How can generate a point using rotation from other point in n-dimensional space?

Question

Suppose we have a point $P_1 = [X_1,X_2, X_3, ... X_n]$ in n-dimension space, How does the rotation of point $P_1$ get another point $P_2= [Y_1, Y_2, Y_3, ...Y_n]$? There is a method to do multiple rotations to form a well-known curve starting from $P_1$, for example figure

Answer 1

From Powers of complex numbers I take these pictures

and

which spiral a point around $O$. By taking negative powers you can go other way around as well.

For dimensions higher than 3 I think almost every invertible matrix will do.