|
In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological space's shape or structure. It is commonly denoted by χ (Greek letter chi). Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
The Greek alphabet (Greek: ) is an alphabet consisting of 24 letters that has been used to write the Greek language since the late 8th or early 9th century BC. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel...
Look up Χ, χ in Wiktionary, the free dictionary. ...
The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom the concept is named, was responsible for much of this early work. In modern mathematics, the Euler characteristic arises from homology and connects to many other invariants. For the game magazine, see Polyhedron (magazine). ...
In geometry, a Platonic solid is a convex regular polyhedron. ...
Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
Polyhedra
The Euler characteristic χ was classically defined for polyhedra, according to the formula  where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. For any polyhedron homeomorphic to a sphere the Euler characteristic turns out to be In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ...
An edge between two vertices For edge in graph theory, see Edge (graph theory) In geometry, an edge is a one-dimensional line segment joining two zero-dimensional vertices in a polytope. ...
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ...
This word should not be confused with homomorphism. ...
For other uses, see Sphere (disambiguation). ...
 This result is known as Euler's formula, and can be applied not only to polyhedra but also to embedded planar graphs. A proof is given below. In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ...
Examples of convex polyhedra The surface of any convex polyhedron is homeomorphic to a sphere and therefore has Euler characteristic 2, by Euler's formula. This fact can be used to show that there are only five Platonic solids (regular polyhedra): In geometry, a Platonic solid is a convex regular polyhedron. ...
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
Image File history File links Tetrahedron. ...
A hexahedron is a polyhedron with six faces. ...
Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ...
Image File history File links Hexahedron. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
Image File history File links No higher resolution available. ...
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. ...
Image File history File links This is a lossless scalable vector image. ...
[Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...
Image File history File links Icosahedron. ...
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 a 20-year-old Cauchy, is as follows. Image File history File links V-E+F=2_Proof_Illustration. ...
Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...
Remove one face of the polyhedron. By pulling the edges of the missing face away from each other, deform all the rest into a planar network of points and curves, as illustrated by the first of the three graphs for the special case of the cube. (The assumption that the polyhedron is homeomorphic to the sphere at the beginning is what makes this possible.) After this deformation, the regular faces are generally not regular anymore — in fact, they are not even polygons. 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. In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ...
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:
- 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.
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. David Eppstein is a computer scientist at the Computer Science Department, Donald Bren School of Information and Computer Sciences, University of California, Irvine. ...
Proof and Refutations is a book by the philosopher Imre Lakatos expounding his view of the progress of mathematics. ...
Imre Lakatos (November 9, 1922 – February 2, 1974) was a philosopher of mathematics and science. ...
Examples of non-convex polyhedra Nonconvex polyhedra can have various Euler characteristics: In geometry, the term star polyhedron does not seem to have been propely defined, even though it is in common use. ...
In geometry, the tetrahemihexahedron is a concave uniform polyhedron, indexed as U4. ...
Image File history File links Tetrahemihexahedron. ...
In geometry, the octahemioctahedron is a nonconvex uniform polyhedron, indexed as U3. ...
Image File history File links Download high resolution version (639x640, 15 KB) Summary Uniform polyhedron, Octahemioctahedron, U3 Licensing I, the creator of this work, hereby release it into the public domain. ...
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. ...
Image File history File links Download high resolution version (639x639, 15 KB) Summary Image of uniform polyhedron, U20. ...
Formal definition The polyhedra discussed above are, in modern language, two-dimensional finite CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic can be defined as the alternating sum In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ...
In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...
 where kn denotes the number of cells of dimension n in the complex.
More generally still, for any topological space, we can define the nth Betti number bn as the rank of the n-th homology group. The Euler characteristic can then be defined as the alternating sum Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ...
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 one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
 This quantity is well-defined if the Betti numbers are all finite and if they are zero beyond a certain index n0. This definition subsumes the previous ones.
Another generalization of the classical Euler characteristic — 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 ith sheaf cohomology group of . Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ...
In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...
Properties
Homotopy invariance Since the homology is a topological invariant (in fact, a homotopy invariant — two topological spaces that are homotopy equivalent have isomorphic homology groups), so is the Euler characteristic. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
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. ...
Inclusion-exclusion principle If M and N are any two topological spaces, then the Euler characteristic of their disjoint union is the sum of their Euler characteristics, since homology is additive under disjoint union: In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...
