Editing Hexahedron (section)

==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>