![]() The group generated ends up being the same, so you might as well work with the hyperplanes at an angle of $\pi/p$. The composition would have the same order with an angle of $m\pi/p$, where $m$ is relatively prime to $p$. To get the product of reflections to have order $p$, you want their hyperplanes of reflection at an angle of $\pi/p$ to each other, since the composition of the two reflections is a rotation by twice the angle between them. Given a Coxeter diagram with $n$ nodes, you can construct reflections in $n$-dimensional space to realize the Coxeter group.įor convenience, we'll identify reflection isometries with their hyperplane of reflection, so if $\rho$ is a reflection, then $\rho$ also means the hyperplane fixed by $\rho$. This is not my area of expertise, but it addresses the parts of your question about angles, and the mysterious extra circle in your diagram for the cube. So, nodes in the graph are not-adjacent exactly when the product of the corresponding generators has order 2, which for involutions means they commute: Maybe $(\rho_1 \rho_2)^3 = (\rho_1 \rho_3)^4 = (\rho_2 \rho_3)^5 = 1$.īy convention, edges that would be labeled "2" are omitted, and any "3" labels are left off, so we'd actually have. There is a node for each generator, and an edge between the two labeled with the order of their product.įor instance, if you have a group generated by three reflections $\rho_1$, $\rho_2$, and $\rho_3$, then you know that $\rho_i^2 = 1$ (the order of each reflection is two), but the order of $\rho_1 \rho_2$, $\rho_1 \rho_3$, and $\rho_2 \rho_3$ could be anything. The Coxeter diagram tells you that information. To know what this group is like, you need to know more than just how many generators there are: you need to know the relationships between the generators. Any collection of reflections (in Euclidean space, say) will generate a group. The diagrams are a way of describing a group generated by reflections. ![]() Where is a good place to start understanding this?.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?.Why is this information useful? How does it relate to better-known geometric properties? What are the applications of the diagram?. ![]() What information is the Coxeter diagram communicating?.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. 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. 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. The information on Wikipedia has not helped me. 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. ![]()
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