Maximal Equidistant Spacings

Abstract

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.

Figures (13)

• Figure 1: An example of three different equidistant spacings in $\mathbb{R}^2$ with different values for $I$. Details given in Example \ref{['exp:perf-spacings']}. In each figure, the class label is indicated by color. For example, the two red points in Figure \ref{['fig:eps-first-examp:2']} are $Y_1$ for that equidistant spacing. Notice how there is no requirement for the spacing of the points within one class.
• Figure 2: An example showing several ways in which a equidistant spacing produces other equidistant spacings in uninteresting ways. This figure shows eight equidistant spacings total. The figures are made up of four pairs. In the first pair, $\mathcal{Y}^{(1)}$ and $\mathcal{Y}^{(2)}$, are related by rototranslation. That is $Y^{(1)}_i = pY^{(2)}_i$ where $p \in E(2)$. The second pair, $\mathcal{Y}^{(3)}$ and $\mathcal{Y}^{(4)}$, are related by permuting class labels. That is, $Y_i^{(3)} = Y^{(4)}_{\sigma(i)}$ for some permutation $\sigma$ on $\left \{ 1,\dots,I\right \}$. The third pair, $\mathcal{Y}^{(5)}$ and $\mathcal{Y}^{(6)}$, are related by what we call a squash and stretch, so named because the green points are 'squashed' together, and the red points 'streched' out. Intuitively points of one class are 'squished' together while points of another class are 'stretched' apart to preserve equidistant spacing.
• Figure 3: Two examples of the recoloring trick. The first example shows $\mathcal{Y}^{(1)}$, plotted in subfigure \ref{['fig:recoloring:1']} recolored to $\mathcal{Y}^{(2)}$, plotted in \ref{['fig:recoloring:2']}. The second shows $\mathcal{Y}^{(3)}$, plotted in subfigure \ref{['fig:recoloring:3']} recolored to $\mathcal{Y}^{(4)}$, plotted in \ref{['fig:recoloring:4']}.
• Figure 4: An example of the construction in Lemma \ref{['lem:ortho-reflection-lemma']} in three dimensions for $I = 4$. (a) An illustration of $\mathcal{Y} \in \mathcal{E}\mathcal{S}(4,\mathbb{R}^n)$ where the red dot denotes $Y_4$. (b) By applying Lemma \ref{['lem:ortho-reflection-lemma']} we union the red dot with its reflection across its orthogonal projection onto the affine space containing the other three dots. This produces a spacing $\mathcal{Y}'$ that maximizes $\mathcal{Y}$.
• Figure 5: Examples of gluings with glue sites shown with a black $*$. Subfigures \ref{['fig:examp:gluings:Y1:2d']} - \ref{['fig:examp:gluings:Y1-vee-Y2:2d']} show gluings in $\mathbb{R}^2$, and subfigures \ref{['fig:examp:gluings:Y3:3d']} - \ref{['fig:examp:gluings:Y3-vee-Y4:3d']} show gluings in three dimensions.

Theorems & Definitions (149)

• Definition 2.1.1: Equidistantly Spaced, $\mathcal{E}\mathcal{S}$
• Example 2.1.2
• Example 2.1.3: Operations that Preserve Equidistant Spacings
• Remark 2.2.1: Notational Shorthand: I
• Lemma 2.2.2: Orthogonality Lemma
• proof
• Definition 2.2.3: Minimal Affine Set
• Lemma 2.2.4: The Recoloring Trick
• proof
• Remark 2.2.5: Necessity for $J \geq 2$ in the Recoloring Trick