# Angle between lines drawn from 4-simplex vertices to its center

For triangle, the angle between lines drawn from vertices to its center is 120 degrees.

I have already read the answer to question Angle between lines joining tetrahedron center to vertices and there the angle is approximately 108,5 degrees

I am principally interested to know what happens in 4-dimensional case, what's the angle there? I have a hypothesis for an interaction model in 4D, and would appreciate an answer as it allows me to calculate whether the hypothetical structure is going to be stable or decay because of its constituents.

No need to answer this, but out of curiosity: I understand simplexes exist in even higher dimensions, so when the number of dimensions increases, is the angle going only nearer and nearer to 90 degrees, or even below it? Is there a generic formula to this?

• You can place the vertices of a simplex at the positive-unit points on coordinate axes "one dimension up". For instance, the $2d$ triangle in $3d$: $(1,0,0), (0,1,0), (0,0,1)$; the $3d$ tetrahedron in $4d$: $(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)$; etc. The center of each figure is simply the average of the coordinates: $\frac13(1,1,1)$, $\frac14(1,1,1,1)$, etc. Do you know how to calculate (the cosine of) the angle you want from that information?
– Blue
Jun 27 '19 at 13:51

Think of an equilateral triangle with vertices at unit distance from the center. If you draw vectors from the center to the vertices then by symmetry the resultant must match a rotation of itself, possible only for zero resultant. Also by symmetry if you pick one vector the components of every other vector along your chosen one must be the same. In your triangle the second and third vectors must therefore have components of $$-1/2$$ to be equal to each other and give a zero resultant for all vectors. The angle between two unit vectors where one has a component of $$-1/2$$ along the other is then $$\cos^{-1}(1/2)=120°$$.
When you apply this idea to an $$n$$-dimensional simplex the angle works out to $$\cos^{-1}(-1/(n-1))$$, which for large $$n$$ approaches but never reaches a right angle.