Coxeter diagram

Coxeter diagrams or Coxeter–Dynkin diagrams are a compact way of representing a wide variety of polytopes. They are named after Harold Scott MacDonald Coxeter, who first described them, but have been more recently generalized and reformatted by members of the Hi.gher.space forum. They're closely related to Coxeter reflection groups.

A Coxeter diagram consists of a set of nodes joined by numbered edges. Nodes represent mirrors, or hyperplanes; and edges describe the angles between them: an edge with the number k indicates an angle of π/k radians. Nodes can be marked to describe the relation of a point to them, which is then used to build the whole polytope. Traditionally, Coxeter diagrams are shown as drawings. In more recent plaintext extensions of this notation, other symbols may be used to represent different edge lengths, non-linear graphs, different types of snubbing, or lace towers of polytopes that generalize the notion of prisms or pyramids.

Groups
The unmarked graph represents a description of a Coxeter group. A Coxeter group is a symmetry group generated by mirror symmetries. That is it has a generating set of mirror symmetries such that any symmetry in the group can be made by chaining together symmetries from the generating set. As these are mirror symmetries, the generators are involutions. That is each mirror symmetry is its own inverse, or for generator $$g_i$$, $$g_ig_i$$ is equal to the identity ($$e$$). The Coxeter diagram represents this generating set, with nodes representing mirrors and edges representing certain relationships between these mirrors.

If two nodes, $$g_1$$ and $$g_2$$, have an edge between them labeled $$n$$, then $$(g_1g_2)^n = e$$. That is, if you apply $$g_1$$ then $$g_2$$ and repeat that process $$n$$ times everything comes back to where it has started. For mirrors this is the same as the mirrors meeting at an angle of $$\pi/n$$ radians. For example example if two mirrors meet with an angle of $$\pi/2$$ then if you alternate between the mirrors after 4 reflections it returns to the start.

If two nodes do not have an edge between them it is treated the same as them having an edge of 2 connecting them, but it is omitted to reduce clutter. Additionally an edge of 3 may be left unlabeled from the diagram. If an edge is drawn without a label it is assumed to be an edge of 3. Edges can also be labeled &infin; to indicate that there is no relationship between the generators, that is, the mirrors do not intersect.

Polytopes
If you have a symmetry group you can describe a polytope with that symmetry by describing what it looks like within a single fundamental domain. Since the polytope obeys that symmetry, the polytope can be completed by reflections of the fundamental domain. Traditional Coxeter diagrams do this by describing a Coxeter group and modifying it to specify the location of a "seed point" in the fundamental domain relative to the mirrors. Each node in the diagram is given a ring to represent the point is distance $$1/2$$ from that mirror and is left unringed  to indicate that the distance is 0. If all the nodes are left unringed the entire polytope collapses into a single point, so for meaningful polytopes at least one node must be ringed. This seed will correspond to the vertices in the final polytope.

Once the location of the vertex is established the remaining elements are formed perpendicular to the mirrors. For example in the diagram indicates that the point lies on the blue mirror, but is distance $$1/2$$ from the other two mirrors. Two half-edges are added from the vertex to the two mirrors such that they meet with the mirror at a right angle. Since the half-edges are orthogonal to the mirror when reflected they will form full edges.

Not every polytope can be described this way, however extensions can greatly increase the number of important polytopes which can be.

The Coxeter diagram is also not necessarily unique. For example, , and all describe a cube but with different underlying symmetries. Generally the Coxeter diagram with the highest symmetry is considered for a particular polytope, for the cube this is.

Plaintext Notation
Since drawing diagrams or including images can be cumbersome, there is a common shorthand which uses plaintext to represent Coxeter-Dynkin diagrams. These are called plaintext Coxeter diagrams or ASCII coxeter diagrams (although some of these diagrams will use characters outside of the ASCII range). The simplest version of this notation allows the expression of any linear diagram. However various extensions make it more flexible, in some cases the extensions to the plaintext notation are more powerful than the plain graphical notation. (see ).

To convert a linear diagram to ASCII each part of the diagram is replaced in place with some ASCII. Edges are replaced with their number (unlabeled edges are replaced with, and missing edges are  ), and nodes are each replaced with a special characters. The two basic nodes, ringed nodes and unringed nodes, are marked with   and   respectively. Extensions to both notations add additional types of nodes. If an edge is labeled infinity the character  is used in the plaintext diagram even though it is not an ASCII character.

Virtual nodes
In order to allow for loops and bifurcation points in the plaintext notation for Coxeter diagrams, Richard Klitzing added virtual nodes. One first starts by writing the diagram from any node: these are the real nodes. Whenever one needs to specify that a node is connected to a previous one, one writes down an asterisk, followed by a lowercase letter indicating the index of the to be referred node. Indices start at  for the first node and precede in alphabetical order. The 15th index is  which is also used to represent dual ringed nodes, however the   prevents any ambiguity and it is unlikely to have a diagram with enough nodes in the first place.

