Editing Polyhedron (section)

===Duality===
{{main|Dual polyhedron}}
[[File:Dual Cube-Octahedron.svg|thumb|180px|The octahedron is dual to the cube]]

For every convex polyhedron, there exists a dual polyhedron having
* faces in place of the original's vertices and vice versa, and
* the same number of edges.
The dual of a convex polyhedron can be obtained by the process of [[Dual polyhedron#Polar reciprocation|polar reciprocation]].<ref>{{citation
 | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy
 | last2 = Rollett | first2 = A.P.
 | edition = 2nd
 | location = Oxford
 | mr = 0124167
 | publisher = Clarendon Press
 | title = Mathematical models
 | title-link = Mathematical Models (Cundy and Rollett)
 | year = 1961
 | contribution = 3.2 Duality
 | pages = 78–79}}.<!-- Describes only duality by polar reciprocation through the midsphere --></ref>  Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.<ref>{{citation
 | last1 = Grünbaum
 | first1 = B.
 | author1-link = Branko Grünbaum
 | last2 = Shephard
 | first2 = G.C.
 | author2-link = Geoffrey Colin Shephard
 | doi = 10.1112/blms/1.3.257
 | journal = [[Bulletin of the London Mathematical Society]]
 | mr = 0250188
 | pages = 257–300
 | title = Convex polytopes
 | url = http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf
 | volume = 1
 | issue = 3
 | year = 1969
 | access-date = 2017-02-21
 | archive-url = https://web.archive.org/web/20170222114014/http://www.wias-berlin.de/people/si/course/files/convex_polytopes-survey-Gruenbaum.pdf
 | archive-date = 2017-02-22
 }}. See in particular the bottom of page 260.</ref>

Abstract polyhedra also have duals, obtained by reversing the [[partial order]] defining the polyhedron to obtain its [[Duality (order theory)|dual or opposite order]].<ref name=grunbaum-same/> These have the same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition.<ref name=acoptic/>