Although the zonohedra form a subset of this class of zonish solids, it is a small proper subset. As the figure at right indicates, zonish polyhedra may have faces with an odd number of sides, e.g., triangles and pentagons, which are not possible in zonohedra. This example consists of 122 faces: 12 pentagons, 20 triangles, 60 squares, and 30 golden rhombi (arranged as in the rhombic triacontahedron). As the color indicates, it can be seen as an icosidodecahedron enlarged by the addition of six (tan colored) zones.
The key idea of zonish polyhedra is a generalization of the well known method of constructing a zonohedron in which one zone is added at a time to a smaller zonohedron, starting from a point or a parallelepiped. The only difference is that here we allow any polyhedron to be a "seed" for the process. So a zonish polyhedron is characterized by a convex seed polyhedron (oriented in space) and a set of zonal directions (a "star") which is used to expand the seed. If the seed is a zonohedron, the result is again a zonohedron, but when the seed is chosen not to be a zonohedron, new "zonish" polyhedra are formed. For a symmetric result, the seed and the star directions should be of the same symmetry group.
To illustrate the process, we begin with an icosidodecahedron as seed, and add one zone in a direction parallel to a 5-fold axis. The zonal edge length was chosen to be equal to the seed edge length so the ten new faces are squares rather than rectangles. The result is a Johnson solid, known as elongated pentagonal gyrobirotunda, which is formed by splitting the seed into two hemispheres and inserting edges parallel to the the chosen zonal edge direction. Adding second zone along the direction of a second 5-fold axis gives a result with two zones. Two opposite faces, where the two zones cross, are golden rhombi. The addition of a third zone can be done in two ways. If the three zones all surround some triangle, we get an oblate result. If they have no triangle in common, we get a prolate result. Continuing until zones are constructed for all six axes of 5-fold symmetry gives the six-zone result illustrated above. Note that the final construction depends only on the seed and the set of zonal directions, not the order in which they are used. Also, faces which are not from the seed are always centrally symmetric and part of a zone.
Paper models of the above example are easy to make as there are only four simple types of faces. It can also be quickly approximated with the zometool, since all the edges are in the directions of icosahedral symmetry axes. Not all edges can be equal with the zometool however, as blue and red sticks are required, so the squares must be realized as rectangles.
As a second example, analogous but with octahedral symmetry, we can begin with the cuboctahedron and expand with four zones in the directions of the axes of 3-fold symmetry. The result is a pleasing pentacontahedron consisting of 8 triangles, 30 squares, and 12 square-root-of-two rhombi (arranged as in the rhombic dodecahedron). I have noticed that a picture of it appears in two books of Peter Pearce: Structure in Nature is a Strategy for Design (p. 49) and Polyhedra Primer (p. 50) but not with this method of construction. (The related pentacontahedron illustrated in those books on the same pages is the 13-zone zonohedron based on the symmetry axes of the cube.)
The above two examples are somewhat special in that they begin with quasi-regular polyhedra and expand in directions orthogonal to their equatorial polygons. As a consequence they are also equivalent to expansions of the seed polyhedron (or their dual rhombic polyhedra) by Mrs. Stott's method of expanding faces. (See W.W. Rouse Ball) If one applies either that method or the zonish expansion (along the three 4-fold axes) to the remaining convex quasiregular polyhedron, the octahedron, the result is the rhombicuboctahedron, yet another zonish polyhedron. However, in most cases, zonish solids and Mrs. Stott's expansions are very different things.
Another similar example, but with only tetrahedral symmetry, uses the tetrahedron as seed and expands in the directions of its four 3-fold axes. The resulting 28-sided solid has 4 triangles, 12 squares, and the same 12 square-root-of-two rhombi. Observe how its faces are a subset of the above pentacontahedron.
Every zonish polyhedron is in a sense intermediate between two related zonohedra, one which it contains and one which contains it:
Just as an n-zone zonohedron is the convex hull of the projection of
an n-dimensional hypercube into 3-space, so a zonish polyhedron with n
zones is the projection of a n+3 dimensional object --- the cross of the
seed and an n-dimensional hypercube --- into 3-space. And incidentally,
the process can be applied to higher-dimensional seed polytopes as well.
If someone writes a 4D viewer Netscape plug-in, I'll make some zonish data
files for it.