This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Simplex

Geometrically, simplices are:

- point,
- closed segment,
- closed triangle,
- closed pyramid, etc.

These are, respectively,

- $0$-dimensional simplex,
- $1$-dimensional simplex,
- $2$-dimensional simplex,
- $3$-dimensional simplex, etc.

The simplest example of $n$-simplex is the polygon in ${\bf R}^n$ with $n+1$ vertices at $(0,0,0,0,...,0), (1,0,0,0,...,0), (0,1,0,0,...,0), ..., (0,0,0,0,...0,1,0), (0,0,0,0,...,0,1)$.

A *geometric $n$-simplex* in ${\bf R}^{n+1}$ is defined as the convex hull (the set of all convex combinations) of $n+1$ points

in general position:

If we treat the simplex as a cell complex, its topology is very simple:

**Theorem.** The $n$-simplex is homeomorphic to the $n$-ball ${\bf B}^n$.

There is however an additional geometric structure; a simplex has *faces*.

**Example.** Suppose $a$ is a $1$-simplex

Then its faces are $v_0$ and $v_1$. They can be easily described algebraically. An arbitrary point in a is a convex combination of $v_0$ and $v_1$:

What about $v_0$ and $v_1$? They are convex combinations too but of a special kind:

**Example.** Suppose ${\tau}$ is a $2$-simplex

$$a = {\rm \hspace{3pt}conv}\{v_0,v_1,v_2 \}.$$

An arbitrary point in a is a convex combination of $v_0,v_1,v_2$:

$$a_0v_0 + a_1v_1 + a_2v_2 with a_0 + a_1 + a_2 = 1.$$

To find all $1$-faces set one of these coefficient equal to $0$:

$b = a_0v_0 + a_1v_1 + a_2v_2$ with $a_0 + a_1 + a_2 = 1$ and $a_1 = 0,$

$c = a_0v_0 + a_1v_1 + a_2v_2$ with $a_0 + a_1 + a_2 = 1$ and $a_0 = 0$.So,

$$a,b,c < {\tau}.$$

To find all $0$-faces set two of these coefficient equal to $0$:

$$v_0 = 1 \cdot v_0 + 0 \cdot v_1 + 0 \cdot v_2, {\rm \hspace{3pt} etc}.$$

An *abstract $n$-simplex* is simply any finite set $A_0A_1...A_n$.

**Exercise.** Prove that the boundary of an $n$-simplex is homeomorphic to ${\bf S}^{n-1}$. Hint: put the sphere inside the simplex.

See also Simplicial complex.

Instead of ${\bf R}^n$, one can carry out this construction in any vector space.