Duals by Rotating the Edges of Polyhedra

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tetrahedron

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There are several ways to construct the dual

D

of a polyhedron

P

. Roughly speaking,

P

and

D

switch faces and vertices and the edges flip around. The dual of

D

is

P

.

Here is a combinatorial definition. The vertices

v

i

of

D

are the centers of the faces

f

i

of

P

. An edge of

D

joins

v

i

and

v

j

if their corresponding faces

f

i

and

f

j

were adjacent in

P

. Each face

f

of

D

corresponds to a vertex

v

of

P

: if

v

was a vertex of the faces

f

1

,

f

2

, … of

P

, then

f

is the polygon with vertices

v

1

,

v

2

, …. This definition leads to skew polygonal faces if

P

is not concave or has holes.

The dual can be constructed by various geometric operations on

P

: truncating either at vertices or edges, augmenting faces, or by stellation.

The number of edges of

P

and of

D

are the same. Could the edges of

P

be transformed to form

D

? Yes: by rotating each edge about the axis from the center of

P

to the midpoint of the edge.