History
Coxeter described diagrams with unlabeled nodes for groups in the early 1930s, but an equivalent graph was devised during the 1940s by Dynkin and by De Witt.

In group theory, there is a special additional meaning applied when the edge is even, as to whether an arrow points one way or the other, viz the four-edge is represented as a double-line with arrow, e.g. ==>==, there is no '5' edge, and the 6-edge is a triple-line representation. The groups o==<==o-o and o==>==o-o are distinct, and represented by the letters C and D. The group represented by o-o==>==o-o is the same as with the arrow reversed.

Such represents a physical kaleidoscope, where the separate mirrors are labeled A, B, C, ... and the edges represent the angles between them. The arrow-head notation does not apply, which is why there is no 'D' symmetry. Instead, such additional symmetries called for not in Lie theory, are simply added to the end of the Lie list, which is why {5,3} and {5,3,3} appear at G.

The node markings predate the graph by some 20 years in publication. This is due to Alicia Boole Stott's construction of the fifteen uniform polychora by using a series of expansions. Starting with C_600, one can 'expand' the edges, hedra, and faces outwards, without changing the size of the original figure. Doing this to the edges of a cube, for example, would produce a truncated cube (specifically a larger cube, with the edge-lengths as for the smaller one, the result is new faces appearing at the vertices). Expanding the faces outwards would lead to additional faces replacing the edges (generally rectangles), and triangles at the old vertice, resulting in a rhombocuboctahedron.

In order to "get rid of the cube", one has to "contract" the vertex to nothing. In this case, the edges and squares of the original cube are made to shrink to zero. The truncated cube would then shrink to a cuboctahedron, and the rhombocuboctahedron would become an octahedron. (the squares of the cube would disappear, and the rectangles into digons, which become line segments.)

Wright(?) rearranged this growth to eliminate the contraction, by supposing a start from a zero-size cube (ie r4o3o), where r denotes adjacent vertices o|o on the end of a zero-edge cube. The same operations as Stott uses still work, but one expands r to x, rather than contracting x to o.

Wythoff found a way of implementing this construction with mirrors. Instead of expanding a kind of element, the vertex would be held at a distance from the mirror. Because it is a simple shape kaleidoscope, we see that a point can be set 0 or 1 from any mirror. This can be represented by a parallelogram set into the kaleidoscope, each vertex of the parallelogram is either on some mirror, or off it, in every combination.

This is the paper that Coxeter noticed in 1935, and proceed to give the construction of the omnitruncated 4_21, along with an anotated history, which is what is given here. However, Coxeter's notation does not show that he has linearised the graph.

Snub nodes
Nodes can also be "hollowed" (alternated). Traditionally, these nodes are marked with a empty circle. In the plaintext notation, they're marked with, which can stand for "snub", "semiation", or for the surname of Alicia Boole Stott who introduced this further node type to Coxeter. In this case, a point is also placed off a mirror, but only vertices generated by an even amount of reflections through the alternated mirrors are accounted for. This results in the snub polytopes.

Alicia Boole Stott suggested to Coxeter, that removal of the mirrors in a marked diagram, but leaving the marks, would lead to an alternation of vertices. In this, the removal of the mirrors would mean removal of the black dot, but leaving the outer circle marking the node. thus:. The removal of a node in this way, would remove half of the vertices, replacing these with simplex-faces, being the vertex-figure. The original polygons at the vertex would be replaced with polygons of half as many vertices.

is the general snub polygon, arising from alternating the edges of, and then removing half of the vertices, so that the polygon P is restored, but rotated relative to the symmetry. A solution is always possible. Note that a snub digon   is an edge rotated relative to the x-y axis, being the diagonal of a general rectangle.

derives from the alternation of. This leads to three snub polygons,,  , and. The values of $x$, $y$, and $z$ are then set to make the resulting edges of the alternated polygons equal. This is three equations in three variables, which is always solvable, although often adding a further cubic to the equation.

is generally not solvable. There are four free variables, $w$, $x$, $y$, $z$ but six edges demanding to be equal,,  ,   along with  ,  , and. The general case has no solution, which means , which topologically has 10 icosahedra, 20 octahedra, and 60 tetrahedra, is never uniform.

The only known solution for a uniform snub in four or higher dimensions, is of the form where the vertex-figure is either a regular simplex, or a pyramid with that as a base. This leads to the half-figures (e.g. half-cube), and to the Coxeter snubs.

In ... where $q$ is even, the figure represents simple removal of alternate vertices of ..., so is a tetrahedron,  is a hexadecachoron,  is a hemipenteract, and so forth. is a triangle, is the triangular tiling, such that the snub faces all pointing 'up', and the hemiated hexagons become those triangles pointing 'down'.

