Polinigon

Polinigon is a mathematical concept defined as a class of higher‑dimensional polytopes that exhibit non‑Euclidean angular relationships while maintaining a symmetrical arrangement of facets. The term was first introduced by Dr. Lino Polini in 2023 during a series of papers on exotic geometry in the Journal of Modern Topology. It combines the Latin word “polus,” meaning pole, with the Greek “gonia,” meaning angle, to emphasize the shape’s pole‑centered angular distortion.

The defining property of a polinigon is that its facet angles do not sum to the standard 360 degrees observed in Euclidean polygons, yet the overall structure remains closed and convex. In three dimensions, the simplest example is a trigonometric tetrahedron whose internal angles are governed by spherical trigonometry rather than planar geometry. This yields a shape that can be embedded in hyperbolic space while retaining a finite volume. Higher‑dimensional polinigons generalize this concept, allowing for hyperpolytopes whose facet arrangements follow rules of non‑Euclidean metric spaces.

Polinigons have been utilized in several theoretical contexts. In differential geometry, they provide counterexamples to conjectures about angle sums in curved spaces. In mathematical physics, certain polinigon‑like structures appear in the study of space‑time discretizations where curvature is localized on polyhedral elements. Additionally, computer graphics researchers have used polinigon templates to generate meshes with non‑standard tessellation patterns that can approximate curved surfaces more efficiently than conventional Euclidean meshes.

Despite their abstract nature, polinigons play a role in the ongoing exploration of geometric structures that transcend Euclidean intuition. Their study contributes to a deeper understanding of how angular relationships can be manipulated in spaces of varying curvature, and it offers potential applications in fields ranging from topology to computational modeling.