Polyhedron

Polyhedron
About
Rank	3
Plural	Polyhedra
Facet name	Face
Counts
Regular convex polyhedra	5
Regular star polyhedra	4
Regular finite compounds	1
Regular tesselations	3
Regular finite skew polyhedra of full rank	9
Regular skew honeycombs	3
Regular skew apeirotopes	27
Hierarchy
← Polygon Polyhedron Polychoron → 2 3 4

For the convex set, see Convex polytope § Definition.

A polyhedron or 3-polytope is a polytope of rank 3. The facets of polyhedra are polygons, which are called its faces.

Definition

Polyhedra can be defined to be polytopes of rank 3, however there is no universal agreed-upon definition of a polyhedron since the meaning of what a polytope is will differ depending on the context. In nearly all cases, polyhedra will have the structure of an abstract polytope, but sometimes this is not required (as an example, holyhedra do not have the abstract structure of a polyhedron).

The term polyhedron may occasionally be used as a synonym for polytope. For example associahedra, despite its suffix, are not necessarily rank 3.

A polyhedron compound is a compound of rank 3, or equivalently an arrangement of polyhedra.

Regular polyhedra

Main article: Regular polyhedron

A regular polyhedron is a polyhedron whose flags are identical under its symmetry group. All regular polyhedra are uniform, isogonal, isotoxal, isohedral, and noble. In 3D Euclidean space, there are 9 finite regular polyhedra and 3 tilings that are not skew, 9 finite skew polyhedra, and 27 regular skew apeirohedra.

External links

• Wikipedia contributors. "Polyhedron".
• Weisstein, Eric W. "Polyhedron" at MathWorld.