                            Dual models

    Refer to Dual Models by Magnus J. Wenninger (C.U.P, ISBN 0-521-24524-9)
or Mathematical Models by Cundy and Rollett for more details than can be
given here.

    Stellate can make the duals of uniform polyhedra. A dual model has faces
corresponding to the original models vertices and vertices corresponding to
the originals faces. See the example in the Draw file Reference.DualFig.

    For example:
an icosahedron has 12 vertices, each surrounded by 5 3-sided faces, 20 in all;
a dodecahedron has 20 vertices, each surrounded by 3 5-sided faces, 12 in all.
    So the icosahedron converts to the dodecahedron by swapping vertices and
faces: they are dual models.

    A uniform polyhedron has all vertices the same so the dual of a uniform
polyhedron has all faces the same. Stellate takes all the vertices of the
original uniform polyhedron and calculates the planes of its dual. These
planes are all the same (apart from a possible reflection, see below)
and they lie at the same distance from the origin with normals in the
directions of the originals vertices. Stellate makes them into a stellation
diagram and allows you to select areas in it in just the same way as a
normal solid.

    Since the vertices of the dual (corresponding to the planes of the
original solid) are at a distance inversely proportional to the distance of
the originals planes there is a difficulty with those uniform polyhedra
which have planes through the centre (Hemi-models e.g. W67). The vertices
corresponding to these planes are at infinity. Stellate arbitrarily cuts
them off with an invented second component. Try W67 and W68 to see this
effect.

Wenninger numbers of hemi-models are:
 67, 68, 78, 89, 91, 100, 102, 106, 107, 119.

           Making a dual with Stellate

    First make the uniform polyhedron; for example, select W7, the truncated
octahedron from the Wenninger number menu, then choose Auto Select in the
Parameters window. Then Plot Solid gives you the 3D view of the
truncated octahedron. Click Menu over the 3D window and move over the
arrow opposite Dual; this shows that the dual, named Tetrakis Hexahedron
has already been defined (the dual has to have a name before it can be
created). Move back and click on Dual to open the stellation diagram of
the tetrakis hexahedron.
   If you had used Plot Solid immediately from the Parameters window,
the program assumes that you are not interested in stellation diagrams and
would show the tetrakis hexahedron solid when you clicked on Dual; then
you would have to close the 3D window with <Adjust> to see the duals
stellation diagram.
   Try making the first stellation of the tetrakis hexahedron by selecting
the 3 triangles immediately surrounding the central triangle in the
stellation diagram.

   The duals stellation diagram and the resulting solid may be saved in
the usual way but saved line instructions will contain both the original
polyhedron and the dual so that they can both be recreated by Stellate
later.

           Reflected faces.

Some uniform polyhedra have vertices of two kinds; by definition they must
have the same faces at each vertex, but the order of these faces can invert.
The Platonic solids have only one kind of face so this can not happen. It
also does not happen with Archimedean solids with two types of faces but it
can in those with 3 types.

   The Rhombicosidodecahedron (W14) has a pentagon, a square, a triangle and
a square in that order. Inverting the order of these makes no difference.
   The Rhombitruncated Cuboctahedron (W15) on the other hand has a square, a
hexagon and an octagon at half its vertices while at the other half it has a
square, an octagon and a hexagon. That is, the order of the faces inverts.

   This means that the dual model has two kinds of faces which are mirror
images of each other and it will not fit together with identical faces. 

   Click on the Reflect icon in the stellation diagram panel to try to
correct this problem. 

   Wenningers models needing Reflect for their duals are:
15, 16, 79, 84, 93, 98, 101, 103, 108, 109.