Polyhedra

Polyhedra are three-dimensional solids bounded by polygonal faces, with straight edges and vertices. Each face is a polygon, and the surface is composed of a finite number of such faces. They can be convex or non-convex and may have various degrees of symmetry.

They are commonly classified as convex or non-convex and as regular, semi-regular (Archimedean), or irregular (such as Johnson solids). Regular polyhedra have congruent regular polygonal faces and the same number of faces meeting at every vertex.

There are five regular polyhedra, known as the Platonic solids: the tetrahedron, cube (hexahedron), octahedron, dodecahedron, and icosahedron. They are often described by Schläfli symbols {p,q}, where p is the number of sides of each face and q is the number of faces meeting at each vertex. Examples include {4,3} for the cube, {3,4} for the octahedron, {5,3} for the dodecahedron, and {3,5} for the icosahedron.

Archimedean solids are semi-regular polyhedra with faces that are more than one type of regular polygon but with identical vertex configurations; there are 13 such solids. Johnson solids are strictly convex polyhedra with regular polygonal faces that are not uniform.

Polyhedra exhibit duality: every polyhedron has a dual in which faces and vertices are interchanged. The cube and the octahedron are duals, as are the dodecahedron and the icosahedron.

Euler’s formula V − E + F = 2 applies to convex polyhedra and to polyhedra homeomorphic to a sphere; it generalizes in more complex topologies. Nets, or unfoldings, represent planar layouts of a polyhedron’s surface and have practical use in design and modeling. Polyhedra appear in geometry, crystallography, architecture, computer graphics, and 3D printing.