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.

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.

Proof 1: Interdigitating Trees

Proof 2: Induction on Faces

Proof 3: Induction on Vertices

Proof 4: Induction on Edges

Proof 5: Divide and Conquer

Proof 6: Electrical Charge

Proof 7: Twin Electrical Charge

Proof 8: Sum of Angles

Proof 9: Spherical Angles

Proof 10: Grab’s Theorem

Proof 11: Ear Decomposition

Proof 12: Shelling

Proof 13: Triangle Elimination

Proof 14: Noah’s Ark

Proof 15: Binary Homology

Proof 16: Binary House Partition

Proof 17: Valuations

Proof 18: Hyperplane Preparations

Proof 19: Integer-Level Enumeration

Proof 20: Euler tours

All Proofs

References

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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+Et^{2}+Ft^{3}+t^{4}, 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).

From the Geometry Junkyard,

computational

and leisure geometry pointers.

David Eppstein,

Principle Neighborhood,

ICS,

UC Irvine.

Semi-automatically

filtered

from a frequent source file.

Final change: .