Wythoffian operation

A Wythoffian operation is a polytope operation that can be represented on a conventional Coxeter-Dynkin diagram.

Applied to the regular polytope s in a given dimension, the Wythoffian operations produce the "Archimedean polytopes" of that dimension (named as an extension of the Archimedean solids - convex, finite, non-prismatic, and maintaining the regulars' symmetry). Wythoffian operations can also be applied to some uniform polytope s to produce other uniform polytopes, although this may produce degenerate cases such as multiple covers.

Shorthand and naming
Each named Wythoffian operation (except for loose cases like "expanded" or "omnitruncated") corresponds to a specific ringing of nodes and how far it is from the terminus of a Coxeter diagram, regardless of the diagram's size. That is, rectification always means node #1 is ringed, and bitruncation always means nodes #1 and #2 are ringed. (This may become confusing on branched and looped diagrams - for instance, the birectified pentacontatetrapeton has two #2 nodes ringed because the indexing starts at the single-node branch in the middle.)

Which nodes are ringed in a Coxeter diagram can be represented by numbers corresponding to their indices in the diagram, starting at #0. Thus we can say "t012" instead of "the #0, #1, and #2 nodes are ringed" or drawing out the entire diagram.

Since a polytope's Coxeter diagram has a number of nodes equal to the polytope's dimension, an n-dimensional polytope can only use operations with "prefix numbers" (positions of ringed nodes) less than n. (It makes sense since 0 is included in those numbers.) In addition, the names of some operations are typically not used until even higher dimensions than this rule may suggest. For example, rectified polygons are simply called "dual" even though a polygon's diagram can technically be rectified, and even though node 2 of the rectified hexacosichoron is ringed it is not referred to as a "birectified hecatonicosachoron."

When the ringing of a Coxeter diagram has a symmetry to it (ignoring the weights of the connections), the name for the resulting figure can be a combination of the regular and its dual. Examples include the icosidodecahedron (3D, ) and the hexeractihexacontatetrapeton (6D, ).

Examples of Wythoffian operations and their CDs
The Johnson/Ruen adjectives are meant to describe the "distances" between the first ringed node and each other ringed node. A prefix indicates which node is the first one ringed, then the largest distance between rings is listed (the distance between the first and last rings), followed by each other ring's distance from the first.

When the Johnson/Ruen adjectives differ from the Bowers adjectives, the former is more descriptive of the Coxeter diagram.

The Coxeter diagram can also be read in the opposite direction. This is often done when most of the nodes are on the far side. When this is done, the polytope's name is based on the dual. An operation with three or more rings is reduced to multiple two-ring operations that all start at the first ringed node.

There can, of course, be bi-, tri-, or even further versions of these from the terminus of the diagram. Some operations do not correspond to an exact number and position of ringed nodes, but are well-defined nonetheless.

It would not make sense for a polytope to be bidualed, triexpanded, or even biomnitruncated. If one wanted to describe the polytope ... as "biomnitruncated" instead of writing out all the operations in full, they could just call it a "t1,2,3,...,(n-1)-simplex,". In fact this is a case showing up the precedence of the Bowers terminology above the Johnson one. In fact simply turn over the graph into ... and use the appropriate t0,(n-2) prefix and replace therein the "small" by "great" and you are done.

Abstract definition
Wythoffian operations may be generalized to abstract polytopes, which need not be regular. The simplest version of this generalization takes as input a rank-$$n$$ abstract polytope $$\mathcal{P}$$ and a set $$T \subseteq \{1, \ldots, n\}$$ of unringed nodes in the Coxeter diagram, which is assumed to be linear and numbered left to right. For example, $$T=\{1, \ldots, n-1\}$$would represent the dual, as the only ringed node would be node $$n$$. The unringed nodes are given rather than the ringed ones because, instead of performing Stott expansion on the original polytope by "adding" its various rectates, this definition begins with the omnitruncate (where all nodes are ringed) and then "collapses" elements to lower ranks based on the unringed nodes.