Coxeter observed in ... the edges can be 'directed'. This means that all edges can be cut to the same ratio. The removal of a vertex then places new vertices on these edges, each of which have been cut in this ratio. For, the ratio of 1:f gives an icosahedron, and is the icosahedron,  is the snub disicositetrachoron, and  is the snub disicositetrachoric tetracomb.

In the general case, by means of mere vertex alternation the edge lengths of the resulting polytopes will be different from one another. However, in special cases, like the snub cube or the snub disicositetrachoron, they can be made uniform by adjusting the placement of the point accordingly. (This depends on the ratio of positional degree of freedom of the seed point vs. the count of obtained different edge types.)

Holosnub nodes
Norman Johnson suggested that all vertices could be alternated, into an odd and even figure. This results in a local application of usual snubbing, i.e. every alternate vertex is maintained, every other is omitted. Some times, this snub construction could result in a recurrent edge path in overlapping vertices, one of which is accounted for, and the other one of which isn't. If we carry out the construction as described above anyways, we create a special type of snub called a holosnub. To represent these a new node is introduced, or   (German ezsett) in plaintext. For example, produces a pentagram,  a small ditrigonary icosidodecahedron, and  a small ditetrahedronary hexacosihecatonicosachoron. As it turns out, whenever a snub node of either type is connected to a non-snub node by means of an odd link mark, then every vertex finally will be maintained and as well replaced by snub facets. Thus holosnubs in contrast to usual snubs do not halve the vertex count. Turning this observation to a rule now allows to apply holosnubbing even when all those connecting link marks are even as well. This then would result in compounds though. For example, the is a stella octangula, which reduces by symmetry to  as well. Because 4 is even, the double-cover reduces to two separate (but inverted) tetrahedra.

Early Notations
The brief for the original notation was to allow all CD diagrams to be 'linearised', in such a way that Stott construction could be implemented by vectors. The extent of covered groups were those as far as the paracompact (finite content) groups, including the compact (finite extent) groups in hyperbolic geometry. These represent those groups, for the removal of any node results in at least euclidean or finite groups.

A branch connects 'parent' to 'child'. When a node has more than two marked nodes, the parent-child has to be broken, and specific grand-parent and grand-child nodes are implemented. A grandparent branch connects the child node to a node further back.

Three levels of grandparent branches are required, x3c3b3a3oNn The branch N is a grandparent branch, this would normally connect node n to the node marked 'o'. Instead, the branch runs from c, b, or a as N is C, B, or A. Note that a direct connection to N would be a '3' branch. This allows most groups to be shown in the 'icosahedral' format, eg 3_21 is {3,3,3,3,3,A}

Two levels of grandchild branches are needed. This connects a leading node to node 3, 4 in the chain as nNx3e3g3o3o3o.. N would normally connect n to x, the use of E and G allows for n-e and n-g branches. This allows for 'dodecahedral format', ie {5,3,3,3...}, so 2_31 is oGx3o3o3o3o3o

Loops are implemented by a structural node, which indicates a return to either node 0 (z) or 1 (zz).

The intent was to allow the polytope to be expressed as a vector, by giving the structure varying weights. For example, allowing the letters s, q, f to stand for 3, 4, 5 branches, one can represent a polytope, as for example 10.22s2.21s44q2 as if the vector (10.22, 2.21, 44, 2) were applied to coordinate system ssq. The other extent is to simply replicate Coxeter's 3_21. by using a number to represent an unbroken chain of 3's and the letter above, along with Q (4), F (5), H(6), and V (5/2) for non-3 branches. 3_21 becomes, eg 5B. Marked nodes are indicated by a /, so /5B or 1/4B for its rectate. This is more legible than o3x3o3o3o3oBo or o3x3o3o3o3o *d3o.

When using with "Wythoff notation" (Wythoff had no part in it), the letters S and R are for 3 and 2 branches, along with 'i' to give the supplement figure (ie P/(P-D)).

Duals
Some polytopes don't have a Coxeter-Dynkin Diagram but have a dual which does. In these cases there is extended notation to show these polytopes in terms of their duals.

To do this two new node types are introduced (ASCII  ) and  (ASCII  ). The diagram is then the same as the diagram of it's dual but with ringed nodes replaced with  and snub nodes  replaced with. Unringed nodes are left unchanged. These nodes cannot be mixed with their dual counterparts in the same diagram.

Complex polytopes
Complex polytopes can be represented by numbering the nodes.

The non-intersection marker
The non-intersection symbol, Ø (a crossed O, although it was chosen to look like a "no" sign) is used to notate non-simplicial domains.

