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.

Wikipedia
  • 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.

Wiktionary –>

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