Platonic solids but dropping convexity: Can we still show that two Platonic solids of the same $V, E, F$ are similar in $\mathbb{R}^{3}$?

Question

Note.

This question was borne out of a slight ambiguity in the definition of Platonic solids (whether it assumes convexity). When I first made this question I did not think convexity was part of the definition of Platonic solids. Since then, I am interested to see whether "convexity" is even needed when trying to prove there are only five Platonic solids.

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Problem.

I was looking into the proof that there are only five Platonic solids in Basic Concepts of Algebraic Topology by F.H. Croom at page 29, Theorem 2.7. To clarify,

• We define a Platonic solid as a simple, regular polyhedron homeomorphic to $S^{2}$.
• We define a simple polyhedron to be a polyhedron that does not self-intersect itself.
• We define a regular polyhedron to be a polyhedron whose faces are regular polygons all congruent to each other and whose local regions near the vertices are all congruent to each other.

Using homology theory, one can prove that the Euler formula $V-E+F=2$ must hold for Platonic solids. Then by using Euler's formula and invoking a counting argument, we find that there are five possible tuples $(V, E, F)$. This is a beautiful proof, but I am unsatisfied with a question: How do we know there can't be two non-similar Platonic solids that have the same $(V, E, F)$-tuple?

Almost all sources I've looked at seem to assume it is obvious that two Platonic solids with the same $(V, E, F)$-tuple are similar, and it is not obvious to me.

Does anyone have any suggestions for how to prove this? Alternatively, does anyone know of a reference where this is proved rigorously?

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Progress.

One poster invoked Cauchy's rigidity theorem, which is proven in, e.g., Proofs From the BOOK by Aigner & Zeigler. One can show that any two Platonic solids that have the same $(V, E, F)$-tuple must be combinatorically equivalent. However, in order for the theorem to apply, we need to show that our Platonic solids are convex. I can't seem to think of any rigorous argument for why the Platonic solids have to be convex. And actually, you don't need to show that the entire polyhedron is convex. If I'm not mistaken, the proof for Cauchy's rigidity theorem only relies on the vertices of the polyhedron being convex. So really it suffices to show the vertices are convex.

I've been able to prove uniqueness for Platonic solids that have three faces meet at each vertex. This is because vertices with three faces are "rigid," so there is only one way obtain such a vertex and the resulting unique vertex is convex. Hence Cauchy's rigidity theorem applies.

Next, I've been able to prove uniqueness for the case where four faces meet at each vertex. This is because there is no way to join four congruent faces at a vertex without making the vertex convex. Thus, Cauchy's rigidity theorem applies here as well.

The icosahedron remains and it has been giving me a lot of trouble. I can reduce my problem to the following question: Given a polyhedron with properties #1-#6, is the polyhedron necessarily the standard icosahedron? The properties are:

1. $(V, E, F) = (12, 30, 20)$.
2. No self-intersections.
3. Each vertex is locally congruent in the following sense: If you take an open ball centered at a vertex, intersect it with the polyhedron, the results will be identical up to isometries.
4. All faces are regular polygons congruent to each other.
5. The polyhedron is homeomorphic to $S^{2}$.
6. The polyhedron is a subset of $\mathbb{R}^{3}$.

Answer 1

Edit: This answer invokes rotational symmetry about the vertices, which isn't required in the OP, so this is not a complete answer without a proof that such symmetry ought to hold.

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Given $V,E,F$, you can compute the number of sides $n$ of each face ($2E/F$) and the degree $k$ of each vertex ($2E/V$).

By regularity, a Platonic solid with degree $k$ should be invariant under a rotation by $360/k$ degrees about a vertex. This forces the arrangement of regular $n$-gons at a vertex to be "rigid", because this rotational symmetry along with a specific internal angle of each polygonal face forces a unique relative position of the edges from a vertex, up to rotation of the whole configuration in space. See Eric Wofsey's comment for more elaboration. (Also note that the uniqueness of such a configuration is reliant on there being no self-intersections between the polygons around the vertex; if you allow this, the Kepler-Poinsot polyhedra result.)

Furthermore, this rigid configuration about a vertex is fully determined by the position of three of the edges in the configuration; two edges are enough to determine the configuration up to a reflection, and the third tells us whether the vertex is convex or not (whether the corner points "out" or "in").

So, if we start with a vertex and an orientation of its local neighborhood, we get a rigid structure of the polygons meeting at a single vertex. This gives us a ring of "boundary" vertices, where the neighboring polygons have not yet been fully specified. We repeatedly choose a boundary vertex where two boundary polygons meet, which has at least 3 fixed edges attached to it and therefore forces a unique attachment of polygons from that vertex by the above paragraph. By adding in these polygons to our rigid structure, this ring of boundary vertices grows outward, and we can repeat the process. So long as we perform this extension on the boundary vertices closest (in the graph-theoretic sense) to our starting vertex, we will eventually specify the local neighborhood of every vertex in the graph (possibly after infinitely many operations; we aren't assuming that things will naturally close up here.)

The result of this procedure is a surface without boundary, so if the polyhedron is homeomorphic to the $2$-sphere, this process must have terminated after finitely many points and closed up into a unique finite solid.

(Of course, this does not show that any such polyhedron does exist - see this question for an algebraic topology proof of existence. Essentially, the vertex-extending process described above forms a covering space of $S^2$, and so must actually yield the sphere.)

Answer 2

Another way to interpret the $5$ Platonic solids is that they are the only configurations of at least $3$ regular polygons around each vertex satisfying that the total sum of angles at that vertex is less than $180^{\circ}$ Also note that each Platonic solid is uniquely determined by the number of faces around each vertex and the number of sides of each face because all Platonic solids are vertex-transitive and rotationally symmetric about each vertex. Then, note that (a) the number of sides to each face is determined by $n_{f} = \frac{2e}{f}$, and (b) the number of faces around each vertex is determined by $n_{v} = \frac{2e}{v}$. Thus, each Platonic solid is uniquely determined by the choice of $(v, e, f)$, or any two of them given Euler's relation. $\blacksquare$

Answer 3

If I understand you correctly, then you want your polyhedron to be an embedding of the sphere with a well-defined interior. So we can consider it as the boundary of this interior (this will be important later).

We can now speak about the (generalized) Gaussian curvature of this boundary. By the Gauss-Bonnett theorem, the total curvature of this surface is $2\pi$ times the Euler characteristic of the sphere, hence positive. But in a polyhedron, the curvature is concentrated in the vertices (i.e. is zero everywhere else). The "curvature" at a vertex is better know as its angular defect, and since all vertices are locally the same, they all must have the same angular defect. But since these values add up to a positive value, each such angular defect must be positive.

In sum, the curvature is non-negative everywhere. Now there seems to be the following theorem: the boundary of a shape is convex if and only if it has non-negative curvature everywhere (see the answer to this question). Hence your polyhedron is convex.

Once we know this (and since you already know that the combinatorics is unique) you can apply Cauchy's rigidity theorem to conclude the uniqueness of the Platonic solids.

Answer 4

OK so we can imagine two solids, each with the same triple of $(V,E,F)$ each has the same number of faces. And you calculate the number of edges that each face has as $2E/F$ (each edge can be in two faces). So they have the same number of faces and each face has the same number of edges. How many edges meet at each vertex? This is $2E/V$ (each edge hits two vertices). So these two polyhedra have the same number of faces, each with the same number of sides and the same number of edges meeting at each vertex so they must be the same polyhedron. We could build each to the precise formula of faces, number of edges per face and number of edges per vertex and we have no room to make choices. The recipe precisely defines the polyhedron.