Different edge lengths
A further extension to these diagrams was introduced by Wendy Krieger, which isn't paralleled by the traditional Coxeter diagram notation. It allows for non-unit edge sizes in the diagram representation, by interpreting ringed and unringed nodes as denoting distances rather than incidences. That is, the symbol x represents a point at half-unit distance from a mirror (so as to create a unit edge), the symbol o represents a distance of zero, and other symbols create edges of other sizes. The most common edge sizes are shown in the table below, along with the characters commonly used to represent them, and with the origins of such characters.

For example, the vertex figure of the icosidodecahedron may be written as x2f.

Other letters, most commonly y and z, are used as variables. In some contexts, x is also treated as a variable. For example, the rectangle could be represented as x2y, and the cuboid could be represented as x2y2z.

Compounds
A compound of polytopes of the same symmetry can be denoted by merging notations into one. Since they share a symmetry the only difference between diagrams written in terms of that symmetry is the values of the nodes. So the compound is written placing multiple nodes between each edge, one for each of the polytopes in the compound

For example, the cube, and an octahedron,  , can be combined into the compound. This doubles up the node symbols, the first in each set applies to the first figure, and the second applies to the second figure.

Laced polytopes
The diagram merging can also be used to denote several different types of laced polytopes. To denote a laced polytope, add &#, representing the extra dimension, to the end of the symbol for the compound, and additional markers depending on the type of laced polytope.


 * Lace prisms are denoted by &#x. They can be created by lacing two component polytopes with laces of unit edge length (x). For example, you can lace together a triangle, x3o, and a hexagon, x3x, to get a triangular cupola, xx3ox&#x
 * Lace simplices are a natural extension of lace prisms. They are denoted by the same notation, but with three or more components. They created by taking the layers and lacing every pair, as in a simplex: this is the natural extension and does not require a further qualifier. The simplexes themselves o...o&#x are the easiest examples. Another example is oxx&#x for the square pyramid, made out of a point and two dyads laced together. Lace simplices derive from the vertex figure of WME polytopes, having a lacing simplex with as many vertices as marked nodes, and a cross symmetry equal to the unmarked nodes.  It is not impossible to have no mirrors at all, as the omnitruncate shows.
 * Lace tegums are the duals of lace prisms, denoted by &#m. (what are the duals of lace simplices called?)
 * Lace towers are created by lacing multiple parallel layers within an axial stack: they get suffixed by a t. An example is xxo3oxx&#xt for the cuboctahedron, which is built from a triangle, on top of a (diametral) hexagon, on top of a triangle rotated with respect to the first. The lace tower is read as &(an extra axis of symmetry), #(kill the symmetry), x(lacing of edge x), t(all layers in a tower).
 * Lace rings, which lace polytopes in a cyclic order, become represented by means of the suffix r. Simple examples include oxox&#xr for the octahedron, and oxxo&#xr for the triangular prism. Duoprisms, like the square-pentagonal duoprism xxxx5oooo&#xr, are also examples of lace rings.
 * Wout Gevaert also considered the tegum sum. suffixed with &#z, take the convex hull s of the compound. For example, the rhombic dodecahedron, xo4oo3oq&#z is the convex hull of the cube-octahedron compound xo4oo3oq mentioned previously. Note that the individual layers themselves here no longer are true boundary facets of the resulting polytope.

Wythoff symbol
Wythoff symbols are simplified versions of Coxeter diagrams used for polyhedra. They are formatted as either "p q r |", "p q | r" or "p | q r" where p, q and r refer to the order of symmetry of the axes. The Wythoff symbol represents a triangular Coxeter diagram where p, q and r are on the edges. The numbers to the left of the bar refer to edges opposite ringed nodes, while the numbers to the right of the bar repesent edges opposite unringed nodes. The symbol "| p q r" is misleading, as it represents not a diagram with all nodes unringed, but a diagram with all nodes snubbed. For example, the truncated tetrahedron is represented by 2 3 | 3.

Schläfli symbol
Schläfli symbols are used to denote regular polytopes. They consist of a bracketed list of numbers, such as $$\{4,3,3\}$$ for the tesseract. $$\{n\}$$ represents an $\sqrt{5}$-sided polygon, while in higher dimensions the facet is the polytope represented by all the numbers except the last, and the last number represents how many facets meet at a peak. Schläfli symbols can be easily converted to Coxeter diagrams. To convert you create a linear diagram with the first node ringed and then label the edges in between with the values of the Schläfli symbol in order. So the Schläfli symbol $$\{n,p,\dots,z\}$$ becomes the Coxeter diagram .... This includes cases where the Schläfli symbol has fractional or infinite values.

Just as with Coxeter diagrams, various extensions to Schläfli symbols have been created to represent additional polytopes. With extensions neither system is strictly more expressive. For example the mucube[[ has no Coxeter diagram but it does have the extended Schläfli symbol ($\{4,6\mid4\}$), and the [[disdyakis dodecahedron has no Schläfli symbol but it does have a Coxeter diagram,.