Editing Polyhedron (section)

== Other families of polyhedra ==
===Space-filling polyhedra===
{{Main|Honeycomb (geometry)}}
A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a [[#Dehn invariant|Dehn invariant]] equal to zero. Some honeycombs involve more than one kind of polyhedron.

=== Flexible polyhedra ===
{{main|Flexible polyhedron}}
It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By [[Cauchy's theorem (geometry)|Cauchy's rigidity theorem]], flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result is known as the bellows theorem.<ref>{{citation
 | last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine
 | last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)
 | contribution = 23.2 Flexible polyhedra
 | doi = 10.1017/CBO9780511735172
 | isbn = 978-0-521-85757-4
 | mr = 2354878
 | pages = 345–348
 | publisher = Cambridge University Press, Cambridge
 | title = Geometric Folding Algorithms: Linkages, origami, polyhedra
 | title-link=Geometric Folding Algorithms
 | year = 2007}}.</ref>

=== Ideal polyhedron ===
{{main article|Ideal polyhedron}} 
Convex polyhedra can be defined in three-dimensional [[hyperbolic space]] in the same way as in Euclidean space, as the [[convex hull]]s of finite sets of points. However, in hyperbolic space, it is also possible to consider [[ideal point]]s and the points within the space. An [[ideal polyhedron]] is the convex hull of a finite set of ideal points.<ref name=thurston>{{citation
 | last = Thurston | first = William P. | authorlink = William Thurston
 | isbn = 0-691-08304-5
 | mr = 1435975
 | publisher = Princeton University Press, Princeton, NJ
 | series = Princeton Mathematical Series
 | title = Three-dimensional geometry and topology. Vol. 1
 | volume = 35
 | year = 1997
 | page = 128
 | url = https://books.google.com/books?id=9kkuP3lsEFQC&pg=PA128
}}</ref> Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space.

=== Lattice polyhedron ===
Convex polyhedra in which all vertices have integer coordinates are called [[convex lattice polytope|lattice polyhedra]] or [[integral polyhedron|integral polyhedra]]. The [[Ehrhart polynomial]] of lattice a polyhedron counts how many points with [[integer]] coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of [[combinatorics]] and [[commutative algebra]].<ref name=stanley-97>{{citation
 | last = Stanley | first = Richard P. | author-link = Richard P. Stanley 
 | year = 1997
 | title = Enumerative Combinatorics, Volume I
 | edition = 1
 | publisher = Cambridge University Press
 | pages = 235–239
 | isbn = 978-0-521-66351-9
}}</ref> An example is [[Reeve tetrahedron]].<ref name=k>{{citation
 | last = Kołodziejczyk | first = Krzysztof
 | doi = 10.1007/BF00150027
 | issue = 3
 | journal = Geometriae Dedicata
 | mr = 1397808
 | pages = 271–278
 | title = An "odd" formula for the volume of three-dimensional lattice polyhedra
 | volume = 61
 | year = 1996| s2cid = 121162659
 }}</ref>

There is a far-reaching equivalence between lattice polyhedra and certain [[Algebraic variety|algebraic varieties]] called [[Toric variety|toric varieties]].<ref>{{citation |last=Cox |first=David A. |title=Toric varieties |date=2011 |publisher=American Mathematical Society |others=John B. Little, Henry K. Schenck |isbn=978-0-8218-4819-7 |location=Providence, R.I. |oclc=698027255}}</ref> This was used by Stanley to prove the [[Dehn–Sommerville equations]] for [[simplicial polytope]]s.<ref name=stanley-96>{{citation |last=Stanley |first=Richard P. |title=Combinatorics and commutative algebra |date=1996 |publisher=Birkhäuser |isbn=0-8176-3836-9 |edition=2nd |location=Boston |oclc=33080168}}</ref>

=== Polyhedral compound ===
A [[polyhedral compound]] is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the [[list of Wenninger polyhedron models]].

=== Zonohedron ===
A [[zonohedron]] is a convex polyhedron in which every face is a [[polygon]] that is symmetric under [[rotation]]s through 180°. Zonohedra can also be characterized as the [[Minkowski sum]]s of line segments, and include several important space-filling polyhedra.<ref>{{citation
 | last = Taylor | first = Jean E. | author-link = Jean Taylor
 | doi = 10.2307/2324178
 | issue = 2
 | journal = [[American Mathematical Monthly]]
 | mr = 1144350
 | pages = 108–111
 | title = Zonohedra and generalized zonohedra
 | volume = 99
 | year = 1992| jstor = 2324178 }}.</ref>

=== Orthogonal polyhedron ===
{{main article|Orthogonal polyhedron}}
[[File:Soma_cube_figures.svg|thumb|Some [[orthogonal polyhedra]] made of [[Soma cube]] pieces, themselves [[polycube]]s]]
{{anchor|Orthogonal polyhedra}}Polyhedra are said to be [[orthogonal polyhedra|orthogonal]] because all of their edges are parallel to the axes of a Cartesian coordinate system. This implies that all faces meet at [[right angle]]s, but this condition is weaker: [[Jessen's icosahedron]] has faces meeting at right angles, but does not have axis-parallel edges. Aside from the [[rectangular cuboid]]s, orthogonal polyhedra are nonconvex. They are the three-dimensional analogs of two-dimensional orthogonal polygons, also known as [[rectilinear polygon]]s.  Orthogonal polyhedra are used in [[computational geometry]], where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a [[polygonal net]].<ref>{{citation
 | last = O'Rourke | first = Joseph | author-link = Joseph O'Rourke (professor)
 | contribution = Unfolding orthogonal polyhedra
 | doi = 10.1090/conm/453/08805
 | mr = 2405687
 | pages = 307–317
 | publisher = Amer. Math. Soc., Providence, RI
 | series = Contemp. Math.
 | title = Surveys on discrete and computational geometry
 | volume = 453
 | year = 2008| isbn =978-0-8218-4239-3 | doi-access = free
 }}.</ref> [[Polycube]]s are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar [[polyomino]]es.<ref>{{citation
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | date = November 1966
 | issue = 5
 | journal = [[Scientific American]]
 | jstor = 24931332
 | pages = 138–143
 | title = Mathematical Games: Is it possible to visualize a four-dimensional figure?
 | volume = 215| doi = 10.1038/scientificamerican1166-138
}}</ref>