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 0simplex is a point,
a 1simplex is a line segment,
a 2simplex is a triangle,
a 3simplex is a tetrahedron,
a 4simplex is a 5cell.Specifically, a ksimplex is a kdimensional 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 nsimplex 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.
“militaryindustrial 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 ndimensional space.
Simplex (noun)
A simple word, one without affixes.