The regular polyhedra were known at least since the time of the ancient Greeks. The names of the more complex ones are purely Greek. But despite their being known for close to two millenia no one apparently noticed the fact that the sum of the number of faces F and the number of vertices V less the number of edges E is equal to two for all of them; i.e.,
V - E + F = 2
The value of (V-E+F) is usually denoted by the Greek letter Chi (Χ). Thus Χ(cube)=2.
It was the Swiss mathematician Leonhard Euler who recognized and published this fact. The value of two is said to be the Euler characteristic of each of the polyhedra. This value is not changed by stretching or shrinking any side or face or even shrinking a side or face to zero. This means that the Euler characteric is a topological invariant because it is not altered by any continuous mapping.
Another French-Swiss mathematician, Simon Lhuilier (1750-1840), found a slight generalization of Euler's formula to take into account polyhedra having holes. Lhuilier's formula is
V - E + F = 2 − 2G = 2(1− G)
where G is the number of holes in the polyhedron. Thus the Euler characteristic is 2 for a regular polyhedron but 0 for a torus-like polyhedron.
https://www.sjsu.edu/faculty/watkins/eulerpoincare2.htm
Leonhard Euler (/ˈɔɪlər/ OY-lər;[b] 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
https://en.m.wikipedia.org/wiki/Leonhard_Euler
The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra.
Proof of Euler's formula
First steps of the proof in the case of a cube
There are many proofs of Euler's formula. One was given by Cauchy in 1811, as follows. It applies to any convex polyhedron, and more generally to any polyhedron whose boundary is topologically equivalent to a sphere and whose faces are topologically equivalent to disks.
Remove one face of the polyhedral surface. By pulling the edges of the missing face away from each other, deform all the rest into a planar graph of points and curves, in such a way that the perimeter of the missing face is placed externally, surrounding the graph obtained, as illustrated by the first of the three graphs for the special case of the cube. (The assumption that the polyhedral surface is homeomorphic to the sphere at the beginning is what makes this possible.) After this deformation, the regular faces are generally not regular anymore. The number of vertices and edges has remained the same, but the number of faces has been reduced by 1. Therefore, proving Euler's formula for the polyhedron reduces to proving V − E + F = 1 for this deformed, planar object.
If there is a face with more than three sides, draw a diagonal—that is, a curve through the face connecting two vertices that are not yet connected. Each new diagonal adds one edge and one face and does not change the number of vertices, so it does not change the quantity V − E + F . (The assumption that all faces are disks is needed here, to show via the Jordan curve theorem that this operation increases the number of faces by one.) Continue adding edges in this manner until all of the faces are triangular.
Apply repeatedly either of the following two transformations, maintaining the invariant that the exterior boundary is always a simple cycle:
Remove a triangle with only one edge adjacent to the exterior, as illustrated by the second graph. This decreases the number of edges and faces by one each and does not change the number of vertices, so it preserves V − E + F .
Remove a triangle with two edges shared by the exterior of the network, as illustrated by the third graph. Each triangle removal removes a vertex, two edges and one face, so it preserves V − E + F .
These transformations eventually reduce the planar graph to a single triangle. (Without the simple-cycle invariant, removing a triangle might disconnect the remaining triangles, invalidating the rest of the argument. A valid removal order is an elementary example of a shelling.)
At this point the lone triangle has V = 3, E = 3 , and F = 1, so that V − E + F = 1 . Since each of the two above transformation steps preserved this quantity, we have shown V − E + F = 1 for the deformed, planar object thus demonstrating V − E + F = 2 for the polyhedron. This proves the theorem.
