Simplex invariants?

Question

Let $k=s$ be a positive definite symmetric function and a simililarity over the natural numbers such that $k(a,a) = 1 $ for all natural numbers $a$.

A similarity $s:X\times X \rightarrow \mathbb{R}$ is defined in Encyclopedia of Distances as:

1. $s(x,y) \ge 0 \forall x,y \in X$

2. $s(x,y) = s(y,x) \forall x,y \in X$

3. $s(x,y) \le s(x,x) \forall x,x \in X$

4. $s(x,y) = s(x,x) \iff x=y$

We can define the metric $d(x,y) = \sqrt{k(x,x)+k(y,y)-2k(x,y)}=\sqrt{2(1-k(x,y)}$.

Every $3$ point metric space can be embedded in $\mathbb{R}^2$ as a triangle, (See https://math.stackexchange.com/questions/3393140/whats-the-name-of-this-surface-a2b2c22abc-1-0 ), and hence for each triple of distinct natural numbers $x,y,z$ we get a triangle.

For three points $x,y,z$ in a metric space, we can define (using the law of cosines) the following quantity:

$$S(x,y,z) = \frac{d(x,y)^2+d(y,z)^2-d(x,z)^2}{2d(x,y)d(y,z)}$$

We then have:

$$\pi = \arccos(S(x,y,z))+\arccos(S(z,x,y))+\arccos(S(y,z,x))$$

wich I consider to be an invariant, because it does not depend on the metric space $(X,d)$. (and has a nice effect of yielding some amusing formulas for $\pi$ https://math.stackexchange.com/questions/4183066/some-formulas-for-pi ).

Now my naive question is, if it is possible to define other invariants for each simplex corresponding to a subset of $n$ natural numbers, which is independent on the function $k=s$ chosen.

The only thing we know are the properties above about $k=s$ or $d$.

(We can also assume that the determinant of the Gramian matrix $k(x_i,x_j)$ for each subset $x_i$ of pairwise distinct $x_i,x_j$ is not zero.)

Answer 1

I have found an answer:

The right generalisation of this question from triangles to simplices is the question of dihedral angles:

In the book "Matrices and Graphs in Geometry" by Miroslav Fiedler, we have the following generalisation of the sum of angles equals $\pi$ in triangles:

Here is some Sagemath-Code which implements this idea.

For example for $3$ points we have the following algebraic variety given by $\det = 0$ :

$$0 = 2 \, c_{12} c_{13} c_{23} - c_{12}^{2} - c_{13}^{2} - c_{23}^{2} + 1$$

which is the Cayley nodal cubic.

For $4$ points we have the following algebraic variety given by $\det = 0$: $$0 = c_{14}^{2} c_{23}^{2} - 2 \, c_{13} c_{14} c_{23} c_{24} + c_{13}^{2} c_{24}^{2} - 2 \, c_{12} c_{14} c_{23} c_{34} - 2 \, c_{12} c_{13} c_{24} c_{34} + c_{12}^{2} c_{34}^{2} + 2 \, c_{12} c_{13} c_{23} + 2 \, c_{12} c_{14} c_{24} + 2 \, c_{13} c_{14} c_{34} + 2 \, c_{23} c_{24} c_{34} - c_{12}^{2} - c_{13}^{2} - c_{14}^{2} - c_{23}^{2} - c_{24}^{2} - c_{34}^{2} + 1$$