TITLE: What is the Coxeter diagram for?
QUESTION [44 upvotes]: I understand that  Coxeter diagrams are supposed to communicate something about the structure of  symmetry groups of  polyhedra, but I am baffled about what that something is, or why the Coxeter diagram is clearer, simpler, or more useful than a more explicit notation.  The information on Wikipedia has not helped me.
Wikipedia tells me, for example, that the Coxeter diagram for a cube is , but I don't understand why it is this, in either direction; I don't understand either how you could calculate the Coxeter diagram from a knowledge of the geometry of the cube, or how you could get from the Coxeter diagram to an understanding of the geometric properties of the cube.
I gather that the three points represent three reflection symmetries, and that the mutual angles between the three reflection planes are supposed to be $45^\circ, 60^\circ, $ and $90^\circ$, but I can't connect this with anything I know about cubic symmetry. Nor do I understand why it is perspicuous to denote these angles, respectively, with a line marked with a 4, an unmarked line, and a missing line.
My questions are:

What information is the Coxeter diagram communicating?
Why is this information useful? How does it relate to better-known geometric properties? What are the applications of the diagram?
What makes it a good notation?  Is it used for its concision, or because it is easy to calculate with, or for some other reason?
Where is a good place to start understanding this?

REPLY [4 votes]: Following Nich Matteo's answer, I find this's straightforward to find $\rho_0, \rho_1, \rho_0$ in a cube.
$\rho_0$ is reflecting by plane OAD.
$\rho_1$ is reflecting by plane OBC.
$\rho_2$ is reflecting by plane OAB.

$ (\rho_1  \rho_2)^3=1$ can be illustrated by the following six reflections:

$ (\rho_0  \rho_1)^4=1$ can be illustrated by the following 8 operations:
 
quote from wiki: In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other.
The binary operations also satisfy:
(1) closure property: If $\rho_i \rho_j$ is a group element, then $\rho_j \rho_i$ must be a group element.
(2) associative property: $\rho_0 \rho_1 \rho_2 = \rho_2 \rho_0 \rho_1 $.