In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, and more specifically in algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy and polyhedral combinatorics, the Euler characteristic (or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology's shape or structure regardless of the way it is bent. It is commonly denoted by χ (Greek letter The Greek alphabet is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early 8th century BC. It is the first and oldest alphabet in the narrow sense that it notes each vowel and consonant with a separate symbol. It is as such in continuous use to this day. The letters were also used to represent chi Chi is the 22nd letter of the Greek alphabet, pronounced as /ˈkaɪ/ in English. Its value in Ancient Greek was an aspirated velar stop /kʰ/ (in the Western Greek alphabet: /ks/)).

The Euler characteristic was originally defined for polyhedra A polyhedron is a geometric solid in three dimensions with flat faces and straight edges. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, "many") + -edron (form of έδρα, "base", "seat", or "face") and used to prove various theorems about them, including the classification of the Platonic solids In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and angles. Leonhard Euler Leonhard Paul Euler[citation needed] was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər (like "Oiler") in English and [ˈɔʏlɐ] in German; the pronunciation /ˈjuːlər/ EW-lər is incorrect, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from homology In mathematics , homology (in Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group and connects to many other invariants.

Contents 1 Polyhedra 1.1 Planar graphs 1.2 Proof of Euler's formula 2 Topological definition 3 Properties 3.1 Homotopy invariance 3.2 Inclusion-exclusion principle 3.3 Product property 3.4 Covering spaces 3.5 Fibration property 4 Relations to other invariants 5 Examples 6 Generalizations 7 See also 8 Notes 9 Further reading 10 External links

Polyhedra

The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula

where V, E, and F are respectively the numbers of vertices In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces (typically triangles) in 3D models, where each such point is given as a vector (corners), edges In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects and faces In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix -hedron is derived from the Greek word hedra which means face in the given polyhedron. Any convex polyhedron's surface has Euler characteristic

This result is known as Euler's formula. This corresponds to the Euler characteristic of the sphere (which is 2), and applies identically to spherical polyhedra. An illustration of the formula on some polyhedra is given below.

Name	Image	Vertices V	Edges E	Faces F	Euler characteristic: V − E + F
Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids. The tetrahedron is the only convex polyhedron that has four faces		4	6	4	2
Hexahedron A hexahedron is a polyhedron with six faces. A regular hexahedron, with all its faces square, is a cube or cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube is dual to the		8	12	6	2
Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex		6	12	8	2
Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges. Its dual polyhedron is the icosahedron. If one were to make every one of the Platonic solids with edges		20	30	12	2
Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids		12	30	20	2

The surfaces of nonconvex polyhedra can have various Euler characteristics;

Name	Image	Vertices V	Edges E	Faces F	Euler characteristic: V − E + F
Tetrahemihexahedron		6	12	7	1
Octahemioctahedron		12	24	12	0
Cubohemioctahedron		12	24	10	−2
Great icosahedron		12	30	20	2

A modified form of Euler's formula, using polyhedral density (D) and polygon density of the vertex figures In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off (d_v) and faces (d_f) was given by Arthur Cayley Arthur Cayley was a British mathematician. He helped found the modern British school of pure mathematics, and holds both for convex polyhedra (where the correction factors are all 1), and the regular non-convex Kepler–Poinsot polyhedrons:

d_vV − E + d_fF = 2D

Further, projective polyhedra all have Euler characteristic 1, corresponding to the real projective plane In mathematics, the real projective plane is a non-orientable two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space without intersecting itself. It has Euler characteristic 1, hence a demigenus of 1, while toroidal polyhedra all have Euler characteristic 0, corresponding to the torus In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle. In most contexts it is assumed that the axis does not touch the circle (in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit). Other types.

Planar graphs

The Euler characteristic can be defined for connected planar graphs by the same V − E + F formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face.

The Euler characteristic of any planar graph is 2. For via stereographic projection In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: the plane maps to the two-dimensional sphere, such that the graph maps to a polygonal decomposition of the sphere, which has Euler characteristic 2. This viewpoint is implicit in Cauchy's proof of Euler's formula given below.

Proof of Euler's formula

First steps of the proof in the case of a cube

The first rigorous proof of Euler's formula, given by Cauchy Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation groups in abstract algebra. A profound in 1811, is as follows.

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, 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. As such, 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 aren't connected yet. This adds one edge and one face and does not change the number of vertices, so it does not change the quantity V − E + F. Continue adding edges in this manner until all of the faces are triangular.

Apply repeatedly either of the following two transformations:

1. 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.
2. 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.

Repeat these two steps, one after the other, until only one triangle remains.

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 Nineteen Proofs of Euler's Formula by David Eppstein. Multiple proofs, including their flaws and limitations, are used as examples in Proofs and Refutations by Imre Lakatos Imre Lakatos was a philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.^[1]

Topological definition

The polyhedral surfaces discussed above are, in modern language, two-dimensional finite CW-complexes In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. (When only triangular faces are used, they are two-dimensional finite simplicial complexes In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts . Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.) In general, for any finite CW-complex, the Euler characteristic can be defined as the alternating sum

where k_n denotes the number of cells of dimension n in the complex.

