-: Dynkin Matrices :-

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The Dynkin symbol is a drawing of the fundemental region of a reflection group, for groups that have a region in the shape of a simplex. We shall encounter fundemental groups that are not simplexes elsewhere.

The symbol is quite powerful. Not only is it a discribing frame, rather like 'truncated dodecahedron', but the frame itself lends to calculations.

Representations

The most common form for this to be shown is in its graphic form, like

                      o
                      |
                      |                           5
  o----o----o----o----o----o----o     o----o----o----o

The symbols do not lend themselves to inline writing, esepcially if there are marked nodes.

Coxeter in his Regular Polytopes uses the notation 4₂₁ for the former, and {3,3,5} for the latter. Note that he has no notation for central or several central nodes: the books Regular Complex Polytopes and Twelve Essays deal with these, but resort to drawing the pictures or writing the pictures inline, eg @---o---o.

Jonathan Bowers uses a style of printer's greeking, reducing the branches to nothing. So the figure on the left is oooo8o. Because he deals in one kind of decoration (@), one needs only four symbols for the none-over-node: 8, 6, 9 and 5. Non-three branches are shown as a pucutation eg oo"o.

Wendy Krieger uses a pseudo-regular trace, making the symbol into a kind of regular figure. The notion came from something like 2₂₁, but written. Here the idea is to make everything regular, and then use the inlining symbols.

                   /---------\                           5
  o---o---o---o---o---o---o   o  Dynkin Symbol o---o---o---o

                            B                            5
  o---o---o---o---o---o---o---o   Trace        o---o---o---o

    S   S   S   S   S   S   B      = 6B          S   S   F     = 2F

 , 3 , 3 , 3 , 3 , 3 , B                 , 3 , 5 

  o 3 o 3 o 3 o 3 o 3 o 3 o B o                o 3 o 3 o 5 o

One sees that the dynkin symbol has been converted into a regular-like chain thing. This line is a trace of the symbol. What it does do is to allow us to order the nodes without any effort. Calling a group by any trace-name orders the nodes.

Because the earliest form had no symbol for a '3' branch, it became that we use the numbers direct, eg 6B means six three branches in a simple chain, followed by a B branch.

The chief novelty of this is the way the A, B, C branches work. A subject node removes the subject further backwards. So a branch is read as a verb connecting subject to object. An object branch has a deferred object.

  object branch           subject branch
  deferred object        andvanced subject

    /------=\                /-------\
   o    o----o----o----o----o----o    o

   o====o----o----o----o----o----o====o
     E    3    3     3    3   3     A    =  E5A


   /------------\                 /------------\
  o====*....o....o.....~~~..o....o....o....*====o
            E    G          C    B    A      B

       Object Node              Subject Node

Loop-Node

Loop-nodes were introduced to allow for loops to be unfolded, and written as if it were a chain. In practice, we just unfastened a link and spread the chain out.

Such nodes are written by a trailing z or : node. The forms are SSS:, {3,3,3:} or o3o3o3z or 3:.

Decorations

A decoration is a motif added to the symbol, not so much to change the symmetry, but to give the underlying kaleidoscope something to play with.

Edge-Paedel

The most common use for this is to place edge-paedels, or marks to indicate an edge crosses this mirror. For some users this is the only marking.
      Nodes were originally indicated by a circled dot, but the lack of such a symbol means we use something like (o), or the commercial at, @, both of which look like a circled dot.
      Jonathan Bower's greeking notation presumes only paedel-nodes are present, since stacked nodes would need a square of symbols in addition to the lining ones.

Stott-Paedel

Mrs Stott suggested using an empty circle node (ie keeping the circle, but removing the dot, such as ( ).
      Proffessor Johnson's double-density holosnubs are sometimes written as a double-circle, and typed (( )).

Complex-Paedel

Coxeter introduced the notation of writing a small number under a node, to indicate that instead of being dyadic, it was n-adic, (that is, for example, an edge had three vertices.
      Such notation is used in Regular Complex Polytopes.

Margin-Paedel

No one seems to have treated the need for a margin-paedel, or the representation of the duals.

The Electrification Standard

Back in the days of the seventies, when the world of computers was young, i did many things to try and squeeze the last byte out of whatever i would imagine the technology to be.

Since programming would also be a part of the new world, i decided to draw up standards from the begininning, and these became a series of electrification standards. So there were standards for prime-data, and standards for the polytopes.

The long-term dream was that you could take a symbol like /SF, feed it into a program, and from the symbol alone, it should deduce most of the metrical properties, and make a good stab at the numerical ones as well. That is, it could tell you it was an icosahedron, and that its various radii and volumes. most of this is now met

verb: Dr Klitzing interpreted the branch-nodes as having an inplied pronoun, rather like as if the sentence stopped before the B branch and a new sentence referring tho the third-last node was invoked.

computers: In the days of the early computers, people spent a great deal of time trying to pack large amounts of information into small data fields, because data was cheap. We called ourselves then Data Processing
      Today, computer and computer storage abounds. We now call computers information technology, and call any box-shifter a computer technician.
      On the other hand, i have, while being on the computer help-desk, suggested good old pencil and paper solutions (because of audit-trails and lack of replication). Now that's information-techology - finding the right technology for information!

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