### 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.

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

 km,n,... nodes 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 ```
 km,n,... nodes 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 *) ```
 km,n,... nodes 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)