Polyhedron

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Polyhedron
About
Rank3
PluralPolyhedra
Facet nameFace
Counts
Regular convex polyhedra5
Regular star polyhedra4
Regular finite compounds1
Regular tesselations3
Regular finite skew polyhedra of full rank9
Regular skew apeirotopes27
Hierarchy
← Polygon Polyhedron Polychoron →
2 3 4

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

Definition[edit | edit source]

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[edit | edit source]

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[edit | edit source]