---
tag: YX1DJK2QAG
kind: concept
name: Simplex
msc: [51M20, 52B11]
specializes: [02AS45T3H9]
sources:
  - url: https://en.wikipedia.org/wiki/Simplex
same_as:
  wikidata: Q331350
  wikipedia: Simplex
---

## Summary

An $n$-simplex is the [[convex hull|0FQSDC2EQ2]] of $n + 1$ points in general position, the simplest convex shape of dimension $n$.
A point, a segment, a [[triangle|0CTNFGYMM2]] and a [[tetrahedron|JFX8ASQS3E]] are the simplices of dimensions $0$ to $3$.
Simplices are the building blocks of [[simplicial complexes|V8GZ0W1ZZ5]] in topology and the cells of triangulations in geometry.
Many notions of triangle geometry, such as the circumsphere and the centroid, extend to simplices.

## Explanations

### L1 human

Start with a point.
Add a second point and fill in between: a segment.
Add a third point off the line and fill in: a triangle.
Add a fourth point off the plane and fill in: a tetrahedron.
Each step adds one point in a new direction, and the $n$-simplex is what you get after $n$ steps.