# Duals by Rotating the Edges of Polyhedra

Duals by Rotating the Edges of Polyhedra

There are several ways to construct the dual of a polyhedron . Roughly speaking, and switch faces and vertices and the edges flip around. The dual of is .

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Here is a combinatorial definition. The vertices of are the centers of the faces of . An edge of joins and if their corresponding faces and were adjacent in . Each face of corresponds to a vertex of : if was a vertex of the faces , , … of , then is the polygon with vertices , , …. This definition leads to skew polygonal faces if is not concave or has holes.

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The dual can be constructed by various geometric operations on : truncating either at vertices or edges, augmenting faces, or by stellation.

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The number of edges of and of are the same. Could the edges of be transformed to form ? Yes: by rotating each edge about the axis from the center of to the midpoint of the edge.

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