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Was (wer) ist dual to - definition
![Canonical [[dual compound]] of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common [[midsphere]].](https://commons.wikimedia.org/wiki/Special:FilePath/Dual compound 6-8 max.png?width=200 "Canonical [[dual compound]] of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common [[midsphere]].")
![The [[Infinite-order apeirogonal tiling]], {∞,∞} in red, and its dual position in blue](https://commons.wikimedia.org/wiki/Special:FilePath/Infinite-order apeirogonal tiling and dual.png?width=200 "The [[Infinite-order apeirogonal tiling]], {∞,∞} in red, and its dual position in blue")
![topological dual]].<br>Images from [[Kepler]]'s [[Harmonices Mundi]] (1619)](https://commons.wikimedia.org/wiki/Special:FilePath/Ioanniskepplerih00kepl 0271 crop.jpg?width=200 "topological dual]].<br>Images from [[Kepler]]'s [[Harmonices Mundi]] (1619)")
![The [[square tiling]], {4,4}, is self-dual, as shown by these red and blue tilings](https://commons.wikimedia.org/wiki/Special:FilePath/Kah 4 4.png?width=200 "The [[square tiling]], {4,4}, is self-dual, as shown by these red and blue tilings")
 = \varphi(x_1 + x_2) = \varphi(x)</math>'' for any ''<math>\varphi</math>'' in the dual space.](https://commons.wikimedia.org/wiki/Special:FilePath/Double dual nature.svg?width=200 "''x''<sub>1</sub> + ''x''<sub>2</sub>}}.
The addition +′ induced by the transformation can be defined as ''<math>[\Psi(x_1) +' \Psi(x_2)](\varphi) = \varphi(x_1 + x_2) = \varphi(x)</math>'' for any ''<math>\varphi</math>'' in the dual space.")
Wikipedia
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
Duality preserves the 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 – the (convex) Platonic solids and (star) Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an isotoxal polyhedron (one in which any two edges are equivalent [...]) is also isotoxal.
Duality is closely related to polar reciprocity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
