Something after after duals?

Question

So if I have a polyhedron, x. Is it possible for a shape to be inscribed inside x (y). And inside y is z, but inside z is x. Is there a shape that could be x? If so what is it?
Also, x, y, and z are different shapes.

Answer 1

Well, you can come up with sporadic examples: Start with an octahedron, whose dual is a cube.

Inside a cube, you can inscribe a tetrahedron (by choosing "every other" vertex of the cube).

Then, use the midpoint of each edge of the tetrahedron to inscribe an octahedron. (Note that the final octahedron will have a different orientation than what you started with; it isn't a scaled version of the first octahedron, as would happen with a double dual).

In general, let's say we call $d(P)$ the dual of a polyhedron $P$. Now, $d(P)$ has certain nice properties related to the dual of a vector space, giving us that $d \circ d (P) = d^2(P) = P$; we go into the dual space for the dual, and back to our space for the double dual (although our space and the dual are isomorphic, so we can stay in our space if we please).

It seems that, if we call your desired map $t$ and demand that $t^3(P) = P$, then it will be hard for $t$ to have properties that are as nice as those of $d$, as we can't use the dual space in an obvious way.