outlined below, attempts to detect the Type I pairs in a catalog.

Suppose a catalog has \(N\) objects, and let \(P = N(N-1)/2\) denote the number of pairs generated from this set. Compute all \(P\) distances between pairs of objects, and order them from smallest to largest. Let \(\Delta_i\) denote the difference between the \((i+1)\)st distance and the \(i\)th distance. Notice \(\Delta_i \geq 0\) for all \(i\text{,}\) and \(i\) runs from 1 up to \(P-1\text{.}\)

Now, \(\Delta_i = 0\) for some \(i\) if two different pairs of objects in the catalog have the same separation. It could be that unrelated pairs happen to have the same distance, or that the two pairs responsible for \(\Delta_i = 0\) are of the form \(\{p,q\}\) and \(\{T(p),T(q)\}\) (Type I pairs). (We're assuming there are no type II pairs in the catalog.)

Let \(Z\) equal the number of the \(\Delta_i\)'s that equal zero. Then

denotes the proportion of the differentials that equal zero. This single number is a measure, in some sense, of the likelihood of living in a multiconnected universe.

In a real catalog involving estimations of distances, one wouldn't expect Type I pairs to produce identical distances, so instead of using \(Z\) as defined above, one might let \(Z_\epsilon\) equal the number of the \(\Delta_i\)'s that are less than \(\epsilon\text{,}\) where \(\epsilon\) is some small positive number. For more details on this method, see [28].