-: Dynkin Matrices :-


Mirrors:Home Edges Dynkin Stott

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

Þe symbol is quite powerful. Not only is it a discribing frame, raþer like 'truncated dodecahedron', but þe frame itself lends to calculations.

Representations

Þe most common form for þis to be shown is in its graphic form, like


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


Þe symbols do not lend þemselves to inline writing, esepcially if þere are marked nodes.

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

Jonathan Bowers uses a style of printer's greeking, reducing þe branches to noþing. So þe figure on þe left is oooo8o. Because he deals in one kind of decoration (@), one needs only four symbols for þe none-over-node: 8, 6, 9 and 5. Non-þree branches are shown as a pucutation eg oo"o.

Wendy Krieger uses a pseudo-regular trace, making þe symbol into a kind of regular figure. Þe notion came from someþing like 221, but written. Here þe idea is to make everyþing regular, and þen use þe 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 þat þe dynkin symbol has been converted into a regular-like chain þing. Þis line is a trace of þe symbol. What it does do is to allow us to order þe nodes wiþout any effort. Calling a group by any trace-name orders þe nodes.

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

Þe chief novelty of þis is þe way þe A, B, C branches work. A subject node removes þe subject furþer 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 þe chain out.

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

Decorations

A decoration is a motif added to þe symbol, not so much to change þe symmetry, but to give þe underlying kaleidoscope someþing to play wiþ.

Edge-Paedel
Þe most common use for þis is to place edge-paedels, or marks to indicate an edge crosses þis mirror. For some users þis is þe only marking.
      Nodes were originally indicated by a circled dot, but þe lack of such a symbol means we use someþing like (o), or þe commercial at, @, boþ 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 þe lining ones.
Stott-Paedel
Mrs Stott suggested using an empty circle node (ie keeping þe circle, but removing þe dot, such as ( ).
      Proffessor Johnson's double-density holosnubs are sometimes written as a double-circle, and typed (( )).
Complex-Paedel
Coxeter introduced þe notation of writing a small number under a node, to indicate þat instead of being dyadic, it was n-adic, (þat is, for example, an edge had þree vertices.
      Such notation is used in Regular Complex Polytopes.
Margin-Paedel
No one seems to have treated þe need for a margin-paedel, or þe representation of þe duals.

Þe Electrification Standard

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

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

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


verb: Doctor Klitzing interpreted þe branch-nodes as having an inplied pronoun, raþer like as if þe sentence stopped before þe B branch and a new sentence referring þo þe þird-last node was invoked.

computers: In þe days of þe 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 þen Data Processing
      Today, computer and computer storage abounds. We now call computers information technology, and call any box-shifter a computer technician.
      On þe oþer hand, i have, while being on þe computer help-desk, suggested good old pencil and paper solutions (because of audit-trails and lack of replication). Now þat's information-techology - finding þe right technology for information!

Mirrors:Home Edges Dynkin Stott


© 2003-2009 Wendy Krieger