In Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). One of the most prominent combinatorialists of recent times was... combinatorial Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics, Sperner's lemma states that every Sperner coloring of a This article is about measurement by the use of triangles: for other usages of the term triangulation, see triangulation (disambiguation). In trigonometry and elementary geometry, triangulation is the process of finding a distance to a point by calculating the length of one side of a triangle, given measurements of angles... triangulation of an n-dimensional This article is about the mathematics concept. In communications, simplex refers to a one-way communications channel. See duplex, simplex communication. In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely... simplex contains a cell colored with a complete set of colors. The initial result of this kind was proved by Emanuel Sperner (9 December 1905 - 31 January 1980) was a German mathematician known for his two lemmas. He was born in Waltdorf (near Nysa, now in Poland), and died in Salzburg-Laufen. He was a student at Hamburg University. He was appointed Professor in Königsberg in 1934, and subsequently... Emanuel Sperner, in relation with proofs of Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states: If U is an open subset of Rn and f : U → Rn is an injective continuous map, then V = f(U) is open and f is a homeomorphism between U and... invariance of domain. Sperner colorings have been used for effective computation of See also fixed-point arithmetic. In mathematics, a fixed point of a function is a point that is mapped to itself by the function. For example, if f is defined on the real numbers by f(x) = x2 − 3x + 4, then 2 is a fixed point of f, because... fixed points, and in A root-finding algorithm is a numerical method or algorithm for finding a value x such that f(x) = 0, for a given function f. Here, x is a single real number called the root. When x is a vector, algorithms to find x such that f(x) = 0 are... root-finding algorithms.

According to the Soviet Mathematical Encyclopaedia (ed. Ivan Matveevich Vinogradov (September 14, 1891–March 20, 1983) was a Russian mathematician, who was one of the creators of modern analytic number theory, and also the dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. He graduated from the University... I.M. Vinogradov), a related 1929 theorem (of Knaster, Karol Borsuk (May 8, 1905 - January 24, 1982) was a Polish mathematician born in Warsaw. He received his masters degree and doctorate from Warsaw University in 1927 and 1930, respectively. His main interest was topology. See Borsuk_Ulam theorem. See also: Zygmunt Janiszewski, Stanislaw Ulam. External links Biography of Karol... Borsuk and Stefan Mazurkiewicz (b. September 25, 1888, in Warsaw - June 19, 1945 in Grodzisk Mazowiecki, Poland) was a Polish mathematician, who worked in mathematical analysis, topology and probability. Member of PAU. He was a student of Wacław Sierpiński. The Hahn-Mazurkiewicz theorem is a basic result on... Mazurkiewicz) has also become known as the Sperner lemma - this point is discussed in the English translation (ed. M. Hazewinkel).

Two-dimensional case

The two-dimensional case is the one referred to most frequently. It is stated as follows:

Given a For alternate meanings, such as the musical instrument, see triangle (disambiguation). A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. Types of triangles Triangles can be classified according to the lengths of their sides... triangle ABC, and a triangulation T of the triangle. The set S of vertices of T is colored with three colors in such a way that

1. A, B and C are colored 1, 2 and 3 respectively
2. Each vertex on an edge of ABC is to be colored only with one of the two colors of the ends of its edge. For example, each vertex on AC must have a color either 1 or 3.

Then there exists a triangle from T, whose vertices are colored with the three different colors.

Multidimensional case

In the general case the lemma refers to a n-dimensional simplex

.

We consider a triangulation T which is a disjoint division of into smaller n-dimensional simplices. Denote the coloring function as f : S → {1,2,3,...,n, n+1}, where S is again the set of vertices of T. The rules of coloring are:

1. The vertices of the large simplex are colored with different colors, i. e. f(A_i) = i for 1 ≤ i ≤ n+1.
2. Vertices of T located on any given k-dimensional subface

are colored only with the colors

.

Then there exists a simplex from T, whose vertices are colored with all n+1 colors.

Proof

We shall first address the two-dimensional case. Consider a graph G built from the triangulation T as follows:

The vertices of G are the members of T plus the area outside the triangle. Two vertices are connected with an edge if their corresponding areas share a common border, which is colored 1-2.

Note that on the interval AB there is an odd number of borders colored 1-2 (simply because A is colored 1, B is colored 2; and as we move along AB, there must be an odd number of color changes in order to get different colors at the beginning and at the end). Therefore the vertex of G corresponding to the outer area has an odd degree. But it is known that in a finite graph there is an even number of vertices with odd degree, and therefore there is an odd number of vertices with odd degree corresponding to members of T.

It can be easily seen that the only possible degree of a triangle from T is 0, 1 or 2, and that the degree 1 corresponds to a triangle colored with the three colors 1, 2 and 3.

Thus we have obtained a slightly stronger conclusion, which says that in a triangulation T there is an odd number (and at least one) of full-colored triangles.

A multidimensional case can be proved by induction on the dimension of a simplex. We apply the same reasoning, as in the 2-dimensional case, to conclude that in a n-dimensional triangulation there is an odd number of full-colored simplices.

External link

• Sperner's Lemma (http://www.cut-the-knot.org/Curriculum/Geometry/SpernerLemma.shtml) (Java)