Is a set of unit vectors uniquely determined by its distances? Let $X = \{x_1, \dots, x_n\}$ be a finite set of points in $\mathbb{R}^d$. We can associate to $X$ its multiset of distances
$$ D_X := \{  \lVert x_i - x_j \rVert : 1 \le i,j \le n \} \qquad \text{(read as a multiset)} $$
where $\lVert \cdot \rVert$ denotes the Euclidean norm, and each pair $(i,j)$ counts once in the multiset, so that $\#D_X = n^2$. A natural question is to ask whether $X$ can be uniquely reconstructed from $D_X$ up to Euclidean isometry (translation, rotation, and reflection). The answer turns out to be no, and there are even infinitely many counterexamples in dimension $d=1$.
However, suppose we restrict the points $x_i$ to lie on the unit sphere $S^{d-1} \subset \mathbb{R}^d$. Is this constraint sufficient to ensure that $X$ is uniquely determined by $D_X$ up to Euclidean isometry?

Even for $d = 2$ there are such sets of points. Thus, for $ n = 4: \lbrace  0,1,2,5 \rbrace   $ and $\lbrace  0,1,5,6 \rbrace \mod 8$.
You can also specify homometric systems with equal sets of distances.
For $n = 6$, you can find at least 5, and for $n = 12$ - at least 18 homometric sets of points on the circle  with equal sets of distances.
