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The generalized Schlafli symbol
Posted:
Jan 20, 2004 12:59 PM
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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 generalized-Sclafli-ness 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
zig-zags 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 \
Q---X---R
| \ | / |
Y---F' |
| |
T-------S
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 2-dimensional element F or F', so we call them 2-neighbors,
and in the symbol, join A and B in the 2-column.
The two flags QXF' and RXF' differ in their 0-dimensional elements
(Q and R), and so are 0-neighbors, while QXF' and QYF' are 1-neighbors,
differing in the 2-dimensional elements X and Y.
In general, we join two distinct flag-types in the d-column just
when they have representatives that are d-neighbors. For the
cuboctahedron, it happens that the 0-neighbor and 1-neighbor 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' |
| |
T-------S
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 0-neighbors b, b 1-neighbors c, and c 2-neighbors d,
accounting for all the vertical joins (the 1-neighboring relation being
represented by two vgertical lines in the 1-column).
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 d-dimensional element by putting your thumb over what I'm
now calling "the d-column".
This continues to work. For example, deleting the 2-column 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 1-third of its full symmetry.
Killing the 1-column 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 0-column 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 d-neighbors and e-neighbours.
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
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