page_title: Dual polyhedron
section_title: Self-dual polytopes and tessellations
hierarchical_section_title: Dual polyhedron / Dual polytopes and tessellations / Self-dual polytopes and tessellations
caption_attribution_description: English: Two square tilings. Esperanto: Du kvadrataj kahelaroj.
context_page_description: In geometry, any polyhedron is associated with a second dual figure, 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 are also 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 also belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of an isotoxal polyhedron is also isotoxal.
  Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
context_section_description: The primary class of self-dual polytopes are regular polytopes with palindromic Schläfli symbols. All regular polygons, {a} are self-dual, polyhedra of the form {a,a}, 4-polytopes of the form {a,b,a}, 5-polytopes of the form {a,b,b,a}, etc.
  The self-dual regular polytopes are:
  All regular polygons, {a}.
  Regular tetrahedron: {3,3}
  In general, all regular n-simplexes, {3,3,...,3}
  The regular 24-cell in 4 dimensions, {3,4,3}.
  The great 120-cell {5,5/2,5} and the grand stellated 120-cell {5/2,5,5/2}
  The self-dual (infinite) regular Euclidean honeycombs are:
  Apeirogon: {∞}
  Square tiling: {4,4}
  Cubic honeycomb: {4,3,4}
  In general, all regular n-dimensional Euclidean hypercubic honeycombs: {4,3,...,3,4}.
  The self-dual (infinite) regular hyperbolic honeycombs are:
  Compact hyperbolic tilings: {5,5}, {6,6}, ... {p,p}.
  Paracompact hyperbolic tiling: {∞,∞}
  Compact hyperbolic honeycombs: {3,5,3}, {5,3,5}, and {5,3,3,5}
  Paracompact hyperbolic honeycombs: {3,6,3}, {6,3,6}, {4,4,4}, and {3,3,4,3,3}
caption: The square tiling, {4,4}, is self-dual, as shown by these red and blue tilings
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