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The Coxeter-Elte-Gosset polytopes   km,n,...

It was the professional lawyer and amateur mathematician T. Gosset, who in 1900 published his investigation of regular polytopes of dimensions greater than 3 and of semi-regular ones, i.e. the regular-facetted and vertex-uniform polytopes. He even included alike tesselations, which he already considered as degenerate polytopes. Gosset esp. gets the credit for finding the polytopes 22,1, 32,1, and 42,1.

E.L. Elte listed in 1912 the then rediscovered semi-regulars of Gosset, but he further allowed recursively for likewise semi-regular k-faces of up to 2 types (shapes) each. This then already encompassed the main representatives of the En family, except for 14,2.

The symbols km,n (and lateron also kl,m,n) finally where assigned by H.S.M. Coxeter, who rediscovered and completed their full list. These symbols originally where shortcuts for his extended Schläfli symbols, which then are restricted to have a chain of 3's on the left and two (or more) chains of 3's at the right. k then is the count of those at the left and the indices would represent the right counts each.

{
3,...,3 3,...,3,...,3
}
= km,n
3,...,3

Accordingly 0 as an index is to be excluded here. But it might well occure as a leading number. The sum of that leading number and all the indices clearly equates to one but the number of nodes of the corresponding Dynkin diagram, and therefore represents one but the number of dimensions of embedding space for polytopes, respectively represents the number of dimension of the according euclidean tesselations. The order of the indices obviously is irrelevant.

These extended Schläfli symbols are to be translated into Dynkin diagrams as bifurcated diagrams, each number line then represents one leg, and the left number line is the one, of which the end node is being marked. E.g.

{
3,3 3,3,3
}
= 23,1 =
                     o---3---o---3---o
                    /                 
                   3                  
                  /                   
(o)--3---o---3---o                    
                  \                   
                   3                  
                    \                 
                     o                
3

Note that all Coxeter-Elte-Gosset polytopes are quasiregular by definition. Conversely, not all quasiregulars occur here. Not only that all diagrams using others than the link mark 3 are missing, even within the bifurcated diagrams all quasiregulars with ringed nodes other than the end ones or than the bifurcation spot are missing as well.

External
links
wikipedia   esp. k2,1: wikipedia   2k,1: wikipedia   1k,2: wikipedia  

* The thus marked ones happen to be flat euclidean tesselations and their embedding space dimensions thus is one less than the number of nodes.


nodes km,n,...
2
01 - x3o  {3}
3
02   - x3o3o  tet
01,1 - o3x3o  oct
4
03   - x3o3o3o  pen
02,1 - o3x3o3o  rap
01,1,1 - o3x3o *b3o  ico
11,1   - x3o3o *b3o  hex
5
04   - x3o3o3o3o  hix
03,1 - o3x3o3o3o  rix
02,2 - o3o3x3o3o  dot
02,1,1 - o3x3o3o *b3o  nit
12,1   - x3o3o3o *b3o  hin
21,1   - o3o3o3x *b3o  tac
01,1,1,1 - o3x3o *b3o *b3o  icot *)
11,1,1   - x3o3o *b3o *b3o  hext *)
6
05   - x3o3o3o3o3o  hop
04,1 - o3x3o3o3o3o  ril
03,2 - o3o3x3o3o3o  bril
03,1,1 - o3x3o3o3o *b3o  brox
13,1   - x3o3o3o3o *b3o  hax
31,1   - o3o3o3o3x *b3o  gee
02,2,1 - o3o3x3o3o *c3o  ram
12,2   - o3o3o3o3o *c3x  mo
22,1   - x3o3o3o3o *c3o  jak
nodes km,n,...
7
06   - x3o3o3o3o3o3o  oca
05,1 - o3x3o3o3o3o3o  roc
04,2 - o3o3x3o3o3o3o  broc
03,3 - o3o3o3x3o3o3o  he
04,1,1 - o3x3o3o3o3o *b3o  bersa
14,1   - x3o3o3o3o3o *b3o  hesa
41,1   - o3o3o3o3o3x *b3o  zee
03,2,1 - o3o3x3o3o3o *c3o  rolin
13,2   - o3o3o3o3o3o *c3x  lin
23,1   - x3o3o3o3o3o *c3o  laq
32,1   - o3o3o3o3o3x *c3o  naq
02,2,2 - o3o3x3o3o *c3o3o  ramoh *)
22,2   - x3o3o3o3o *c3o3o  jakoh *)
8
07   - x3o3o3o3o3o3o3o  ene
06,1 - o3x3o3o3o3o3o3o  rene
05,2 - o3o3x3o3o3o3o3o  brene
04,3 - o3o3o3x3o3o3o3o  trene
05,1,1 - o3x3o3o3o3o3o *b3o  bro
15,1   - x3o3o3o3o3o3o *b3o  hocto
51,1   - o3o3o3o3o3o3x *b3o  ek
04,2,1 - o3o3x3o3o3o3o *c3o  buffy
14,2   - o3o3o3o3o3o3o *c3x  bif
24,1   - x3o3o3o3o3o3o *c3o  bay
42,1   - o3o3o3o3o3o3x *c3o  fy
03,3,1 - o3o3o3x3o3o3o *d3o  rolinoh *)
13,3   - o3o3o3o3o3o3o *d3x  linoh   *)
33,1   - x3o3o3o3o3o3o *d3o  naquoh  *)
nodes km,n,...
9
08   - x3o3o3o3o3o3o3o3o  day
07,1 - o3x3o3o3o3o3o3o3o  reday
06,2 - o3o3x3o3o3o3o3o3o  breday
05,3 - o3o3o3x3o3o3o3o3o  treday
04,4 - o3o3o3o3x3o3o3o3o  icoy
06,1,1 - o3x3o3o3o3o3o3o *b3o  barn
16,1   - x3o3o3o3o3o3o3o *b3o  henne
61,1   - o3o3o3o3o3o3o3x *b3o  vee
05,2,1 - o3o3x3o3o3o3o3o *c3o  ribfoh *)
15,2   - o3o3o3o3o3o3o3o *c3x  bifoh  *)
25,1   - x3o3o3o3o3o3o3o *c3o  bayoh  *)
52,1   - o3o3o3o3o3o3o3x *c3o  goh    *)
10
09     - x3o3o3o3o3o3o3o3o3o  ux
08,1   - o3x3o3o3o3o3o3o3o3o  ru
07,2   - o3o3x3o3o3o3o3o3o3o  bru
06,3   - o3o3o3x3o3o3o3o3o3o  tru
05,4   - o3o3o3o3x3o3o3o3o3o  teru
07,1,1 - o3x3o3o3o3o3o3o3o *b3o  brade
17,1   - x3o3o3o3o3o3o3o3o *b3o  hede
71,1   - o3o3o3o3o3o3o3o3x *b3o  ka
etc. (An and Dn diagrams only)


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