To illustrate how this collapse works, consider the truncated icosahedron, with CD diagram. However, the orientation of the diagram in this case would suggest a bitruncated dodecahedron instead, where $$n=3$$, $$\mathcal{P}=$$ doe, and $$T=\{1\}$$. The omnitruncate of doe is grid, which may be constructed abstractly from doe as follows.


 * First, the elements of grid (except the bottom) are defined to be the chains of doe which do not include the "improper" top and bottom elements. In this case:
 * the top element of grid is the empty set (which is vacuously a chain of doe);
 * the faces of grid are chains containing a single proper element of doe (the 20 hexagons are vertices, the 30 squares are edges, and the 12 decagons are faces);
 * the edges of grid are chains with two proper elements of doe (namely 60 vertex-edge chains, 60 vertex-face chains, and 60 edge-face chains);
 * and the vertices of grid are 3-element chains i.e. flags of doe (minus the improper elements), of which there are 120.
 * Next, the chains are ordered by reverse inclusion, i.e. for all chains $$A$$, $$B$$ of doe, $$A$$ is contained in $$B$$ within grid if and only if $$B \subseteq A$$.

Now, node 1 of the diagram must be "collapsed" to obtain the desired polytope ti. Concretely, this looks like shortening the edges of grid that lie between squares and decagons until they become points. Abstractly, it is necessary to understand what happens to each element of grid during this process. The elements of a Wythoffian truncate may be found by taking subsets $$X$$ of the nodes $$\{1, \ldots, n\}$$, where each $$X$$ corresponds to an element type of the omnitruncate. Specifically, this element type is a chain containing elements of $$\mathcal{P}$$ whose ranks are in $$\{1, \ldots, n\} \setminus X = \overline{X}$$. Next, $$X$$ is broken into contiguous runs, which amounts to "factoring" the element as a prism product of other (non-prismatic) polytopes. The "factors" which contain only unringed nodes (i.e. the runs which are subsets of $$T$$) are effectively points, which do not contribute to the rank of the collapsed element. As a result, these are discarded, leaving the $$\Gamma$$-minimal set $$X_\Gamma$$ with respect to the linear diagram $$\Gamma$$. The size of $$X_\Gamma$$ gives the rank of the collapsed element, and for all $$X, Y \subseteq \{1, \ldots, n\}$$, if $$X_\Gamma = Y_\Gamma$$, then the element types of the omnitruncate corresponding to $$X$$ and $$Y$$ collapse to the same thing. The following table continues the example for ti: From the resulting set of possible $$X_\Gamma$$, it can be seen that ti has one vertex type $$\varnothing$$, two edge types $$\{2\}$$ and $$\{3\}$$, two face types $$\{1, 2\}$$ and $$\{2, 3\}$$, and one top element $$\{1, 2, 3\}$$. However, since multiple different element types of grid collapse to the same type of the truncate, the duplicate elements must be removed. For this, we find the maximum $$X$$ that collapses to a given $$X_\Gamma$$. In this case, it is easy to look through the table and find the maxima, but it is more convenient in general to construct them from $$X_\Gamma$$ itself. This may be done by adding back all nodes which are both unringed (i.e. elements of $$T$$) and not adjacent to any element of $$X_\Gamma$$ (as otherwise, it would change some of the retained contiguous runs), which form the $$\Gamma$$-complement of $$X$$, denoted $$X^\Gamma$$. The maximum set $$X_\Gamma \cup X^\Gamma = X\langle\Gamma\rangle$$ is the $$\Gamma$$-completion of $$X$$, which determines which element type of the omnitruncate should become the element of the truncate given by $$X_\Gamma$$. The element type is given in the same way as a generic $$X$$ in the omnitruncate: as a chain whose element ranks are in $$\overline{X\langle\Gamma\rangle}$$. The ti example is once again continued in the next table: Thus:
 * Finally, a bottom element is added to obtain the full abstract polytope of grid. Although the bottom element may be any set other than the aforementioned chains, the set of all elements of doe is a good choice, as the ordering definition naturally extends to include it.