More generally still, for any topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology, we can define the nth Betti number b_n as the rank In mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space over the rational numbers one would need to "contain" it; or alternatively how large a free abelian group it can contain as a subgroup of the n-th singular homology In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of topological invariants of a topological space X, the so-called homology groups Hn. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is group. The Euler characteristic can then be defined as the alternating sum

This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index n₀. For simplicial complexes, this is not the same definition as in the previous paragraph but a homology computation shows that the two definitions will give the same value for χ.

Properties

As a corollary of Poincaré duality, the Euler characteristic of any closed odd-dimensional manifold is zero. This applies more generally to any compact In mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a closed and bounded subset of a Euclidean space stratified space all of whose strata are odd-dimensional. Furthermore, the Euler characteristic behaves well with respect to many basic operations on topological spaces, as follows.

Homotopy invariance

Since the homology is a topological invariant (in fact, a homotopy invariant — two topological spaces that are homotopy equivalent have isomorphic In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups homology groups), so is the Euler characteristic.

For example, any convex polyhedron is homeomorphic to the three-dimensional ball In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general, so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The, which has Euler characteristic 2. This explains why convex polyhedra have Euler characteristic 2.

Inclusion-exclusion principle

If M and N are any two topological spaces, then the Euler characteristic of their disjoint union The elements of the disjoint union are ordered pairs . Here i serves as an auxiliary index that indicates which Ai the element x came from. Each of the sets Ai is canonically embedded in the disjoint union as the set is the sum of their Euler characteristics, since homology is additive under disjoint union:

More generally, if M and N are subspaces of a larger space X, then so are their union and intersection. In some cases, the Euler characteristic obeys a version of the inclusion-exclusion principle:

This is true in the following cases:

• if M and N are an excisive couple. In particular, if the interiors In mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S of M and N inside the union still cover the union.^[2]

• if X is a locally compact space, and one uses Euler characteristics with compact In mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. Thus a closed and bounded subset of a Euclidean space supports, no assumptions on M or N are needed.

• if X is a stratified space all of whose strata are even dimensional, the inclusion-exclusion principle holds if M and N are unions of strata. This applies in particular if M and N are subvarieties of a complex A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the algebraic variety In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in classical algebraic geometry.^[3]

In general, the inclusion-exclusion principle is false. A counterexample In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy". Because this statement makes the claim that a certain property holds for all students, even a single example of a diligent student will is given by taking X to be the real line In mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space. It is the Euclidean space of dimension one, and can be thought of as a vector space , a metric space, a topological space, or simply as a linear continuum, M a subset In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B consisting of one point and N the complement In discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another. The terms "absolute" and "relative" complement refer to more specific applications of the concept, with universal complements referring to elements of M.

Product property

Also, the Euler characteristic of any product space In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the M × N is

These addition and multiplication properties are also enjoyed by cardinality In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers of sets A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; see [1].

Covering spaces

For more details on this topic, see Riemann–Hurwitz formula.

Similarly, for an k-sheeted covering space In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p a[›] from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p. This means that for each point x in X, there is associated an ordered pair, , where U is a neighborhood of x and one has

More generally, for a ramified covering space, the Euler characteristic of the cover can be computed from the above, with a correction factor for the ramification points, which yields the Riemann–Hurwitz formula.

Fibration property

The product property holds much more generally, for fibrations with certain conditions.

If is a fibration with fiber F, with the base B path-connected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property:^[4]

This includes product spaces and covering spaces as special cases, and can be proven by the Serre spectral sequence on homology of a fibration.

For fiber bundles, this can also be understood in terms of a transfer map – note that this is a lifting and goes "the wrong way" – whose composition with the projection map is multiplication by the Euler class of the fiber:^[5]

Relations to other invariants

The Euler characteristic of a closed orientable In mathematics, an orientation on a real vector space is a choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed , respectively. However, the choice of orientation is surface In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be can be calculated from its genus The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed g (the number of tori in a connected sum decomposition of the surface; intuitively, the number of "handles") as

The Euler characteristic of a closed non-orientable surface can be calculated from its non-orientable genus k (the number of real projective planes in a connected sum decomposition of the surface) as

For closed smooth manifolds, the Euler characteristic coincides with the Euler number, i.e., the Euler class of its tangent bundle evaluated on the fundamental class of a manifold. The Euler class, in turn, relates to all other characteristic classes of vector bundles.

For closed Riemannian manifolds, the Euler characteristic can also be found by integrating the curvature; see the Gauss–Bonnet theorem for the two-dimensional case and the generalized Gauss–Bonnet theorem for the general case.

A discrete analog of the Gauss–Bonnet theorem is Descartes' theorem that the "total defect" of a polyhedron, measured in full circles, is the Euler characteristic of the polyhedron; see defect (geometry).

Hadwiger's theorem characterizes the Euler characteristic as the unique (up to scalar multiplication) translation-invariant, finitely additive, not-necessarily-nonnegative set function defined on finite unions of compact convex sets in Rⁿ that is "homogeneous of degree 0".

