Additivity of angles in higher dimensions Suppose $x_i$'s are IID random samples from $d$-dimensional isotropic Gaussian centered at zero and normalized to have $E[\|x\|^2]=1$
Suppose we have
$$a=x_1 \\ b=a+x_2 \\ c=b+x_3$$
As $d$ increases, the following is observed
$$\cos(a,c) \approx \cos(a,b)\cos(b,c)$$
Where
$$\cos(x,y)=\frac{\langle x, y\rangle}{\|x\| \|y\|}$$
Why?

notebook
individual cosine distribution:

We can continue this pprocess with $d=c+x_4$, $e=d+x_5$, and this "three-way" cosine similarity seems to approach integer values among all triplets of vectors


This was answered on Mathoverflow by Iosif Pinelis. Using Chebychev's inequality he showed that isotropic normal RV's $X,Y,Z$
\begin{equation}
 U:=X,\quad V:=X+Y,\quad W:=X+Y+Z, 
\end{equation}
\begin{equation}
 \cos(U,V)=\frac{U\cdot V}{\sqrt{U\cdot U}\sqrt{V\cdot V}}\sim_P\frac1{\sqrt2}, 
\end{equation}
\begin{equation}
 \cos(V,W)=\frac{V\cdot W}{\sqrt{V\cdot V}\sqrt{W\cdot W}}\sim_P\frac2{\sqrt6}, 
\end{equation}
\begin{equation}
 \cos(U,W)=\frac{U\cdot W}{\sqrt{U\cdot U}\sqrt{W\cdot W}}\sim_P\frac1{\sqrt3}.  
\end{equation}
\begin{equation}
 \cos(U,V)\cos(V,W)\sim_P \cos(U,W). 
\end{equation}
