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Topic: tiling notation
Replies: 10   Last Post: Jan 19, 2004 10:47 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Coxeter's books...
Posted: Dec 31, 2003 8:56 PM
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On 27 Dec 2003, Paul McCarthy wrote:

> I don't know of any sources on the Internet 'freely' available that
> discuss the extended Schlafli symbol. There may be papers which you
> can get over the Internet if you, your school, or business subscribes
> to one of the Internet technical, academic papers services. Usually
> it is by paid subscription. In fact, I wouldn't be surprised if John
> Conway has written some papers describing the extended Schlafli
> symbol.


[John Conway speaking] No. He hasn't yet written anything, and since
he originated the new form of the symbol, you won't be able to
find it in any of...

> The only sources I know of are three of Coxeter's books, particularly
> "Regular Polytopes". Coxeter also discusses the extended Schlafli
> symbol in "Introduction to Geometry" and "Regular Complex Polytopes".


The new symbol is based on a prior approximation by Andreas Dress,
who however called his version a "Delaney symbol", because he based it on
some idea(s?) he got from reading a paper of Delaney's. Olaf DelGado and
Daniel Huson have written a few things about thisDress-Delaney symbol.

However, I don't really recommend anyone to read about that version,
because the new one is so visual and so much easier to understand.
If I weren't starting early tomorrow morning on a 17-day trip to
New Zealand, I'd write a quick description here and now, but in the
circumstances you'll have to wait for that.

> Also... only in my personal opinion,... I don't consider Coxeter to be
> one of the easiest people to read.


And I've always regarded him as a really great mathematical writer!
The difference in our perceptions is probably due to the fact that
the things he writes about tend to have longer absorbtion-times than
the topics discussed by other writers to the same type of audience.

> If someone's going to explain the extended Schlafli symbol here, I'd
> rather someone else do that.


I've just remembered that in fact I did write a brief description
which I think went to the polyhedron list, so you might be able to
find it fairly quickly.

> If memory serves me, the extended Schlafli symbol is kinda like
> taking a fold-out of a n+1 dimensional polytope.


I gather you're talking only of the GSS (by the way, it's in no
way an EXTENDED SS, but rather a GENERALIZED one) of a tessellation
of n-dimensional space. But this space can be Euc, Hyp or Sph, so
all polytopes have GSS's. What it really is, is best described in
terms of a "fold-up" rather than a "fold-out" - in fact it's just
a coded form of the orbifold of the polyhedron or tessellation.

> I can try to explain the extended Schlafli symbol more here, but I
> suspect you're likely going to want to consult Coxeter anyway, as I
> **believe** he is the originator of the extended Schlafli symbol.


As you see, that belief is wrong, and so probably the "ESS" you'd
attempt to explain would be one that differs greatly from my GSS
(whose existence, properties and simplicity totally surprised Coxeter
when I told him about it at his 95th birthday meeting in Februrary 2002,
by the way).

Regards, John Conway

PS. If you haven't been able to locate the brief description I
mentioned (and maybe even if you have), write to me again (but not
too soon, because then it will get sunk under all the messages I'll
not be answering while I'm on my way to NZ - say after 10 days).

JHC





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