Illustrated here is a "transpolyhedron" which is based on the truncated icosidodecahedron and its dual. But the illustration can't capture the fact that when you view it, the different faces are continuously growing and shrinking
Transpolyhedron is a term coined by Haresh Lalvani (see the references) to describe a transitional form, intermediate between a polyhedron and its dual. They display the duality relationship in a very graphic manner, related to, but distinct from the way that a compound of a polyhedron and its dual displays the relationship.
A transpolyhedron contains all the faces of a given polyhedron, plus all the faces of its dual, plus a set of connecting rectangles---one for each edge of the original and/or dual. The relative sizes of the original and dual faces are adjustable, with the rectangles chosen to fill the intervening spaces. In the illustration, the decagons, hexagons, and squares derive from the original truncated icosidodecahedron; the triangles are from the dual; and the greenish-blue rectangles fill the gaps.
This page presents transpolyhedra in a way which is not just transitional, but also transformational. All these objects are defined with inherent movement, which causes them to oscillate between the original form and the dual form, displaying all the intermediate transpolyhedral forms.
Note: To start the transformation, click on the initial polyhedron within the viewer. You can spin it as it transforms, by dragging in the background sky area.