Examples

The Euler characteristic can be calculated easily for general surfaces by finding a polygonization of the surface (that is, a description as a CW-complex) and using the above definitions.

Name	Image	Euler characteristic
Interval		1
Circle		0
Disk		1
Sphere		2
Torus (Product of two circles)		0
Double torus		−2
Triple torus		−4
Real projective plane		1
Möbius strip		0
Klein bottle		0
Two spheres (not connected) (Disjoint union of two spheres)		2 + 2 = 4
Three spheres (not connected) (Disjoint union of three spheres)		2 + 2 + 2 = 6

Any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore its Euler characteristic is 1. This case includes Euclidean space of any dimension, as well as the solid unit ball in any Euclidean space — the one-dimensional interval, the two-dimensional disk, the three-dimensional ball, etc.

The n-dimensional sphere has Betti number 1 in dimensions 0 and n, and all other Betti numbers 0. Hence its Euler characteristic is 1 + ( − 1)ⁿ — that is, either 0 or 2.

The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere — either 0 or 1.

The n-dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property. How many pentagons and hexagons does it take to make a Soccer ball? Assume we use N hexagons and L pentagons; then we have F = N + L faces. Every pentagon (hexagon) has 5 vertices (6 vertices), and each one is shared between 3 faces, hence we have V = (5L + 6N) / 3 vertices. Similarly, every pentagon (hexagon) has 5 edges (6 edges), and each one is shared between 2 faces, hence we have E = (5L + 6N) / 2 edges.

Since the sphere M has Euler characteristic 2, it must be that L = 12. The result is that we always need 12 pentagons on a football/soccer ball; the number of hexagons is in principle unconstrained (but for a real football/soccer ball one obviously chooses a number that makes the ball as spherical as possible). One can also apply this result to fullerenes.

Generalizations

For every combinatorial cell complex, one defines the Euler characteristic as the number of 0-cells, minus the number of 1-cells, plus the number of 2-cells, etc., if this alternating sum is finite. In particular, the Euler characteristic of a finite set is simply its cardinality, and the Euler characteristic of a graph is the number of vertices minus the number of edges.^[6]

More generally, one can define the Euler characteristic of any chain complex to be the alternating sum of the ranks of the homology groups of the chain complex.

A version used in algebraic geometry is as follows. For any sheaf on a projective scheme X, one defines its Euler characteristic

where is the dimension of the i-th sheaf cohomology group of .

Another generalization of the concept of Euler characteristic on manifolds comes from orbifolds. While every manifold has an integer Euler characteristic, an orbifold can have a fractional Euler characteristic. For example, the teardrop orbifold has Euler characteristic 1 + 1/p, where p is a prime number corresponding to the cone angle 2π / p.

The concept of Euler characteristic of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and largest elements; call them 0 and 1. The Euler characteristic of such a poset is defined as the integer μ(0,1), where μ is the Möbius function in that poset's incidence algebra.

This can be further generalized by defining a Q-valued Euler characteristic for certain finite categories, a notion compatible with the Euler characteristics of graphs, orbifolds and posets mentioned above. In this setting, the Euler characteristic of a finite group or monoid G is 1/|G|, and the Euler characteristic of a finite groupoid is the sum of 1/|G_i|, where we picked one representative group G_i for each connected component of the groupoid.^[7]

See also

• List of uniform polyhedra
• List of topics named after Leonhard Euler

Notes

1. ^ Imre Lakatos: Proofs and Refutations, Cambridge Technology Press, 1976
2. ^ Edwin Spanier: Algebraic Topology, Springer 1966, p. 205.
3. ^ William Fulton: Introduction to toric varieties, 1993, Princeton University Press, p. 141.
4. ^ Spanier, Edwin Henry (1982), Algebraic Topology, Springer, ISBN 978 0 38794426 5, http://books.google.com/books?id=h-wc3TnZMCcC , Applications of the homology spectral sequence, p. 481
5. ^ Gottlieb, Daniel Henry (1975), "Fibre bundles and the Euler characteristic", Journal of Differential Geometry 10 (1): 39–48, http://www.math.purdue.edu/~gottlieb/Bibliography/17FibreBundlesAndtheEulerCharacteristic.pdf
6. ^ Olaf Post calls this a "well-known formula": Post, Olaf (2009), "Spectral analysis of metric graphs and related spaces", Limits of graphs in group theory and computer science, Lausanne, Switzerland: EPFL Press, pp. 109–140, arXiv:0712.1507 .
7. ^ Tom Leinster, "The Euler characteristic of a category", Documenta Mathematica, 13 (2008), pp. 21–49

Further reading

• Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
• H. Graham Flegg: From Geometry to Topology. Dover 2001, p. 40

External links

• Weisstein, Eric W., "Euler characteristic" from MathWorld.
• Weisstein, Eric W., "Polyhedral formula" from MathWorld.
• Euler characteristic in the Encyclopaedia of Mathematics

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