parallelenty

Parallelenty is a fictional mathematical concept describing an equivalence relation among geometric shapes based on their projections along a fixed set of directions. Coined in speculative mathematics and science-fiction literature, parallelenty captures the idea that two objects may be treated as equivalent when their shadows, viewed from several prescribed directions, align up to translations along those directions. The term blends geometry's notion of parallelism with a systematic, category-like suffix.

Formally, let D = {d1,...,dk} be a finite set of unit vectors in Euclidean space. For a compact set A ⊆ R^n, denote P_di(A) the orthogonal projection of A onto the hyperplane perpendicular to di. Two objects A and B are parallelent if there exists a vector t = (t1,...,tk) such that for every i, P_di(B) equals P_di(A) translated by ti di. The tuple t is the parallelenty descriptor of (A,B). A parallelenty class consists of all objects whose projections along each di differ by a di-translation in a fixed set of scalars up to a common equivalence.

Properties: If D contains at least two nonparallel directions, parallelenty becomes a strong invariant: objects in the same class have the same multi-view projections up to the allowed translations. Parallelenty respects translations along directions in D, but not arbitrary rotations unless those rotations preserve D. The framework extends to curves and surfaces by applying the projection operator pointwise.

Applications: In theory and imagined practice, parallelenty provides a tool for multi-view shape matching, 3D reconstruction from silhouettes, and alignment of high-dimensional data analyzed via fixed projection directions. In worldbuilding contexts, it can model how observers in different frames perceive the same object as consistently as the projections allow.