 * the vertices of ti are the 60 edge-face chains of doe;
 * the edges of ti are the 60 vertex-face chains and 30 edges of doe;
 * the faces of ti are the 12 faces and 20 vertices of doe;
 * and the top element of ti is the empty chain of doe.

Any two given chains are ordered by inclusion of their corresponding $$X_\Gamma$$ and by whether the chains' union is itself a chain, so:


 * the vertices of ti are contained by both edge types, since $$\varnothing \subseteq \{2\}$$ and $$\varnothing \subseteq \{3\}$$;
 * the first edge type of ti is contained by both face types, since $$\{2\} \subseteq \{1, 2\}$$ and $$\{2\} \subseteq \{2, 3\}$$;
 * the second edge type of ti is contained by only the second face type, since $$\{3\} \subseteq \{2, 3\}$$ but not $$\{1, 2\}$$;
 * and both face types of ti are contained by its top element, since $$\{1, 2\} \subseteq \{1, 2, 3\}$$ and $$\{2, 3\} \subseteq \{1, 2, 3\}$$.

Finally, as with the omnitruncate, a bottom element is added to complete the abstract polytope for ti.

This construction may be further generalized to non-linear Coxeter diagrams $$\Gamma$$, simply by replacing "contiguous run" by "connected component of $$\Gamma$$" when finding $$X_\Gamma$$ and specifying "adjacent" to mean "connected by an edge of $$\Gamma$$" when finding $$X^\Gamma$$. However, not all possible graphs on $$n$$ vertices produce valid polytopes.

Description and Use
The Krieger diagram is a variation of the Coxeter-Dynkin diagram by replacing the marked nodes by connections to an additional vertex node. This is in part because the marked nodes represent mirror edges or edges running from the vertex to its image in the marked node. These connections resemble the mirror-angle branches of the Coxeter diagram.

The Construction of Surtopes
The surtopes, or elements of the polytope, are all themselves described in the Krieger Diagram as a connected set of nodes, including the vertex node. The number of nodes needed to make a surtope of a given rank r is r+1. This is the same number as the vertices of a simplex of that dimension. Because the diagram can also give a count as the quotient of the symmetry group over-all against the symmetry group of the surtope, the method is to calculate the symmetry of the room that contains the interior of the surtope.

Relative to a surtope, there are three kinds of nodes.


 * s(urtope) nodes. These are the nodes that make up the surtope.
 * w(all) nodes. These are nodes that have a connection to an s-node, but are not s-nodes.  These represent mirrors that move the surtope onto an entirely different copy of itself.  w-nodes include any unused vertex-nodes, since all vertex nodes are connected to the nulloid.
 * a(round) nodes. These are the nodes that produce mirrors that contain the surtope.  As one might dance around a maypole, or the earth spins around its axis, these mirrors produce symmetries that do not change the surtope.

These nodes are relative to a polytope.


 * m(irror) nodes. These are the normal mirror-nodes represented by dots in the Wythoff symbol.  These are normally lettered for reference, from left to right.  It is the same system that Klitzing uses for virtual nodes (a separate copy to allow junctions and loops to link to the correct node).
 * v(ertex) nodes  These are vertex-nodes.  Each vertex creates a separate image in the kaleidoscope, the nodes are connected by lacing.  In an antiprism, the resulting lacing edges produce a zigzag as if the top and bottom were laced together.
 * n(amon or nulloid) nodes These are nodes that represent the surtope when the arrounding-figure (eg vertex-figure, edge-figure) is being found.  the w-nodes become v-nodes, and the a-nodes represent the mirrors of the surtope-arounding.

For the interior of the surtope to lie entirely inside a single room, all symmetries generated by the s and a nodes are counted. The a nodes do not contribute to the symmetry of the surtope itself, but rather to the room it is contained in.

