Simplex

The generalization of a tetrahedral region of space to $n$-D. The boundary of a $k$-simplex has $k+1$ 0-faces (Vertices), $k(k+1)/2$ 1-faces (Edges), and ${k+1\choose i+1}$ $i$-faces, where ${n\choose k}$ is a Binomial Coefficient.

The simplex in 4-D is a regular Tetrahedron $ABCD$ in which a point $E$ along the fourth dimension through the center of $ABCD$ is chosen so that $EA=EB=EC=ED=AB$. The 4-D simplex has Schläfli Symbol $\{3, 3, 3\}$.

$n$	Simplex
0	Point
1	Line Segment
2	Equilateral Triangular Plane Region
3	Tetrahedral Region
4	4-simplex

The regular simplex in $n$-D with $n\geq 5$ is denoted $\alpha_n$ and has Schläfli Symbol .

See also Complex, Cross Polytope, Equilateral Triangle, Line Segment, Measure Polytope, Nerve, Point, Simplex Method, Tetrahedron

References

Eppstein, D. ``Triangles and Simplices.'' http://www.ics.uci.edu/~eppstein/junkyard/triangulation.html.