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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 \(\{\,\underbrace{3, \ldots, 3}_{3^{n-1}}\,\}\).

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


Eppstein, D. ``Triangles and Simplices.''

© 1996-9 Eric W. Weisstein