geometorialla

Geometorialla is a theoretical construct in geometry used to study the relationship between geometric size and the combinatorial structure of point configurations. It is defined for finite sets of points in Euclidean space and is intended to be a scalar invariant that reflects both spatial spread and the richness of possible triangulations of the configuration.

Definition and construction. For a finite point set S in R^d in general position, let C be the convex hull of S. Let T(S) denote the number of distinct triangulations of C with vertex set S. The geometorialla G(S) is defined as G(S) = Vol_d(C) × T(S), where Vol_d(C) is the d-dimensional volume of C (area in the plane, ordinary volume in three dimensions, etc.). In variations, researchers may replace Vol_d(C) with alternative geometric measures such as total edge length or other size-related quantities, or apply normalization factors to T(S) to account for symmetries.

Properties. G(S) is designed to be invariant under rigid motions of space and to scale predictably under similarity transformations, inheriting the scaling from Vol_d(C). It tends to increase with the addition of points in a way that reflects both the expansion of the convex hull and the growth in triangulation possibilities. The value of G(S) depends strongly on the combinatorial type of S and can be sensitive to small changes in point configuration, making computation nontrivial for larger sets.

Relation to existing concepts. Geometorialla connects convex hull theory, triangulation counts, and geometric invariants. It relates to but remains distinct from standard invariants by combining a geometric measure with a global combinatorial count, offering a lens on geometric complexity and configuration space.

History and usage. The concept is a hypothetical construction discussed in exploratory works on geometric complexity and combinatorial geometry. It serves as a conceptual tool for thinking about how geometry and triangulation complexity interact, rather than a widely adopted standard invariant. See also convex hull, triangulations, Delaunay triangulation, and geometric invariants.