Twenty Proofs of Euler's System: V-E+F=2

Many theorems in mathematics are vital ample that they’ve been
proved repeatedly in surprisingly many completely other ways. Examples of this
embrace
the
existence of infinitely many prime numbers,
the evaluate of
zeta(2),
the elemental theorem of algebra (polynomials indulge in roots), quadratic
reciprocity (a formula for testing whether or no longer an arithmetic progression
comprises a sq.) and the Pythagorean theorem (which basically based on
Wells has no longer less than 367 proofs).
This also in most cases happens for unimportant theorems,
such because the truth that in any rectangle dissected into smaller rectangles,
if every smaller rectangle has integer width or height, so does the
dapper one.

This page lists proofs of the Euler formula:
for any convex polyhedron, the number of vertices and faces collectively
is exactly two more than the number of edges.
Symbolically V?E+F=2.
As an instance, a tetrahedron has four vertices, four faces, and 6 edges;
4-6+4=2.

A version of the formula dates over 100 years earlier than Euler, to Descartes in 1630. Descartes supplies a discrete make of the Gauss-Bonnet theorem, pointing out that the sum of the face angles of a polyhedron is 2?(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 right here’s same to Euler’s formula, nevertheless later authors equivalent to Lakatos, Malkevitch, and Polya disagree, feeling that the glory between face angles and edges is simply too dapper for this to be viewed because the identical formula. The formula V?E+F=2 changed into as soon as (re)stumbled on by Euler; he wrote about it twice in 1750, and in 1752 published the final consequence, with a injurious proof by induction for triangulated polyhedra basically based on placing off a vertex and retriangulating the hole fashioned by its casting off. The retriangulation step doesn’t necessarily preserve the convexity or planarity of the following form, so the induction doesn’t fight by.
Yet any other early try at a proof, by Meister in 1784, is certainly the triangle casting off proof given right here, nevertheless with out justifying the existence of a triangle to preserve.
In 1794, Legendre equipped a full proof, utilizing spherical angles.
Cauchy got into the act in 1811, citing Legendre and in conjunction with incomplete proofs basically based on triangle casting off, ear decomposition, and tetrahedron casting off from a tetrahedralization of a partition of the polyhedron into smaller polyhedra.
Hilton and Pederson provide more references
as correctly as attractive speculation on Euler’s discovery of the formula. Confusingly, other equations equivalent to e^i pi = -1
and a^phi(n) = 1 (mod n)
also trot by the determine of “Euler’s formula”; Euler changed into as soon as a busy man.

The polyhedron formula, clearly, would possibly per chance maybe also be generalized in many vital ways,
some utilizing solutions described below.
One vital generalization is to planar graphs.
To make a planar graph from a polyhedron,
draw a gentle-weight source near one face of the polyhedron, and a plane on the choice facet.

shadow of a polyhedron

The shadows of the polyhedron edges make a planar graph, embedded in
such a manner that the perimeters are straight line segments.
The faces of the polyhedron correspond to convex
polygons that are faces of the embedding. The face nearest the gentle
source corresponds to the outdoors face of the embedding, which is also
convex. Conversely, any planar graph with obvious connectivity
properties comes from a polyhedron in this form.

Among the proofs below employ handiest the topology of the planar graph,
some employ the geometry of its embedding, and a few employ the
3-dimensional geometry of the normal polyhedron.
Graphs in these proofs is no longer going to necessarily be easy:
edges could also merely connect a vertex to itself, and two vertices could also very correctly be linked
by more than one edges. Several of the proofs depend on the Jordan curve
theorem, which itself has more than one proofs; alternatively these will no longer be
fundamentally basically based on Euler’s formula so one can employ Jordan curves with out
concern of circular reasoning.

Please send email
whereas you realize of a proof no longer listed right here.
I would especially esteem proofs bright cohomology theory,
toric kinds, or other better mathematics.
(Helena Verrill has shown that
Euler’s
formula is same to the truth that every toric diversity over GF[p]
has loads of parts equal to 1 (mod p)
nevertheless is calm lacking non-combinatorial proof of the latter truth.)

I imagine it would possibly per chance be that which it’s possible you’ll imagine to create inductions
basically based on the illustration of convex polyhedra as intersections
of halfspaces or convex hulls of parts, nevertheless the must
take care of inputs in non-frequent space would fetch the following
proofs reasonably messy.

There also appears to be a doable connection
to binomials: if one defines a polynomial
p(t) = 1+Vt+Et²+Ft³+t⁴, the Euler
formula would possibly per chance maybe also be interpreted
as announcing that p(t) is divisible by 1+t.
But for simplices of any dimension,
p(t)=(1+t)^d+1 by the binomial formula.
In all chance there is a proof of Euler’s formula that uses
these polynomials at the moment in preference to merely translating
one of the most inductions into polynomial make.
Jim Propp
asks the same questions for infinite-dimensional polytopes,
decoding p(t) as a vitality series
(glimpse also his latest
expansion of these ideas).