Simplex — Definition, Formula & Examples

A simplex is the simplest possible polytope in any given dimension: a point in 0D, a line segment in 1D, a triangle in 2D, a tetrahedron in 3D, and so on. Each

n

-simplex has exactly

n + 1

vertices, with every vertex connected to every other vertex.

An $n$ -simplex is the convex hull of $n + 1$ affinely independent points $\{v_0, v_1, \dots, v_n\}$ in $\mathbb{R}^m$ (where $m \geq n$ ). Equivalently, it is the set of all convex combinations $\sum_{i=0}^{n} \lambda_i v_i$ where $\lambda_i \geq 0$ and $\sum_{i=0}^{n} \lambda_i = 1$ .

Key Formula

f_k = \binom{n+1}{k+1}

Where:

$n$ = Dimension of the simplex
$k$ = Dimension of the sub-faces being counted (0 = vertices, 1 = edges, etc.)
$f_k$ = Number of k-dimensional faces of the n-simplex

How It Works

To build an

n

-simplex, start with

n + 1

points that do not all lie in the same

(n-1)

-dimensional flat (affinely independent). The simplex is the region enclosed by connecting every pair of these points. A 2-simplex (triangle) has 3 vertices, 3 edges, and 1 face. A 3-simplex (tetrahedron) has 4 vertices, 6 edges, 4 triangular faces, and 1 cell. In general, an

n

-simplex has

\binom{n+1}{k+1}

faces of dimension

k

, for each

k

from

0

n

Worked Example

Problem: How many vertices, edges, and triangular faces does a 4-simplex (the 4D analogue of a tetrahedron) have?

Vertices (k = 0): Count the 0-dimensional faces.

f_0 = \binom{5}{1} = 5

Edges (k = 1): Count the 1-dimensional faces.

f_1 = \binom{5}{2} = 10

Triangular faces (k = 2): Count the 2-dimensional faces.

f_2 = \binom{5}{3} = 10

Tetrahedral cells (k = 3): Count the 3-dimensional faces.

f_3 = \binom{5}{4} = 5

Answer: A 4-simplex has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells.

Why It Matters

In linear programming, the simplex method navigates the vertices of a feasible polytope to find optimal solutions — a technique used constantly in operations research and economics. In topology, simplices are the building blocks of simplicial complexes, which are used to study the shape of spaces. The tetrahedron (3-simplex) is also one of the five Platonic solids.

Common Mistakes

Mistake: Confusing an n-simplex with an n-sided polygon.

Correction: An n-simplex lives in n dimensions and has n + 1 vertices. A 3-simplex is a tetrahedron (4 vertices in 3D), not a triangle with 3 sides.