Example
We consider the example, being the truncated icosahedron.

The nodes are lettered from left to right a, b, c. The vertex-node is numbered 1, and has a '3' branch to b and c nodes. In the following table, S signifies the nodes making the surtope, W are wall nodes, and A represents the symmetry that preserves the surtope without change (ie perpendicular to the surtope). The result is the total symmetry, divided by the symmetry orders of S and A in the form G/S/A. This is evaluated and the surtope described. The Result column shows the value as g/s/a, where g is the group order, s is the surtope order, that of the mirrors listed in column S, and a is the order of the A mirrors, The W mirrors play no part in the count, since they connect different copies of the same surtope. Note that we do not consider zero-elements, such as made by 1a or 1ac, since in these cases, the 'a' node is disconnected from the vertex node.

Two surtopes are incident on each other, if the nodes of the lesser one is a subset of the larger one. The edges of the hexagon represented by 1c are not part of the pentagon 1ab, because 1c is not a subset of 1ab.

If in the evaluation of surtopes, the letters become separated, say 1ac where a were marked, it still produces a surtope, being the prism-product of the separate elements 1a and 1c. The calculation is exactly the same as before, where S=1ac, W=b, A=- 120/4/1 = 30 squares. (line × line prisms)

Wythoff Lace Prisms - multiple vertex nodes
A lace prism still has the same number of nodes, but can have more than one vertex node. A Wythoff-Mirror edge polytope is a specific example with one vertex-node. The effect is to replace the apex of the kaleidoscope with a space that can hold a simplex of vertex-nodes, and each one produces a separate image in the kaleidoscope. These are then laced together, the image here is that polygonal antiprisms resemble a drum where the top and bottom are laced together to keep these surfaces taut.

The pentagonal antiprism is xo5ox&#x, this means it has two vertex nodes, x.5o. and .o5.x connected by unit edge lacing. One notes that the pentagons are reversed here, which is where the anti- (opposite) part comes from, The two vertex nodes are 1 and 2, with the direct connection 1-a(5)b-2. This means 1 is directly connected only to a, and 2 only to b, All vertex-nodes are connected to the nullitope, even if they are not used. That means that they can only be s or w nodes.

Vertex figures, etc.
The vertex figure or the various other arroundings of surtopes (edge-figures, etc) can be read from the diagram as well, the nodes are then allocated as such.


 * s nodes become n-nodes. This means all of these nodes are part of the namon or nulloid, and are not calculated.
 * w nodes become v nodes.  Each separate kind of edge becomes a vertex.
 * a nodes become m nodes  That is, the mirrors around the surtope become the symmetry of its figure.

The wall-nodes of the original surtope become numbered as vertex-nodes, and the calculation proceeds as before. The edges follow branches of the original figure, and the respective lengths become $$2 \cos \frac {\pi}{vb}$$ where v is the number of vertex nodes on the branch, and b is the marking of the branch,

The truncated icosahedron above becomes a-(5)--1---2, There is only one mirror, the vertices are at varying distances from it. The overall symmetry here is produced by node 'a', is 2. The edges are 2 cos(pi/1×5) = 1.618, the vertex figure of the pentagon, and 2 cos(pi/2/3) = 1.732 = vertex figure of hexagon. The edge 12 again lies in a single symmetry, the absence of mirrors in A means it is not reflected onto itslef by the mirrors.

Lace Towers and Lace Cities
Richard Klitzing requested a modification to allow the lacing to be described directly, as in 2 cos(pi/b) in the formula above. Further, various "lace towers" have multiple strata of lace prisms, in the same way that one finds sections of polytopes listed in books. In essence the figures produced by the lace towers are separate lace prisms of two layers, blended at the intermediate bases.

Lace cities are essentially lace towers where the apex is multi-dimensional, usually two. Such are fairly useful at revealing the inner relations of sections as any line across a lace city produces a section.

