Simplex
![A regular 3-simplex or tetrahedron](/Images/godic/202501/12/Tetrahedron4025.png")
![The numbers of faces in the above table are the same as in Pascal's triangle, without the left diagonal.](/Images/godic/202501/12/Pascal's_triangle_5.svg4025.png")
![The total number of faces is always a power of two minus one. This figure (a projection of the tesseract) shows the centroids of the 15 faces of the tetrahedron.](/Images/godic/202501/12/Tesseract_tetrahedron_shadow_matrices.svg4025.png")
![](/Images/godic/202501/12/1-simplex_t0.svg4025.png")
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices.
More formally, suppose the k + 1 points ![u_0,\dots, u_k \in \mathbb{R}^{k}](/Images/godic/202501/12/e1d986c9bb9af18a354cf622e8d311af4025.png")
are affinely independent, which means ![u_1 - u_0,\dots, u_k-u_0](/Images/godic/202501/12/d1d2f31c3c9615eda1196617503697a84025.png")
are linearly independent.
Then, the simplex determined by them is the set of points