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}/n^{2}) =
√((n^{2}+n)/(2n^{2})) =
√((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 |