wendy wrote:So the edges of {3,3,5} run through the centres of {5,3,3}.

Wendy missed to specify the type of centers in the above statement.

But you can go along the following lines.

The mere

combinatorical topology of duals and their elements can be obtained as follows:

- vertices of the dual polychoron are obtrained as the set of cell centers of the original
- edge midpoints of the dual polychoron are obtrained as the face centers of the original
- face centers of the dual polychoron are obtrained as the edge midpoints of the original
- cell centers of the dual polychoron are obtrained as the vertices of the original

If you were after a

metrical description instead, you probably would do best apply the construction of

spherical reciprocation:

- choose some sphere around the bulk center of your polychoron and use its radius for unity of measure
- construct the hyperplanes perpendicular to the rays upto the originals vertices, displaced from the bulk center by the inverse of the former vertex distance (wrt. above chosen units!)
- intersection of any pair of these hyperplanes will happen according to the rule that the according originals vertices are connected by an edge or not

--- rk