For additional proofs, see Eppstein (2013). Multiple proofs, including their flaws and limitations, are used as examples in Proofs and Refutations by Lakatos (1976).
https://en.m.wikipedia.org/wiki/Euler_characteristic
The idea of associating algebraic objects or structures with topological spaces arose early in the history of topology. The basic incentive in this regard was to find topological invariants associated with different structures. The simplest example is the Euler characteristic, which is a number associated with a surface. In 1750 the Swiss mathematician Leonhard Euler proved the polyhedral formula V – E + F = 2, or Euler characteristic, which relates the numbers V and E of vertices and edges, respectively, of a network that divides the surface of a polyhedron (being topologically equivalent to a sphere) into F simply connected faces. This simple formula motivated many topological results once it was generalized to the analogous Euler-Poincaré characteristic χ = V – E + F = 2 – 2g for similar networks on the surface of a g-holed torus. Two homeomorphic surfaces will have the same Euler-Poincaré characteristic, and so two surfaces with different Euler-Poincaré characteristics cannot be topologically equivalent. However, the primary algebraic objects used in algebraic topology are more intricate and include such structures as abstract groups, vector spaces, and sequences of groups. Moreover, the language of algebraic topology has been enhanced by the introduction of category theory, in which very general mappings translate topological spaces and continuous functions between them to the associated algebraic objects and their natural mappings, which are called homomorphisms.
https://www.britannica.com/science/topology/Homeomorphism
Long before Euler, in 1537, Francesco Maurolico stated the same formula for the five Platonic solids (see Friedman). Another version of the formula dates over 100 years earlier than Euler, to Descartes in 1630. Descartes gives a discrete form of the Gauss-Bonnet theorem, stating that the sum of the face angles of a polyhedron is 2π(V - 2), from which he infers that the number of plane angles is 2F + 2V - 4. The number of plane angles is always twice the number of edges, so this is equivalent to Euler's formula, but later authors such as Lakatos, Malkevitch, and Polya disagree, feeling that the distinction between face angles and edges is too large for this to be viewed as the same formula. The formula V - E + F = 2 was (re)discovered by Euler; he wrote about it twice in 1750, and in 1752 published the result, with a faulty proof by induction for triangulated polyhedra based on removing a vertex and retriangulating the hole formed by its removal. The retriangulation step does not necessarily preserve the convexity or planarity of the resulting shape, so the induction does not go through. Another early attempt at a proof, by Meister in 1784, is essentially the triangle removal proof given here, but without justifying the existence of a triangle to remove. In 1794, Legendre provided a complete proof, using spherical angles. Cauchy got into the act in 1811, citing Legendre and adding incomplete proofs based on triangle removal, ear decomposition, and tetrahedron removal from a tetrahedralization of a partition of the polyhedron into smaller polyhedra. Hilton and Pederson provide more references as well as entertaining speculation on Euler's discovery of the formula.
The polyhedron formula, of course, can be generalized in many important ways, some using methods described below. One important generalization is to planar graphs. To form a planar graph from a polyhedron, place a light source near one face of the polyhedron, and a plane on the other side.
The shadows of the polyhedron edges form a planar graph, embedded in such a way that the edges are straight line segments. The faces of the polyhedron correspond to convex polygons that are faces of the embedding. The face nearest the light source corresponds to the outside face of the embedding, which is also convex. Conversely, any planar graph with certain connectivity properties comes from a polyhedron in this way.
Euler's Formula, Proof 6: Electrical Charge
This proof is due to Thurston. He writes:
Arrange the polyhedron in space so that no edge is horizontal – in particular, so there is exactly one uppermost vertex U and lowermost vertex L.
Put a unit + charge at each vertex, a unit - charge at the center of each edge, and a unit + charge in the middle of each face. We will show that the charges all cancel except for those at L and at U. To do this, we displace all the vertex and edge charges into a neighboring face, and then group together all the charges in each face. The direction of movement is determined by the rule that each charge moves horizontally, counterclockwise as viewed from above.
In this way, each face receives the net charge from an open interval along its boundary. This open interval is decomposed into edges and vertices, which alternate. Since the first and last are edges, there is a surplus of one - ; therefore, the total charge in each face is zero. All that is left is +2, for L and for U.
Thurston goes on to generalize this idea to a proof that the Euler characteristic is an invariant of any triangulated differentiable manifold.
