Editing Hexahedron

{{Short description|Polyhedron with 6 faces}}
A '''hexahedron''' ({{plural form}}: '''hexahedra''' or '''hexahedrons''') or '''sexahedron''' ({{plural form}}: '''sexahedra''' or '''sexahedrons''') is any [[polyhedron]] with six [[Face (geometry)|faces]]. A [[cube]], for example, is a [[Regular polyhedron|regular]] hexahedron with all its faces [[Square (geometry)|square]], and three squares around each [[Vertex (geometry)|vertex]].

There are seven [[Topology|topologically]] distinct ''convex'' hexahedra,<ref name=dillencourt>{{citation
 | last = Dillencourt | first = Michael B.
 | doi = 10.1006/jctb.1996.0008
 | issue = 1
 | journal = [[Journal of Combinatorial Theory, Series B]]
 | mr = 1368518
 | pages = 87–122
 | title = Polyhedra of small order and their Hamiltonian properties
 | url = https://escholarship.org/uc/item/39f9s8b5
 | volume = 66
 | year = 1996}}</ref> one of which exists in two mirror image forms. Additional non-convex hexahedra exist, with their number depending on how polyhedra are defined. Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.

==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. Different types of cuboids include the ones depicted and linked below.

{| class="wikitable" width=770
|-
!colspan=7|Cuboids
|- align=center
|[[File:Hexahedron.png|110px]]
|[[File:Cuboid no label.svg|110px]]
|[[File:Trigonal trapezohedron.png|110px]]
|[[File:Trigonal trapezohedron gyro-side.png|110px]]
|[[File:Usech kvadrat piramid.png|110px]]
|[[File:Parallelepiped 2013-11-29.svg|110px]]
|[[File:Rhombohedron.svg|110px]]
|-
|[[Cube]]<br>([[Square (geometry)|square]])
|[[Rectangular cuboid]]<br>(three pairs of<br>[[rectangles]])
|[[Trigonal trapezohedron]]<br>(congruent [[rhombus|rhombi]])
|[[Trigonal trapezohedron]]<br>(congruent [[quadrilateral]]s)
|Quadrilateral [[frustum]]<br>(apex-truncated<br>[[square pyramid]])
|[[Parallelepiped]]<br>(three pairs of<br>[[parallelogram]]s)
|[[Rhombohedron]]<br>(three pairs of<br>[[rhombus|rhombi]])
|}

===Others===
There are seven topologically distinct convex hexahedra,<ref name=dillencourt/> 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.

{| class="wikitable" width=660
|- align=center
! Image
| [[File:Hexahedron5.svg|110px]]
| [[File:Hexahedron7.svg|110px]][[image:Hexahedron7a.svg|110px]]
| [[File:Hexahedron2.svg|110px]]
| [[File:Hexahedron6.svg|110px]]
| [[File:Hexahedron3.svg|110px]]
| [[File:Hexahedron4.svg|110px]]
|-
! Name
| [[Triangular bipyramid]]
| 
| [[Pentagonal pyramid]]
|
| 
|  Doubly truncated tetrahedron<ref>{{citation
 | last1 = Kolpakov | first1 = Alexander
 | last2 = Murakami | first2 = Jun
 | doi = 10.1007/s00010-012-0153-y
 | issue = 3
 | journal = Aequationes Mathematicae
 | mr = 3063880
 | pages = 449–463
 | title = Volume of a doubly truncated hyperbolic tetrahedron
 | volume = 85
 | year = 2013| arxiv = 1203.1061
 }}</ref>
|-
! Features
| {{plainlist|1=
*5 vertices
*9 edges
*6 triangles
}}
| {{plainlist|1=
*6 vertices
*10 edges
*4 triangles
*{{nowrap|2 quadrilaterals}}
}}
| {{plainlist|1=
*6 vertices
*10 edges
*5 triangles
*{{nowrap|1 pentagon}}
}}
| {{plainlist|1=
*7 vertices
*11 edges
*2 triangles
*{{nowrap|4 quadrilaterals}}
}}
| {{plainlist|1=
*7 vertices
*11 edges
*3 triangles
*{{nowrap|2 quadrilaterals}}
*1 pentagon
}}
| {{plainlist|1=
*8 vertices
*12 edges
*2 triangles
*{{nowrap|2 quadrilaterals}}
*2 pentagons
}}
|-
! Properties
| [[Simplicial polytope|Simplicial]]
| {{plainlist|1=
*[[Chiral]]
*[[Self-dual polyhedron|Self-dual]]
}}
|{{plainlist|1=
*[[Halin graph|Dome]]
*[[Self-dual polyhedron|Self-dual]]
}}
|
|[[Halin graph|Dome]]
|{{plainlist|1=
*[[Halin graph|Dome]]
*[[Simple polytope|Simple]]
}}
|}

==Concave==
Three further topologically distinct hexahedra can only be realised as ''concave'' [[acoptic polyhedron|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.<ref>{{citation
 | last = Grünbaum
 | first = Branko
 | author-link = Branko Grünbaum
 | contribution = Acoptic polyhedra
 | doi = 10.1090/conm/223/03137
 | mr = 1661382
 | pages = 163–199
 | publisher = American Mathematical Society
 | location = Providence, Rhode Island
 | series = Contemporary Mathematics
 | title = Advances in discrete and computational geometry (South Hadley, MA, 1996)
 | contribution-url = http://faculty.washington.edu/moishe/branko/BG225.Acoptic%20polyhedra.parsed.pdf
 | volume = 223
 | year = 1999
 | isbn = 978-0-8218-0674-6}}; for the three non-convex acoptic hexahedra see p. 7 of the preprint version and Fig. 3, p. 30</ref>

{| class = wikitable
|-
!colspan = 3 | Concave
|- valign="top"
| [[image:Hexahedron8.svg|110px]]
| [[image:Hexahedron10.svg|110px]]
| [[image:Hexahedron9.svg|110px]]
|-
! 4.4.3.3.3.3 Faces<br>10 E, 6 V
! 5.5.3.3.3.3 Faces<br>11 E, 7 V
! 6.6.3.3.3.3 Faces<br>12 E, 8 V
|}
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 [[k-vertex-connected graph|3-vertex-connected graphs]].<ref>{{citation
 | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler
 | contribution = Chapter 4: Steinitz' Theorem for 3-Polytopes
 | isbn = 0-387-94365-X
 | pages = 103–126
 | publisher = Springer-Verlag
 | series = [[Graduate Texts in Mathematics]]
 | title = Lectures on Polytopes
 | volume = 152
 | year = 1995}}</ref>
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.<ref>{{citation
 | last1 = Hong | first1 = Seok-Hee | author1-link = Seok-Hee Hong
 | last2 = Nagamochi | first2 = Hiroshi
 | doi = 10.1007/s00453-011-9570-x
 | issue = 4
 | journal = Algorithmica
 | mr = 2852056
 | pages = 1022–1076
 | title = Extending Steinitz's theorem to upward star-shaped polyhedra and spherical polyhedra
 | volume = 61
 | year = 2011}}</ref>

==References==
<references/>


{{Polyhedra}}
[[Category:6 (number)]]
[[Category:Polyhedra]]