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Hexahedron

Polyhedron with 6 faces

Summary

Polyhedron with 6 faces

A hexahedron (: hexahedra or hexahedrons) or sexahedron (: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.

There are seven topologically distinct convex hexahedra,{{citation

Convex

Cuboid

A hexahedron that is combinatorially equivalent to a cube may be called a cuboid, although this term is often used more specifically to mean a rectangular cuboid, a hexahedron with six rectangular sides. A cuboid possess 8 vertices, 6 faces and 12 edges. Different types of cuboids include the ones depicted and linked below.

Cuboids
[[File:Hexahedron.png	110px]]
Cube
(square)

Others

There are seven topologically distinct convex hexahedra, the cuboid and six others, which are depicted below. One of these is chiral, in the sense that it cannot be deformed into its mirror image.

Image	Name	Features	Properties
[[File:Hexahedron5.svg	110px]]	[[File:Hexahedron7.svg	110px]]110px	[[File:Hexahedron2.svg	110px]]	[[File:Hexahedron6.svg	110px]]
Triangular bipyramid		Pentagonal pyramid
{{plainlist	1=	{{plainlist	1=	{{plainlist	1=	{{plainlist	1=
Simplicial	{{plainlist	1=	{{plainlist	1=

Concave

Three further topologically distinct hexahedra can only be realised as concave acoptic polyhedra. These are defined as the surfaces formed by non-crossing simple polygon faces, with each edge shared by exactly two faces and each vertex surrounded by a cycle of three or more faces.{{citation | author-link = Branko Grünbaum | contribution-url = http://faculty.washington.edu/moishe/branko/BG225.Acoptic%20polyhedra.parsed.pdf

Concave	4.4.3.3.3.3 Faces
10 E, 6 V	5.5.3.3.3.3 Faces
11 E, 7 V	6.6.3.3.3.3 Faces
12 E, 8 V
110px	110px	110px

These cannot be convex because they do not meet the conditions of Steinitz's theorem, which states that convex polyhedra have vertices and edges that form 3-vertex-connected graphs.{{citation For other types of polyhedra that allow faces that are not simple polygons, such as the spherical polyhedra of Hong and Nagamochi, more possibilities exist.{{citation

References

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

6-(number) polyhedra

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