# The structure of realization spaces of polyhedral graphs

Given a polyhedral graph with $v$ vertices, $e$ edges and $f$ faces, each possible realization of the graph as a geometric (convex) polyhedron corresponds to a point in $\mathbb{R}^{3v}$, corresponding to the cooordinates of its vertices. How does the so called realization space $S_1(P) \subset \mathbb{R}^{3v}$ look like consisting of the possible realizations of a given polyhedral graph $P$? How could its dimension $d_1$ be calculated?

There may be better and more natural parametrizations, e.g. by the $2e$ angles of the faces (if it is true that two polyhedra with the same angles are essentially the same, up to translations, rotations and size). How does the subset $S_2(P) \subset \mathbb{R}^{2e}$ look like? Is its dimension in fact $d_2 = d_1 - 7$? (The allowed realizations are constrained by the requirement that the sum of the angles of a given face must be $(e_f - 2)\pi$ with $e_f$ the number of edges of $f$, and that the sum of angles at a given vertex must be less than $2\pi$ because the polyhedron is convex.)

Or by the $e$ lengths of the edges (in this case assuming that two polyhedra with the same edge lengths are essentially the same, up to translations and rotations). Does $S_3(P) \subset \mathbb{R}^{e}$ have dimension $d_3 = d_1 - 6$? (The allowed realizations are constrained by the triangle inequality.)

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@Joriki: Look at this answer: mathoverflow.net/questions/68721/… . Gil talks of $e$ as the "right" dimension. But maybe of another space? –  Hans Stricker Jan 24 '13 at 10:44
This might be related - are you familiar with Thurston's paper on realizations of triangulations of the sphere? In the first section, he studies the space of hexagons in a lattice of equilateral triangles. You might find it useful. –  Neal Jan 24 '13 at 14:27

## 1 Answer

I suggest the place to start is Jurgen Richter-Gebert's 1996 book (which I believe was a revision of his PhD thesis), entitled

Realization spaces of polytopes. (Springer link)

You can also find PDF versions on the web by searching for that title. Also I think that Ziegler's Lectures on Polytopes (Graduate Texts in Mathematics 152, 1994) has a section on realization spaces, summarizing Richter-Gebert's results, but I cannot look that up at the moment.

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