Dual Polyhedron - Dual Polytopes and Tessellations

Dual Polytopes and Tessellations

Duality can be generalized to n-dimensional space and dual polytopes; in 2-dimensions these are called dual polygons.

The vertices of one polytope correspond to the (n − 1)-dimensional elements, or facets, of the other, and the j points that define a (j − 1)-dimensional element will correspond to j hyperplanes that intersect to give a (n − j)-dimensional element. The dual of a honeycomb can be defined similarly.

In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For regular and uniform polytopes, the dual facets will be the polar reciprocals of the original's facets. For example, in four dimensions, the vertex figure of the 600-cell is the icosahedron; the dual of the 600-cell is the 120-cell, whose facets are dodecahedra, which are the dual of the icosahedron.