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 g_ij = (δ_ij + 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
√(∑∑g_ij/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 .
dim altitude
1 1
2 √(3/4) .8660
3 √(2/3) .8165
4 √(5/8) .7906
5 √(3/5) .7746
6 √(7/12) .7638
∞ √(1/2) .7071

dim	altitude
1	1
2	√(3/4)	.8660
3	√(2/3)	.8165
4	√(5/8)	.7906
5	√(3/5)	.7746
6	√(7/12)	.7638
∞	√(1/2)	.7071