# Simplex

In geometry, a simplex (plural simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an k-simplex is an k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points are affinely independent, which means are linearly independent. Then, the simplex determined by them is the set of points . For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices.

A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.

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” simply means any finite set of vertices.

### Other articles related to "simplex":

Simplex - Applications
... the nonnegative components sum to one, in which case the feasible region forms a simplex ... problems can be solved by the simplex algorithm of George Dantzig ... domain and then fit interpolating polynomials to each simplex ...
Localized Epidermolysis Bullosa Simplex
... Localized epidermolysis bullosa simplex (also known as "Weber–Cockayne syndrome," and "Weber–Cockayne variant of generalized epidermolysis bullosa ...
Simplex Graph
... graph theory, a branch of mathematics, the simplex graph κ(G) of an undirected graph G is itself a graph, with one node for each clique (a set of mutually adjacent vertices ... Therefore, the simplex graph contains within it a subdivision of G itself ... The simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph ...
Alchip - History
... In April 2002 Cadence acquired Simplex Solutions, an ASIC design services company ... Alchip was founded six months later by Kinying Kwan and other former Simplex employees as a fabless ASIC supplier ... Simplex Solutions had gained notoriety for designing the graphics ASIC for Sony's PS2 game console ...
Bergmann Simplex
... The Bergmann Simplex was a compact firearm produced in the early 1900s, utilizing innovations from the earlier Bergman Model 1896 pistol, as well as the Mars pistol ... It was chambered for the proprietary Bergmann-Simplex 8mm cartridge ...