##  [Simplex](/simplex-0) 

 Definition

The convex hull of a finite set of affinely independent points; an n-simplex is the convex hull of n+1 affinely independent points and is the simplest possible n-dimensional polytope.

 

 

 

 

 

 





## Principle

Principle

A simplex is the minimal convex building block in a given dimension: any point in the simplex has unique barycentric coordinates relative to its vertices, and every affine map is determined by its action on the vertices.

 

 

 

 

 





## Demonstration

Demonstration

The standard n-simplex in R^{n+1} is {x in R^{n+1} : x_i >= 0, sum_i x_i = 1}. In low dimensions this gives a segment (1-simplex), triangle (2-simplex) and tetrahedron (3-simplex).

 

 

 

 

## Misapplication

Misapplication

Calling the convex hull of n+1 points a simplex without checking affine independence (if dependent the hull is lower-dimensional), or confusing simplices with cells in a simplicial complex that may be glued non-affinely.

 

 

 

 

 





## Consequence

Consequence

Simplices provide barycentric coordinates, simple formulas for volume and orientation, and form the atoms of triangulations used to compute homology, perform integration, and approximate manifolds numerically.

 

 

 

 

## Reversal

Reversal

If the defining points are affinely dependent there is no n-simplex but a degenerate polytope of lower dimension; reversing the simplex idea yields general polytopes that require more vertices and faces.

 

 

 

 

 





## Boundary

Boundary

The notion presupposes an ambient affine space and affine independence; it excludes non-convex or curved analogues (e.g. geodesic simplices on curved spaces require additional structure) and combinatorial simplices without geometry.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between the geometric simplex (an actual convex set with barycentric coordinates) and the purely combinatorial simplicial complex notion where a 'simplex' may denote an abstract set of vertices without embedding.

 

 

 

 

 





## Synthesis

Synthesis

A simplex is the unique convex polytope determined by affinely independent vertices; as the elementary affine cell it underpins triangulations, coordinate representations, and local linear approximations of geometry.