Let 𝐩 be a configuration of n points in ℝ^d for some n and some d ≥ 2. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that has the same endpoints. A path or loop, as a sequence of edges, also has a Euclidean length. In this paper, we study the question of when 𝐩 will be uniquely determined (up to an unknowable Euclidean transform) from a given set of path or loop lengths. In particular, we

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... consider the setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing the paths or loops that gave rise to the lengths. Our main result is a condition on the set of paths or loops that is sufficient to guarantee such a unique determination. We also provide an algorithm, under a real computational model, for performing a reconstruction of 𝐩 from such unlabeled lengths. To obtain our results, we introduce a new family of algebraic varieties which we call the unsquared measurement varieties. The family is parameterized by the number of points n and the dimension d, and our results follow from a complete characterization of the linear automorphisms of these varieties for all n and d. The linear automorphisms for the special case of n = 4 and d = 2 correspond to the so-called Regge symmetries of the tetrahedron.