Inscribable

Inscribable refers to the geometric property of a polygon or polyhedron being able to have all of its vertices lie on the same circle or sphere, respectively. In two dimensions, a convex n‑gon is called inscribable if there exists a circle that touches every vertex of the polygon. Classic examples include all regular polygons and certain classes of right triangles such as the 30‑60‑90 triangle, which can be circumscribed about a circle. A necessary condition for a convex polygon to be inscribable is that each interior angle must satisfy a specific relationship with the opposite side lengths, but no simple universal criterion exists for arbitrary n‑gons.

In three dimensions, the analogue is an inscribable polyhedron. A convex polyhedron is inscribable if a sphere can be found that is tangent to every vertex. Regular Platonic solids like the cube, octahedron, dodecahedron, and icosahedron are inscribable. Johnson solids can also exhibit this property, but it does not hold for all convex polyhedra. Inscribable polyhedra are studied in the context of sphere packings, polyhedral combinatorics, and geometric optimization.

The concept is related to that of circumscribable figures, where a polygon or polyhedron can be circumscribed around a circle or sphere, and to the broader class of self‑dual solvers in discrete geometry. Inscribability has applications in pattern design, architecture, and the study of tiling and packing problems. Further research continues to explore necessary and sufficient conditions for various classes of shapes and their higher‑dimensional generalizations.