Exactly 5 Platonic solids: Where in the proof do we need convexity and regularity?

Question

The famous statement that

Only five convex regular polyhedra exist.

is usually proven as follows:

Let $P$ be a convex regular polyhedron with

• V vertices,
• E edges and
• F faces.

Moreoever, let

• n be the number of edges of each face and
• c be the number of edges which meet at a vertex.

Then $F=\frac{2E}{n}$ and $V=\frac{2E}{c}$ and $c\geq 3, n\geq 3$.

Substituting this into Euler's polyhedron formula $$ V-E+F=2, $$

and doing some easy calculations, one gets that only

$$ (V,E,F)\in\{(4,6,4),(8,12,6),(20,30,12),(6,12,8),(12,30,20)\} $$ are possible combinations.

So far, so good.

But where do we need that $P$ is convex and regular?

(I think the regularity is at least used for $n\geq 3, c\geq 3$, isn't it?)

Answer 1

As already said in the comments, regularity means being composed of equal faces, thus enabling to connect the numbers $V$, $E$ and $F$ by some algebraic relations. This is the left part of Euler's identity $$V-E+F=2$$ Now, convexivity is, in fact, the right-hand side.

In general, for a surface $S$, the formula reads $$V-E+F=\chi(S)$$ where $\chi$ is the Euler characteristic (defined by the equation above or, alternatively, by the alternating sum of dimensions of homology groups).

If a polyhedron is convex, it can be proven that it's boundary is homeomorphic (topologically equivalent) to a sphere $\mathbb{S}^2$, and $\chi(\mathbb{S}^2)=2$, providing the right part of Euler's equation.

So, convex is just a simplification; the classification really works for all polyhedra homeomorphic to a ball. For some other topology, a different classification may arise.

Answer 2

The regularity is used to reduce consideration to a single face or vertex because they are all similar. Convexity comes from Euler's formula used with regularity implicitly. That is, since every vertex is similar to every other vertex, the solid must be convex at every vertex if it is convex at one vertex, otherwise if is not convex at a vertex then all vertices are not convex and Euler's formula must be modified.

You can read the Wikipedia article Euler characteristic for more details about the role of convexity and it also links to Wikipedia article "Proofs and refutations" which is very relevant.

Answer 3

There are five Platonic Solids because their definition restricts them to polyhedra.

A Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex.

If all geometric structures are included the result is the twelve FFELLONIC FORMS, a complete series ranging from a triangle to the densest possible space filling honeycomb.