History from Stott to Coxeter
Alicia Boole Stott, provided the necessary insight into a new construction of polytopes, involving a process of expansion. (1910) The process is to draw some rank of element of a regular polytope outwards, but without changing the size. The effect is to introduce new elements to fill in the hull. For example, the edges of a cube can be radially moved, so that the square faces of the cube become octagons, and new triangle faces appear at the direction of the cube's vertices. The cube is thence converted into a truncated cube. Mrs Stott used this construction to find all but one of the uniform polychora. The notation that Stott used is $$e_1P_6$$ (i.e. the cube or hexahedron, expanded by the edges). The $c_{0}$ replacement for contract a cube to zero appeared later.

Wythoff restructed the Stott expansions into mirrors. (1912) In essence, an expand preserving the edges was to push the edge further from the centre of the kaleidoscope, without altering its size. The polytope by this construction, is the result of dropping half-edges to various mirrors.

H. S. M. Coxeter had devised a representation of Lie groups, where each point represented an order-two operation (e.g. AA = 1), and each drawn branch represented a non-communitive operator, the most common being an order-3 branch, of the form ABA=BAB. The Lie groups thus generated included the symmetric mirror groups. Dynkin and de Witt also independently devised this representation.

The insight in the next step is to connect Wythoff's construction as a motif to the Lie-group graph in a way to represent the construction of bulk of the uniform polytopes. (1938). Because all of the mirror groups are representable by Lie groups, this allowed the representation of the polytopes by marking nodes on the graph. None the same, the inline name of the polytope followed the modified Stott notation, applied to the Schläfli symbol, e.g. $$t_{0,1} \{4,3\}$$. Coxeter is now invoking Kepler's woodworking idioms onto the matter. His 1938 paper (Wythoff Construction) provides the necessary description using the omnitruncated $4_{21}$.

Alicia Stott provided the necessary insight into snubs by the manner of alternation of vertices. In this case, the node is first marked, and then the mirror-point removed, leaving a hollow ring.

The insights of Wendy Krieger
The complete picture of the Coxeter-Dynkin symbol includes a separate vertex-node. This is foreshaddowed by way of the vertex-figure of a figure with a single marked node. The vertex-node is connected by branches to each marked node, these branches represent half-edges. Where there is no branch, the edge is zero. The elements of the figure represented in the symbol then are found by r+1 nodes, where r is the rank of the element [surtope].

The complete list of elements or surtopes is then derived from the list of r+1 nodes, that include the vertex-node. It does not matter if the mirrors become disjoint when the vertex-node is removed: this simply creates a prism product of the disjoint parts. In order to count the number of a derived surtope, one must divide the remaining mirrors into wall-nodes and around-nodes. The wall-node represents a mirror that reflects the surtope onto a different copy of it, and as such are not part of the symmetry of the surtope. The around-nodes are those nodes that are not directly connected to the nodes making up the surtope, these mirrors contain the surtope in full.

The notion that the kaleidoscope cell can contain something more than just a vertex, leads to several vertices connected by full edges. The prototype for this is the polygonal antiprism, where a pair of mirrors captures the two vertices connected by one edge of the zigzag, as well as the usual half-edges. Such resembles lacing as on a drum or on shoes.

Another insight is that the marked diagram represents a position vector. In this way, Stott addition becomes addition of vectors, and the polytopes themselves become position-polytopes (after position-vectors). The decorated diagram becomes a coordinate system, and a scheme is then devised to admit more lengths than 0 or 1. The sorts of numbers that turn up in polygon-theory are algebraic integers, the solutions to monoic polynomials (the number against the term with the highest power is 1 or -1), and so rational numbers will not work here. Letters were then used to designate the chords of interesting polygons.

Being vectors, it is possible to do vector arithmetic, such as determine the radius or some displacement. This is done by way of a matrix-dot product. The matrix consists of the dot-product of the axial unit vectors, the matrix is multiplied by the first vector, the result is then dot-product to the second vector.

