The last post had two images, and this is the dual of the second one. I was therefore surprised when I ran into this while playing around with Stella 4d, a program which allows easy polyhedron manipulation. (See http://www.software3d.com/stella.php for free trial download.)
Why did it surprise me?
Well, isn’t a polyhedron. for one thing. It is a collection of irregular and concentric polygons which intersect, but they don’t meet at edges. This doesn’t normally happen, so it requires explanation. I figured it out pretty quickly.
I’ve been using the loosest possibly definition for “faceting,” not insisting that faces meet at each edge in pairs, and even making some faces invisible in order to see the interior structure of the “polyhedra.” Since this breaks the faceting-rules, it isn’t surprising that the dual would fail to be a true polyhedron.
That’s my guess, anyway.
RobertLovesPi.net 
Hey, I wanted to point out the really unusual fact that the Compound of Five Tetrahedra (W_24) is in fact one of the rare polyhedra whose dual figure is its own enantiomorph! I cannot think of another, can you? Additionally, the vertices of this polyhedron create a dodecahedron, therefore Phi is a crucial length involved in this construction.
Personally, W_24 is the most magical of all polyhedra!
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