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The configuration space of all tetrahedra is $5$-dimensional, perhaps a non-obvious fact. There are $12$ face angles, but the sum of each of the four faces angles is $\pi$, reducing $12$ to $8$ degrees of freedom. There are, however, additional trigonometric relations that must be satisfied by the angles, as cited in the Wikipedia article. This reduces the dimension of the configuration space to $5$ dimensions.



Wikipedia image: product of sines of marked angles are equal.

My question is:

Is there a generalization to other triangulated convex polyhedra?

For example, consider the space of all convex polyhedra that are combinatorially equivalent to a regular octahedron. Here we have $24$ face angles ($4$ per vertex), but then $8$ triangle angles summing to $\pi$ reduces the $24$ to $16$ degrees of freedom. Presumably there are additional trigonometric relations that further reduce the dimension of the configuration space. My guess is to $10$ dimensions.

Perhaps it is better to think in terms of vertex coordinates rather than in terms of face angles?

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    $\begingroup$ Related question with answer on M.SE: math.stackexchange.com/q/4108507/415941. The post gives an intuition for why the dimension is close to the number of edges. And the comments mention that this holds for non-simplicial polyhedra as well. $\endgroup$
    – M. Winter
    Sep 22, 2021 at 8:06

2 Answers 2

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For simplicial polyhedra, one can simply add up the degrees of freedom from placing the vertices in space and substract 7 degrees of freedom for rotation, translation, and scaling to obtain $3v-7$ as in Alexandre Eremenko's answer.

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  • $\begingroup$ Very clear and concise---Thanks! $\endgroup$ Sep 22, 2021 at 21:52
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A convex polyhedron can be considered as a flat metric with conic singularities on the sphere. This metric is completely determined (up to a constant factor) by conformal structure of the sphere with $v$ marked points ($v$ is the number of vertices) and $v$ angles around these points (sums of angles of faces meeting at a vertex). Three points can be fixed and the rest depend of $2$ real parameters each. The angles satisfy one relation (coming from Gauss-Bonnet). So the total number of parameters is $$2(v-3)+v-1=3v-7.$$ For the tetrahedron, $v=4$ and we obtain $5$.

(Since your argument involves only angles, I suppose you counted tetrahedra up to scaling, as I did).

The fact that every flat metric with conic singularities and angles $<2\pi$ at each singularity corresponds to a convex polytope embedded in $R^3$ is non-trivial, but I suppose it is true.

Remark: Your condition that it is "triangulated", if I understand it correctly, is irrelevant: the surface of any convex polytope can be triangulated without adding new vertices.

Remark 2: this answer is less elementary but contains some extra information in comparison with Yoav Kallus's answer: it gives a global parametrization of the set of convex polytopes, not only the dimension count. It also works for hyperbolic or spherical polytopes (assuming that the angles are less than $2\pi$.

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  • $\begingroup$ You can also get the same answer from the vertex coordinates: 3v before rotation (3), translation (3) and scaling (1). $\endgroup$ Sep 22, 2021 at 1:59
  • $\begingroup$ That is a much simpler argument; why don't you post it as an answer? $\endgroup$ Sep 22, 2021 at 2:01
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    $\begingroup$ Done. Regarding your first remark, I think the point is that a triangular bipyramid will have one more degree of freedom than a quadrilateral pyramid because of the extra constraint that the four points on the quadrilateral faces be coplanar. $\endgroup$ Sep 22, 2021 at 2:27
  • $\begingroup$ Very insightful---Thanks. I believe it is Alexandrov's "Gluing theorem" which guarantees that "every flat metric with conic singularities and angles $<2\pi$ at each singularity corresponds to a convex polytope embedded in $\mathbb{R}^3$." $\endgroup$ Sep 22, 2021 at 10:02
  • $\begingroup$ @Joseph O'Rourke: You are right; at the time of writing I just forgot who proved this (Alexandrov or Pogorelov). $\endgroup$ Sep 22, 2021 at 12:22

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