links
esp.
k2,1:
2k,1:
1k,2:
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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 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.
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{
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3,...,3 | 3,...,3,...,3 |
}
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= km,n
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| 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.
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{
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3,3 | 3,3,3 |
}
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= 23,1 =
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o---3---o---3---o
/
3
/
(o)--3---o---3---o
\
3
\
o
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| 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. And, for sure, symmetries with multiple bifurcation points within their diagram or having loops are missing here by definition.
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External links |
esp.
k2,1:
2k,1:
1k,2:
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