Altitude of Regular Simplex

Assume a regular simplex the length of whose edges is 1. For n dimensions we choose a vertex as the origin and the n edges therefrom as basis vectors for barycentric coordinates. The metric tensor for these coordinates is gij = (δij + 1)/2. More on this metric.

Each of the bc coordinates of the center point on the face opposite the origin is 1/n. Recall the barycentric coordinates of any point sum to 1. The altitude is the distance from there to the origin and is
√(∑∑gij/n2) = √((n2+n)/(2n2)) = √((1+1/n)/2).

For an equilateral triangle the altitude is √(3/4) = 0.866 . For a tetrahedron is it √(2/3) = 0.8165 .
dimaltitude
11
2√(3/4).8660
3√(2/3).8165
4√(5/8).7906
5√(3/5).7746
6√(7/12).7638
√(1/2).7071