Complex vs. Simplex

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.