Where it is possible to make the dynkin-symbol into a linear diagram, this is done. The pseudo-regular trace is then a path from node to node, generally following the longest chain of connected nodes, with jumps to create branches whose source (or target) is two to four nodes back in the trace. The range of notations on this notion vary from using lower case letters as comma-separators for the nodes, and the nodes marked by numerical values, eg 1q1s0 (where s, q, f, h stand for branches of 3, 4, 5 and 6), to 1/Q/ (the number is the number of three-branches, the letters are everything else), to o3x4x (which represent alternating nodes and branches).

The necessary matricies were derived by hand during the 1990s, but it was not until after the turn of the century, that the reciprocal matrix represented the supplement cosines of the angles between the mirrors. This particular operation greatly speeds the calculation of radii.

A note on Schläfli and Coxeter-Dynkin
The methods of Kepler, of Schläfli and Gosset, and of Stott and Wythoff represent three diverse methods of deriving polytopes. Coxeter managed in Regular Polytopes to place bits and pieces of all three in apposition, to the extent that readers may be misled into thinking that the methods include the extraneous parts. For example, Schläfli's series-calculation is equated to the determinate of a particular matrix, whereapon other people refer to that matrix by Schläfli's name. You don't need to do matrix determinates to use the series calculation.

The archemedean polyhedra can be derived by a series of truncations as far as the edge, the scattered few are the snubs, named after the lay of the non-triangle faces (ie snub cube, snub dodecahedron). Coxeter retitled Stott's operator from 'e' to 't', and used Stott's operator appended to Schläfli's names to derive representations for the various non-regular figures, although the instances of using these are scarce.

That the various archemedean polyhedra align with the stott constructions, led people like Norman Johnson, to back-engineer how Kepler's names might fit onto the Wythoff construction. The list was somewhat edited by noting that Kepler's truncated cuboctahedron isn't an actual truncate of a cuboctahedron, so it became a rhombicuboctahedron.

Norman Johnson provided names at the behest of Mangus Wenninger Polyhedral Models (1971) to cover the various starry polyhedra. This book seems fairly influential in that that the names and methods were carried forward into higher dimensions, as a series of fake Kepler constructions, to replace the relatively straight-forward operators suggested by Stott, Wythoff and others. Johnson and Jonathan Bowers continued to develop their names in isolation until Wenninger's mail list.

It should be noted that the Kepler operations are totally inadequate to describe what can be described by Stott-addition, to the extent that no one has ever given a reliable construction to anything other than truncate, rectate, cantellate, and cantetruncate (ie Kepler's operators with Johnson names). Everything else is just patterns of dots on the Coxeter-Dynkin diagram.

The proper Wythoff operations correspond to simply inserting or removing individual nodes in the graph, and named in the style of John Conway as (nodes)-ambo (polytope). The following tables then are the fake Kepler-style names appended to the patterns of dots.

Although the Schläfli symbol closely resembles the matching Coxeter-Dynkin graph, the Schläfli symbol has no notion of nodes, to which end one usually couples something like a Stott operator or a Kepler-style operator. In actual operation, the actual numbers in the Schläfli symbol are used. Gosset extended the power of this process somewhat, to deal with branching groups. In the Coxeter-Dynkin symbol, the branches simply serve as the name of an oblique coordinate system, the action takes place with the nodes. It is not an obvious jump until you have seen it done.

Conway Notation
John Conway devised a notation for operators, that are applied to any polyhedron as if it were regular. After each application, the flags of the derived polyhedron are redrawn, and the operation can occur again. In this way, it is not based on mirror-cells. The names largely follow Kepler's names, except for oxo ambi-, xox expand and xxx bevel. The truncated cube is then tC. Generally this produces interesting non-regular figures, there is an application at George Hart's site that produces models from the symbol.

One observes that aa- is the same as e- and ta- is the same as b-, in line with Kepler's observations.