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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 semiregular ones, i.e. the regularfacetted and vertexuniform polytopes. He even included alike tesselations, which he already considered as degenerate polytopes. Gosset esp. gets the credit for finding the polytopes 2_{2,1}, 3_{2,1}, and 4_{2,1}.
E.L. Elte listed in 1912 the then rediscovered semiregulars of Gosset, but he further allowed recursively for likewise semiregular kfaces of up to 2 types (shapes) each. This then already encompassed the main representatives of the E_{n} family, except for 1_{4,2}.
The symbols k_{m,n} (and lateron also k_{l,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 
}

= k_{m,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 
}

= 2_{3,1} =

o3o3o / 3 / (o)3o3o \ 3 \ o 
3 
Note that all CoxeterElteGosset 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  esp. k_{2,1}: 2_{k,1}: 1_{k,2}: 
* The thus marked ones happen to be flat euclidean tesselations and their embedding space dimensions thus is one less than the number of nodes.



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