 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: In combinatorial mathematics, the inclusion-exclusion principle (also known as the sieve principle) states that if A1, ..., An are finite sets, then where |A| denotes the cardinality of the set A. For example, taking n = 2, we get a special case of double counting: in words, we can count the...
 This is true in the following cases: - if M and N are an excisive couple. In particular, if the interiors of M and N inside the union still cover the union.[1]
- 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 algebraic variety.[2]
In general, the inclusion-exclusion principle is false. A counterexample is given by taking X to be the real line, M a subset consisting of one point and N the complement of M. In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ...
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...
In mathematics, the real line is simply the set of real numbers. ...
“Superset†redirects here. ...
In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
Product property Also, the Euler characteristic of any product space M × N is In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...
 These addition and multiplication properties are also enjoyed by cardinality of sets. In this way, the Euler characteristic can be viewed as a generalisation of cardinality; see [1]. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ...
This article is about sets in mathematics. ...
Other 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 stratified space all of whose strata are odd-dimensional. In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. ...
In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are topological manifolds and are required to fit together in a certain way. ...
Relations to other invariants The Euler characteristic of a closed orientable surface can be calculated from its genus g (the number of tori in a connected sum decomposition of the surface; intuitively, the number of "handles") as In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
An open surface with X-, Y-, and Z-contours shown. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In geometry, a torus (pl. ...
In geometric topology, a connected sum of two connected -dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. ...
- χ = 2 − 2g.
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 The fundamental polygon of the projective plane. ...
- χ = 2 − k.
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. In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. ...
In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...
In mathematics, the fundamental class is a homology class [M] associated to a manifold M. It is defined (firstly) in cases when M is a closed manifold of dimension n, and oriented. ...
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is twisted — particularly, whether it possesses sections or not. ...
In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...
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. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...
The Gauss-Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic). ...
In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed Riemannian manifold as an integral of a certain polynomial derived from its curvature. ...
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). Descartes redirects here. ...
For the game magazine, see Polyhedron (magazine). ...
In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. ...
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". In integral geometry (otherwise called geometric probability theory), Hadwigers theorem states that the space of measures (see below) defined on finite unions of compact convex sets in Rn consists of one measure that is homogeneous of degree k for each k = 0, 1, 2, ..., n, and linear combinations of...
Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...
A scalar may be: Look up scalar in Wiktionary, the free dictionary. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
Look up Convex set in Wiktionary, the free dictionary. ...
Examples 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. This is a glossary of some terms used in the branch of mathematics known as topology. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
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.
For other uses, see Sphere (disambiguation). ...
Image File history File links Sphere-wireframe. ...
In geometry, a torus (pl. ...
Image File history File links Torus. ...
In mathematics, a double torus is a topological object formed by the connected sum of two torii. ...
Image File history File links Size of this preview: 569 × 600 pixelsFull resolution (1176 × 1240 pixels, file size: 350 KB, MIME type: image/png) % illustration of a double torus function main() % N = The number of data points. ...
Image File history File links Size of this preview: 792 × 600 pixelsFull resolution (1320 × 1000 pixels, file size: 366 KB, MIME type: image/png) % illustration of a triple torus. ...
The fundamental polygon of the projective plane. ...
Image File history File links No higher resolution available. ...
A Möbius strip made with a piece of paper and tape. ...
Mobius strip created with Mathematica. ...
The Klein bottle immersed in three-dimensional space. ...
Wikipedia does not have an article with this exact name. ...
Image File history File links Sphere-wireframe. ...
Image File history File links Sphere-wireframe. ...
Partially ordered sets 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; let us call them 0 and 1. The Euler characteristic of such a poset is defined as μ(0,1), where μ is the Möbius function in that poset's incidence algebra. In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ...
Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...
The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...
In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ...
See also The following list contains all 75 nonprismatic uniform polyhedra, 11 uniform tessellations in the plane, and a samplings of the infinite set of prisms and antiprisms. ...
It has been suggested that this article or section be merged with List of topics named after Leonhard Euler. ...
References - ^ Edwin Spanier: Algebraic Topology, Springer 1966, p. 205.
- ^ William Fulton: Introduction to toric varieties, 1993, Princeton University Press, p. 141.
|
Feeds |
Contact