Complex vs. Simplex: What’s the Difference?

• Simplex

  In geometry, a simplex (plural: simplexes or simplices) 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.Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices.

  More formally, suppose the k + 1 points

  u

  0

  ,

  …

  ,

  u

  k

  ∈

  R

  k

  {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}}

  are affinely independent, which means

  u

  1

  −

  u

  0

  ,

  …

  ,

  u

  k

  −

  u

  0

  {displaystyle u_{1}-u_{0},dots ,u_{k}-u_{0}}

  are linearly independent.

  Then, the simplex determined by them is the set of points

  C

  =

  {

  θ

  0

  u

  0

  +

  ⋯

  +

  θ

  k

  u

  k

  |

  ∑

  i

  =

  0

  k

  θ

  i

  =

  1

  and

  θ

  i

  ≥

  0

  for all

  i

  }

  .

  {displaystyle C=left{theta _{0}u_{0}+dots +theta _{k}u_{k}~{bigg |}~sum _{i=0}^{k}theta _{i}=1{mbox{ and }}theta _{i}geq 0{mbox{ for all }}iright}.}

  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.

  The standard simplex or probability simplex is the simplex formed from the k + 1 standard unit vectors, or

  {

  x

  ∈

  R

  k

  +

  1

  :

  x

  0

  +

  ⋯

  +

  x

  k

  =

  1

  ,

  x

  i

  ≥

  0

  ,

  i

  =

  0

  ,

  …

  ,

  k

  }

  .

  {displaystyle {xin mathbb {R} ^{k+1}:x_{0}+dots +x_{k}=1,x_{i}geq 0,i=0,dots ,k}.}

  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.

• Complex (adjective)

  Made up of multiple parts; composite; not simple.

  “a complex being; a complex idea”

• Complex (adjective)

  Not simple, easy, or straightforward; complicated.

• Complex (adjective)

  Of a number, of the form a + bi, where a and b are real numbers and i is a square root of −1.

  “complex function”

• Complex (adjective)

  A curve, polygon or other figure that crosses or intersects itself.

• Complex (noun)

  A problem. en

• Complex (noun)

  A network of interconnected systems.

  “military-industrial complex”

• Complex (noun)

  A collection of buildings with a common purpose, such as a university or military base.

• Complex (noun)

  An assemblage of related things; a collection.

• Complex (noun)

  A psychological dislike or fear of a particular thing.

  “Jim has a real complex about working for a woman boss.”

• Complex (noun)

  An organized cluster of thunderstorms.

• Complex (noun)

  A structure consisting of a central atom or molecule weakly connected to surrounding atoms or molecules.

• Complex (noun)

  A complex number.

• Complex (verb)

  To form a complex with another substance

• Complex (verb)

  To complicate.

• Simplex (adjective)

  Single, simple; not complex.

• Simplex (adjective)

  unidirectional

• Simplex (noun)

  An analogue in any dimension of the triangle or tetrahedron: the convex hull of n+1 points in n-dimensional space.

• Simplex (noun)

  A simple word, one without affixes.

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