Maximal Equidistant Spacings

We call a family $\{Y_1,\dots,Y_I\}$ in Euclidean space an equidistance spacing if $\|y_i - y_j\| = 1$ whenever $y_i \in Y_i, y_j \in Y_j$ and $i \neq j$. In other words, choosing a representative from each set produces a complete distance graph (i.e. equilateral set). We say such a spacing is maximal if each $Y_i$ is maximal under inclusion. In this work we characterize maximal equidistant spacings in $\mathbb R^n$. For each equidistant spacing there is an associated center (a point in $\mathbb R^n$) and radius (a non-negative scalar) so that the centers form an orthocentric system. Using arguments from classical geometry we find that the moduli space of maximal equidistant spacings is described in terms of the movement of its center and radii. Using tools from geometric combinatorics, we develop a discrete combinatorial object called the signature. Our classification theorem shows that maximal equidistant spacings are isometric if and only if they have the same signature. We also construct all maximal equidistant spacings in $\mathbb R^1,\mathbb R^2$ and $\mathbb R^3$, outline a procedure for constructing all maximal equidistant spacings in $\mathbb R^n$, and give an algorithm for checking if a locus of points is equidistantly spaced that is linear in the number of points, an improvement over the naive direct quadratic algorithm.