In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, a 4-simplex is a 5-cell. A k-simplex is a k-dimensional polytope, the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0, …, u k ∈ R k are affinely independent, which means u 1 − u 0, …, u k − u 0 are linearly independent; the simplex determined by them is the set of points C =. A regular simplex is a simplex, a regular polytope. A regular n-simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common edge length; the standard simplex or probability simplex is the simplex formed from the k + 1 standard unit vectors, or. In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex; the associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” means any finite set of vertices.

The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum and with the same Latin adjective in the normal form simplex; the regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the other two being the cross-polytope family, labeled as βn, the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn; the convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 is an m-simplex, called an m-face of the n-simplex; the 0-faces are called the vertices, the 1-faces are called the edges, the -faces are called the facets, the sole n-face is the whole n-simplex itself.

In general, the number of m-faces is equal to the binomial coefficient. The number of m-faces of an n-simplex may be found in column of row of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex; the number of 1-faces of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the th tetrahedron number, the number of 3-faces of the n-simplex is the th 5-cell number, so on. In layman's terms, an n-simplex is a simple shape. Consider a line segment AB as a "shape" in a 1-dimensional space. One can place a new point C somewhere off the line; the new shape, triangle ABC, requires two dimensions. The triangle is a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space. One can place a new point D somewhere off the plane; the new shape, tetrahedron ABCD, requires three dimensions. The tetrahedron is a simple shape that requires three dimensions.

Consider tetrahedron ABCD, a shape in a 3-dimensional space. One can place a new point E somewhere outside the 3-space; the new shape ABCDE, called a 5-cell, is called the 4-simplex. This idea can be generalized, that is, adding a single new point outside the occupied space, which requires go


Timpo Toys Ltd. was an English toy company created in 1938 by Salomon "Sally" Gawrylovitz known as Ally Gee. A Jewish refugee from Germany, Gawrylovitz started as Toy Importers Company known as TIMPO in 1938; the company manufactured various toys out of wood and composition until the end of World War II. Following the war, Timpo made hollowcast metal toy soldiers; the firm ceased operations in 1978. The assortment of Timpo Toys consisted of several figurine series, with the American frontier series and the Knight series forming the core of the product range. Since Timpo further developed the series in the course of production, some series could be divided into generations. Overview of the series: American frontier series Cowboy series Native Americans und Apaches series Union Army series Confederate States Army series Mexican series Knight seriesCrusader series Medieval Knights series Vizor Knights series Gold Knights series Silver Knights series Black Knights series Other series Romans series Vikings series Arabs series French Foreign Legions series American Revolutionary War series Inuit series World War II series Farm series Guard series The following pictures show examples for Timpo plastic figures.

Brown, Kenneth Douglas. The British Toy Business: A History since 1700. Hambledon & London. ISBN 1-85285-136-8. Plastic Warrior Special - Timpo, Plastic Warrior magazine Maughan, Michael; the A to Z of TIMPO. CreateSpace Independent Publishing Platform. P. 126. ISBN 9781514315040

2009 Icelandic Cup

Visa-Bikar 2009 is the 50th season of the Icelandic national football cup. It ended with the final on 3 October 2009 at Laugardalsvöllur; the winners qualified for the second qualifying round of the 2010–11 UEFA Europa League. The First Round consists of 32 teams from lower Icelandic divisions; the matches were played between 23 and 25 May 2009. The Second Round includes the 16 winners from the previous round as well as 24 teams from the second and third division; the matches were played on 1 and 2 June 2009. The Third Round include the 20 winners from the previous round and the 12 teams from the Úrvalsdeild; these matches were played on 17 and 18 June 2009. This round consists of the 16 winners of the previous round; these matches were played on 5 and 6 July 2009. This round consists of the 8 winners of the previous round; the semifinal matches took place at Laugardalsvöllur on 12 and 13 September 2009 and involved the four winners from the previous round. The Final took place at Laugardalsvöllur on 3 October 2009 and was contested between the winners of the Semifinal matches.

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