Dual Polyhedron - Dorman Luke Construction

Dorman Luke Construction

For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the Dorman Luke construction. This construction was originally described by Cundy & Rollett (1961) and later generalised by Wenninger (1983).

As an example, here is the vertex figure (red) of the cuboctahedron being used to derive a face (blue) of the rhombic dodecahedron.

Before beginning the construction, the vertex figure ABCD is obtained by cutting each connected edge at (in this case) its midpoint.

Dorman Luke's construction then proceeds:

1. Draw the vertex figure ABCD
2. Draw the circumcircle (tangent to every corner A, B, C and D).
3. Draw lines tangent to the circumcircle at each corner A, B, C, D.
4. Mark the points E, F, G, H, where each tangent line meets the adjacent tangent.
5. The polygon EFGH is a face of the dual polyhedron.

In this example the size of the vertex figure was chosen so that its circumcircle lies on the intersphere of the cuboctahedron, which also becomes the intersphere of the dual rhombic dodecahedron.

Dorman Luke's construction can only be used where a polyhedron has such an intersphere and the vertex figure is cyclic, i.e. for uniform polyhedra.