Dual polyhedron: Difference between revisions
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[[File:Polyhedron pair 6-8.png|thumb|right|upright=1.35|The dual of a [[cube]] is an [[octahedron]]. Vertices of one correspond to faces of the other, and edges correspond to each other.]] | [[File:Polyhedron pair 6-8.png|thumb|right|upright=1.35|The dual of a [[cube]] is an [[octahedron]]. Vertices of one correspond to faces of the other, and edges correspond to each other.]] | ||
In [[geometry]], every [[polyhedron]] is associated with a second '''dual''' structure, | In [[geometry]], every [[polyhedron]] is associated with a second '''dual''' structure, wherein the [[Vertex (geometry)|vertices]] of one correspond to the [[Face (geometry)|faces]] of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.<ref>{{harvtxt|Wenninger|1983}}, "Basic notions about stellation and duality", p. 1.</ref> Such dual figures remain combinatorial or [[Abstract polytope|abstract polyhedra]], but not all can also be constructed as geometric polyhedra.<ref>{{harvtxt|Grünbaum|2003}}</ref> Starting with any given polyhedron, the dual of its dual is the original polyhedron. | ||
Duality preserves the [[Symmetry|symmetries]] of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra{{snd}}the (convex) [[Platonic solid]]s and (star) [[Kepler–Poinsot polyhedra]]{{snd}}form dual pairs, where the regular [[tetrahedron]] is [[#Self-dual polyhedra|self-dual]]. The dual of an [[Isogonal figure|isogonal]] polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an [[Isohedral figure|isohedral]] polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an [[Isotoxal figure|isotoxal]] polyhedron (one in which any two edges are equivalent [...]) is also isotoxal. | Duality preserves the [[Symmetry|symmetries]] of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra{{snd}}the (convex) [[Platonic solid]]s and (star) [[Kepler–Poinsot polyhedra]]{{snd}}form dual pairs, where the regular [[tetrahedron]] is [[#Self-dual polyhedra|self-dual]]. The dual of an [[Isogonal figure|isogonal]] polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an [[Isohedral figure|isohedral]] polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an [[Isotoxal figure|isotoxal]] polyhedron (one in which any two edges are equivalent [...]) is also isotoxal. | ||
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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. | 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. | ||
One way of describing the self-duality of a polyhedron is through a [[permutation]] of the set of vertices and faces that maps every vertex to a face and every face to a vertex, preserving vertex-face incidences. The inverse of a self-duality permutation is another such permutation, and it is natural to expect a self-duality permutation to be an [[Involution (mathematics)|involution]] (a self-inverse permutation), but there exist self-dual convex polyhedra for which all self-duality permutations are not involutions. An example published by Stanislav Jendroľ in 1989 has 14 vertices and 14 faces.<ref>{{citation | |||
| last = Jendroľ | first = Stanislav | |||
| doi = 10.1016/0012-365X(89)90144-1 | |||
| issue = 3 | |||
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]] | |||
| mr = 992743 | |||
| pages = 325–326 | |||
| title = A non-involutory selfduality | |||
| volume = 74 | |||
| year = 1989}}</ref> | |||
==Dual polytopes and tessellations== | ==Dual polytopes and tessellations== | ||
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{{DEFAULTSORT:Dual Polyhedron}} | {{DEFAULTSORT:Dual Polyhedron}} | ||
[[Category:Polyhedra]] | [[Category:Polyhedra]] | ||
[[Category:Duality | [[Category:Duality (mathematics)|Polyhedron]] | ||
[[Category:Self-dual polyhedra| ]] | [[Category:Self-dual polyhedra| ]] | ||
[[Category:Polytopes]] | [[Category:Polytopes]] | ||