stellation

Stellation is a geometric construction in which the faces of a polyhedron are extended in their planes to form a new, often star-shaped, solid. Starting from a given polyhedron, each face defines a plane; by extending these planes outward, their intersections determine the boundary of a stellated polyhedron. The process can yield finite star polyhedra when the planes intersect in a closed surface, or infinite figures when the extensions do not bound a finite region.

In two dimensions, stellation can be applied to polygons by extending the sides until they meet; the classic example is the pentagon extended to form a pentagram.

In three dimensions, the most famous stellations are the Kepler-Poinsot solids: the small stellated dodecahedron, great stellated dodecahedron, great dodecahedron, and great icosahedron.

Historically, the term stellation comes from Latin stellatus, meaning “starred.” The concept originated with Kepler and Poinsot in the 17th to 19th centuries and was later systematized by Coxeter and others in the 20th century, who studied stellations within the broader framework of polyhedral symmetry and compounds.

Stellation is related to, but distinct from, other polyhedral operations such as truncation and augmentation; it is also used as a tool in mathematical visualization and has connections to art and architecture.