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The generalized Schlafli symbol
Posted:
Jan 20, 2004 12:59 PM


On 19 Jan 2004, John Berglund wrote:
> I still haven't found anything about the extended Schlafli symbol. > More help?
It occurred to me after writing my previous message that perhaps your phrase "the extended Schlafli symbol" was intended to refer to something other than my "Generalized Schlafli Symbol" (GSS). A reasonable referent would be Coxeter's use of ringed Schlafli symbols for the polytopes obtained by Wythof's construction from groups generated by reflections, in which case, of course, reference to his books would be quite appropriate.
However, from now on I'll presume you want the GSS. First of all, let me get its origin straight. The idea of the GSS is due to Andreas Dress, who, however, called his version "the Delaney symbol", since it was suggested to him by his reading some work of Delaney's.
I got my version of the symbol by an easy simplification of Dress's, which simplification makes lots of things more clear, including the fact that it generalizes the classical Schlafli symbol. I therefore call it
"The Generalized Schlafli Symbol of Delaney and Dress"
on formal occasions because
1) I regard its generalizedSclafliness as its most important property 2) I don't want to deprive Delaney and Dress of their credit 3) I don't claim any part of it except making that slight simplification
but on less formal occasions I abbreviate this to "generalized Schlafli symbol" or even just "Schlafli symbol".

Now as regards the description I'm faced with the problem of choosing between giving a short but rather unilluminating definition and giving you the wonderful properties that make the symbol interesting. Let me try to do both, starting with some of the latter.
*** INSERTED LATER : it seems after writing this letter that the definitions in terms of flags are distinctly indigestable, so I suggest you bypass them at a first reading, which should concentrate on the examples and their properties. JHC ***
Here are some examples of the symbols:
*p* *q*
is what the standard Schlafli symbol {p,q} for a regular polyhedron becomes. Those four stars don't really need to be drawn  they just help me to describe various points of the symbol.
The symbol for the cuboctahedron is
*3* ** 4  *4* **
and now I'll start to describe the structure and properties of the general symbol. It has a number (here 2) of "rows" and another number (here 3) of "columns" that correspond to the dimensions (here 0,1,2) of elements of the object. The intersections of rows and columns are marked by stars, except that the stars in intermediate dimensions are duplicated. The stars are connected by zigzags of lines that are marked with integers that in general tell you how many things surround a given one in some sense.
I'll label the columns by numbers 0,1,2,... and the rows by letters A,B,C,... thus:
0 1 2 A: *3* ** 4  B: *4* ** .
What these letters correspond to is orbits of flags under the symmetry group, a flag being a collection of mutually incident elements of each dimension. One can alternatively use what I call "flagstones", which are the simplices of the barycentric subdivision. Let me draw a bit of the cuboctahedron: P / \ / \ / F \ QXR  \  /  YF'    TS
In this there are two types A and B of flagstone, typified by QXF and QXF', and distinguished by whether they're parts of triangles (type A) or squares (type B).
[ The corresponding flags are Q, QR, QRP and Q, QR, QRST. ]
The two particular flagstones mentioned are neighbors, meaning that they differ in only one of their vertices, which in this case corresponds to the 2dimensional element F or F', so we call them 2neighbors, and in the symbol, join A and B in the 2column.
The two flags QXF' and RXF' differ in their 0dimensional elements (Q and R), and so are 0neighbors, while QXF' and QYF' are 1neighbors, differing in the 2dimensional elements X and Y.
In general, we join two distinct flagtypes in the dcolumn just when they have representatives that are dneighbors. For the cuboctahedron, it happens that the 0neighbor and 1neighbor of any flag has always the same type as that flag, so there are no more vertical lines in that case.
Let me now draw two representative faces of a square pyramid:
P /\ Wa \ /b F \ Q__cX___R  \d   F'    TS
in which I've labelled representative flagstones
a = PWF, b = QWF, c = QXF, d = QXF'
to help in reading its symbol:
0 1 2 a ** *4*  b *3* **   c ** *3*  d *4* **
Here the neighborhood relations between distinct types of flagstone are that a 0neighbors b, b 1neighbors c, and c 2neighbors d, accounting for all the vertical joins (the 1neighboring relation being represented by two vgertical lines in the 1column).
I've now told you everything except what the little numbers are. Let me now go back to telling you about properties. You should know the "Thumb Rule" for the ordinary SS, namely that you get the types of ddimensional element by putting your thumb over what I'm now calling "the dcolumn".
This continues to work. For example, deleting the 2column of the cuboctahedron symbol gives
*3* *4*
showing that that polyhedron has two types of face, namely a triangle and a square (and in fact that both are embedded regularly). Doing likewise for the square pyramid gives
**  *3*  **
*4*
showing once again that there are just two types of face, again a triangle and a square, but this time only the square is embedded regularly, while the triangle is embedded with only 1third of its full symmetry.
Killing the 1column gives the types of edge. For the cuboctahedron it gives
*3++*  *4++*
showing that there is only one type of edge (and that this lies between a triangle and a square). For the square pyramid, it gives
*++*  3 *++*
*3++*  *4++*
showing that there are two types of edge, one between two triangles, and one between a triangle and a square.
Finally, killing the 0column tells us that the cuboctahedron has only one type of vertex (since its diagram, remains connected), while the square pyramid has two (since it's breaks into two).
Now about the little numbers that correspond to the zigzag lines: any such zigzag goes between two adjacent dimensions, say d and e = d+1. To get the little number, start with any flagstone say F1 of an appropriate type, and take alternately dneighbors and eneighbours. In this way one obtains a closed circle of some even length 2n, say:
F1 <d> F2 <e> F3 <d> F4 <e ... d> F2n <e> F1
and then the little number is n.
I think I've said enough for now. Regards, John Conway



