Coxeter diagram

Coxeter diagrams are a compact way of representing a wide variety of polytopes. They were devised by Harold Scott MacDonald Coxeter, but have been more recently generalized and reformatted by members of the Hi.gher.space forum.

A basic Coxeter diagram consists in a set of n nodes, ringed or unringed (typewriter-friendly represented as x or o respectively), with integers between them. The nodes represent (n–1)–dimensional hyperplanes, thought of as mirrors. A number k between two nodes indicates that the angle between their respective mirrors is π/k. A point can be placed somewhere between these mirrors, at unit distance from the mirrors with ringed nodes, and inside the mirrors with unringed nodes. Reflecting this point over all mirrors will create the vertices of the polytope that the Coxeter diagram represents. Essentially this is the Wythoff kaleidoscopical construction. Moreover the unringed diagram itself is nothing but the representation of the to be dealt with Coxeter reflection group.

E.g. x4o3x represents the Small rhombicuboctahedron. Its face types can be read off by deletion of any single node, here resulting in x4o . (Square), ''x. x (Square as well, because of equal edge sizes, even so having now outer symmetry of an Rectangle only), and . o3x'' (Triangle).

In order to allow for loops and bifurcation points in a typewriter-friendly inline representation of those original graphical devices (diagrams), Richard Klitzing has added the distinction between real nodes (as before) and virtual nodes. So start writing the diagram from any node by means of real nodes and count those. Whenever some recurrence to such a node is being needed this will be given by means of an asterisk together with the position number of the to be referred node in alphabetical form. E.g. x3o3o3*a closes the diagram into a triangular shape back to the first node, then representing the Triangle tiling. In x3o3o *b3o the second part has to be re-attached to the second node, which thus becomes a bifurcation point of the diagram. This then represents the Hexadecachoron.

A further extension to these diagrams was introduced by Wendy Krieger. It allows for non-uniform edge sizes in the diagram representation. In fact the node symbols not only represent that the seed point of construction is off from the corresponding mirror, it might transport also the respective offness distance, or even better: the resulting edge size. Thus x represents a unit edge. q represents an edge of size $\sqrt{2}$, h one of size $\sqrt{3}$, and f represents one of the golden ratio number. Accordingly the Vertex figure of the Icosidodecahedron can be given as x2f.

Still a further extension by Wendy Krieger is the consideration of lace prisms. Lace prisms are a parallel stack of two polytopes, both represented according to a common symmetry group, together with a device for the lacing connections. Thus, when considering the Triangle cupola we have to deal with a top layer x3o (then represented at each node position as first representant) and a bottom layer x3x (then represented at each node position as second representant). And (&) both are connected/laced (#) by unit edges (x) as well. Therefore that cupola simply becomes xx3ox&#x. Thus the faces of this cupola are x.3o. (top Triangle), .x3.x (bottom Hexagon), xx ..&#x (lacing Squares), and ''.. ox&#x'' (lacing Triangles). - Using more than two layers, the same notation then applies for so called lace simplices. If in contrast it is aimed for to deal with an axial stack of more than two layers, then this is called a lace tower and the suffix will become accordingly a &#xt. Similarily lace rings become represented by means of a suffix &#xr. Btw., leaving out the trailing lacing instruction (which clearly also implies the offness of the parallel layers), then such a multylayer symbol could be used to represent the respective compound. E.g. xo3oo3ox just represents the Stella octangula, having for components x.3o.3o. (Tetrahedron) and .o3.o3.x (dual Tetrahedron).

Wrt. convex polytopes Wout Gevaert considered the tegum sum to be accessible within an (extended) Coxeter diagram as well. Here one starts, as before, with some compound, but one consideres the respective hull instead. That is, one introduces once again lacing edges, but the offness of the layers none the less is zero. This is why the according ending is being written as &#zx, where the additional z represents that zero offness. (Note that this offness restriction and the size of the lacings clearly inter relate!) Thus for instance the D4 symmetry representation of the Hexacosichoron can then be given by foxo3ooof3xfoo *b3oxfo&#zx.