Self-dual polytope

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A self-dual polytope is a polytope whose dual is isomorphic to itself. The number of elements of a dual polytope follow a palindromic pattern. The facets of a self-dual polytope are equivalent to the dual of its vertex figure(s). A self-dual polytope that is also isogonal is also isotopic, making it a noble polytope.

Polygons[edit | edit source]

All polygons are abstractly self-dual, as they have the same number of vertices and edges.

Regular polygons in 2D are also self-dual. However regular skew polygons do not necessarily share this property. Finite regular polygons are self-dual in even dimensions and not in odd dimensions. For example the dual of the skew hexagon (3D) is a flat hexagon, so it is not self-dual, however the pentagonal-pentagrammic coil (4D) is self-dual. Infinite regular polygons follow the opposite pattern, being self-dual in odd dimensions and not in even dimensions. For example the zigzag (2D) is not self-dual, its dual is the flat apeirogon, while the triangular helix (3D) is self-dual.

Polychora[edit | edit source]

Of the regular polychora the pentachoron, icositetrachoron, great hecatonicosachoron, and grand stellated hecatonicosachoron are all self-dual.

Higher dimensions[edit | edit source]

For polytopes of arbitrary rank, the regular simplex and the regular hypercubic honeycomb are also self-dual. For rank 5 and up these are the only self-duals among the planar, non-dense regular polytopes.

Properties[edit | edit source]

  • The pyramid of a self-dual polytope is also self-dual.
  • The dual compound of a self-dual regular polytope is itself regular. For example the stella octangula is a regular compound formed from a tetrahedron and its dual.
  • The blend of two self-dual polytopes of the same rank is also self-dual.