Editing Dual polyhedron (section)

==Self-dual polyhedra==
Topologically, a polyhedron is said to be '''self-dual''' if its dual has exactly the same connectivity between vertices, edges, and faces. Abstractly, they have the same [[Hasse diagram]]. Geometrically, it is not only topologically self-dual, but its polar reciprocal about a certain point, typically its centroid, is a similar figure. For example, the dual of a regular tetrahedron is another regular tetrahedron, [[reflection through the origin|reflected through the origin]].

Every polygon is topologically self-dual, since it has the same number of vertices as edges, and these are switched by duality. But it is not necessarily self-dual (up to rigid motion, for instance). Every polygon has a [[regular polygon|regular form]] which is geometrically self-dual about its intersphere: all angles are congruent, as are all edges, so under duality these congruences swap. Similarly, every topologically self-dual convex polyhedron can be realized by an equivalent geometrically self-dual polyhedron, its [[canonical polyhedron]], reciprocal about the center of the [[midsphere]].

There are infinitely many geometrically self-dual polyhedra. The simplest infinite family is the [[Pyramid (geometry)|pyramids]].<ref name="wohlleben">{{citation
 | last = Wohlleben | first = Eva
 | editor-last = Cocchiarella | editor-first = Luigi
 | year = 2019
 | contribution = Duality in Non-Polyhedral Bodies Part I: Polyliner
 | title = ICGG 2018 - Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary - Milan, Italy, August 3-7, 2018
 | series = Advances in Intelligent Systems and Computing
 | volume = 809
 | url = https://books.google.com/books?id=rEpjDwAAQBAJ&pg=PA485
 | publisher = Springer
 | page = 485&ndash;486
 | isbn = 978-3-319-95588-9
 | doi = 10.1007/978-3-319-95588-9
}}</ref> Another infinite family, [[elongated pyramid]]s, consists of polyhedra that can be roughly described as a pyramid sitting on top of a [[prism (geometry)|prism]] (with the same number of sides). Adding a frustum (pyramid with the top cut off) below the prism generates another infinite family, and so on. There are many other convex self-dual polyhedra. For example, there are 6 different ones with 7 vertices and 16 with 8 vertices.<ref>3D [[Java (programming language)|Java]] models at [http://dmccooey.com/polyhedra/SymmetricSelfDuals.html Symmetries of Canonical Self-Dual Polyhedra], based on paper by Gunnar Brinkmann, Brendan D. McKay, ''Fast generation of planar graphs'' [[PDF]] [http://cs.anu.edu.au/~bdm/papers/plantri-full.pdf]</ref>

A self-dual non-convex icosahedron with hexagonal faces was identified by Brückner in 1900.<ref>Anthony M. Cutler and Egon Schulte; "Regular Polyhedra of Index Two", I; ''Beiträge zur Algebra und Geometrie'' / ''Contributions to Algebra and Geometry'' April 2011, Volume 52, Issue 1, pp 133–161.</ref><ref>N. J. Bridge; "Faceting the Dodecahedron", ''Acta Crystallographica'', Vol. A 30, Part 4 July 1974, Fig. 3c and accompanying text.</ref><ref>Brückner, M.; ''Vielecke und Vielflache: Theorie und Geschichte'', Teubner, Leipzig, 1900.</ref> Other non-convex self-dual polyhedra have been found, under certain definitions of non-convex polyhedra and their duals.