proporcingum

Proporcingum is a theoretical construct in geometry and applied mathematics described as a class of transformations that preserve proportional relationships among distances across multiple dimensions. Unlike ordinary scaling, a proporcingum aims to maintain a coherent ratio pattern among all pairs of points, enabling nonuniform scaling without distorting the overall proportional structure of a figure or data embedding.

Etymology and history: The term proporcingum combines common roots for proportion with a generic suffix, and it emerged in early 21st-century speculative geometry literature to describe a generalized mechanism for proportional preservation. Various proposals treat proporcingum as a family of maps parameterized by a scale function, or as a guiding principle for algorithmic design.

Definition and formalism: Let X be a metric space with distance d_X. A map f: X -> Y is a proporcingum with scale function s if, for all x, y in X, the distance in Y satisfies d_Y(f(x), f(y)) = g(d_X(x, y), s), where g is a fixed function ensuring proportionality across pairs according to s. Special cases recover similarities when g(t, s) = s t. The concept generalizes to manifolds, graphs, and data embeddings.

Applications and relation to existing notions: The idea has been explored in computer graphics for shape-preserving nonuniform scaling, in data visualization to maintain perceptual ratios under dimensionality reduction, and in architectural design to preserve proportional aesthetics during dynamic resizing. It relates to, but extends, similarity and affine transforms by allowing scale variance while preserving proportional structure.

Reception and status: Proporcingum remains a largely theoretical construct with limited formal results and no consensus on a canonical definition. It appears mainly in theoretical discussions and speculative exercises rather than as established tools in applied mathematics. See also: proportion, similarity, scale transformation, invariants.