polyhedrons

A polyhedron is a solid in three dimensions whose boundary is composed of polygonal faces joined at straight edges. The faces meet at vertices, and the arrangement of faces determines the shape's geometry. The plural terms polyhedra and polyhedrons are both used in mathematics.

Polyhedra can be convex, having no indentations, or non-convex, including shapes with cavities or self-intersecting features in some definitions. Star polyhedra, such as the Kepler-Poinsot solids, are non-convex and often have overlapping faces.

The five regular convex polyhedra, known as the Platonic solids, have identical regular faces and the same vertex configuration: tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron. Beyond regular forms, many others exist, including Archimedean solids with two or more types of faces, and Catalan solids with identical face shapes but not uniform vertices. Prisms and antiprisms are common families.

A fundamental relation for convex polyhedra is Euler's formula: V − E + F = 2, where V, E, and F denote the numbers of vertices, edges, and faces. This holds for simple polyhedra, and generalizations describe the topology of more complex surfaces.

Polyhedra arise in various contexts, from crystallography and chemistry to architecture and art. They can be studied by nets, which unfold a surface into a planar pattern, or by their symmetry properties and angular defects.