Talk:Schläfli symbol

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[edit] Expanded Schläfli notation?

The explanation on this page needs to be expanded with a description of Schläfli symbols for non-regular polytopes, which include things like t{3,3} for truncation, etc.. Sometimes subscripts are used, like t0,1{3,3,4}, which need to be explained as well. I don't have a formal definition handy, though.

Agreed, some sort of "Expanded Schläfli notation" section should be added which lists these, both what the notations mean AND references to the origins!
  • [1] Mathworld only shows only simple forms.
  • [2] Posted answer by John Conway with the same question! (Conway, 2003, says Coxeter invented "the" notation, although no examples given)
  • [3] interestingly describes (5.5.5) as a Schläfli Symbol, rather than {5,3}. Also "For 3D crystalline nets, Mike O'Keeffe has defined an Extended Schläfli Symbol." (Something very different apparently!)
  • George Olshevsky lists the notation on his website, but doesn't define in his "glossary" [4]
Tom Ruen 23:48, 11 February 2006 (UTC)

If you use George Olshevsky's examples we have: Polyhedra:

  • t{p,q} - truncated {p,q}
  • r{p,q} - runcinated {p,q}
  • rr{p,q} - runcinated runcinated?
  • tr{p,q} - truncated runcinated?
  • sr{p,q} - snub runcinated?

Polychora versions are all beyond me.

For hyperbolic tilings, Don Hatch] doesn't use any shorthand, but named uniform tessellations (on S2, E2, H2) with terms: truncated, omnitruncated, runcinated, bitruncated, snub.

Similarly a bunch of the [[Andreini_tessellation] have named variations of the cubic honeycomb: Bitruncated, Omnitruncated, Cantitruncated, Truncated, Cantellated, Runcitruncated, Rectified. I assume all could be given short hands prefixed to Schläfli {4,3,4}.

Anyway, having definitions for all these pretty terms would seem to be a first goal. I'm just writing here also, rather than starting anything half-wrong. Tom Ruen 00:04, 12 February 2006 (UTC)

Well, Tamfang has already done definitions for most of these terms in uniform polychoron, although I'm still not sure how to map these to the extended Schläfli symbol.—Tetracube 06:54, 12 February 2006 (UTC)
I forgot about those definitions - very good start. I guess I agree it is useless to use notations we can't define. I'm content if you want to remove them, or I can myself for those I copied from George Olshevsky's website. Tom Ruen 19:18, 12 February 2006 (UTC)
Oh no, don't delete them. I think it's more worthwhile to find out what they mean, than to exclude them just because we don't know precisely what they mean. Maybe Tamfang, who apparently has email contact with Norman Johnson and George Olshevsky, can ask for clarification from them.—Tetracube 02:50, 13 February 2006 (UTC)

[edit] TEST CHART for Extended Schläfli symbols for polychora

I scanned through, George Olshevsky's truncation notations atexamples and made a summary chart and test table at: User:Tomruen/uniform polychoron

Basically there's 15 truncation forms, one for parent, one for dual, three in the middle (bitruncated, runcinated, omnitruncated), and 5 others on each side of the parent/duals: truncated, rectified, runcitruncated, cantellated, cantitruncated.

So the notation t...{p,q,r} is a short notation to long name. Like t0,1{3,3,3} is the truncated 5-cell because 0,1 is associated with truncation. Well, at least this is now defined, even if system is still mysterious.

Another test table for polyhedra extended names at: User:Tomruen/Uniform polyhedron

Tom Ruen 23:20, 1 March 2006 (UTC)

[edit] Extended Schläfli symbols

I decided to put up the extended notations as best I determined them for 3d and 4d, even if they are not defined here. I figured it was a start. Tom Ruen 09:53, 4 March 2006 (UTC)

I expanded tables and images on extended Schläfli symbols for 3d and 4d. I'm still not satified with the overall content quality, at least I think all the vital components are represented.

I did question whether this article or uniform polyhedron and uniform polychoron are best for the tables or diagrams. I settled on here because the sections interconnect, with symbols for cells of polychora coming from the polyhedra.

Another option I'm leaning towards is moving extended content to a new article Extended Schläfli symbol perhaps. Well, unfortunately even that phrase isn't perhaps under general terminology enough to justify. So anyway, I think best to keep here for a while!

Anyway, I'm glad if anyone else wants to replace/improve any of my efforts here! Tom Ruen 03:59, 10 March 2006 (UTC)

[edit] Wythoff

Much of this page is taken up by the Wythoff construction. Did Schläfli have anything to do with developing that? Perhaps it should be moved elsewhere, leaving a brief discussion (with links) of the extended S. notation. —Tamfang 04:37, 20 March 2006 (UTC)

I don't have a firm opinion on where things should go or knowledge on historical development. I've only glanced at Wythoff construction page, and it is certainly a possible place to move material, although at present it is focused on polyhedra construction. Anyway, I agree value in question, but don't have a strong reason for doing anything at present. Tom Ruen 04:45, 20 March 2006 (UTC)
Been thinking more of this recently, and accept much content should be moved to Wythoff construction. No time table for me to jump at it, and that article itself needs expanding to higher dimensions as well. Tom Ruen 05:11, 3 August 2006 (UTC)

[edit] sources

I added the reference template, although I'm responsible for much of the content for the "extended" notation. It may be best at some point to split this article with a standard terminology for extended forms. For now, I figured good to at least feel a little guilty here! Tom Ruen 01:11, 29 July 2006 (UTC)

[edit] Uniform polytope move

I moved most of the extended text to uniform polytope, where it best belongs. I reduced the tables to just show the extended notations.

This whole article could use a top-down rewrite perhaps, but at least for now it contains useful information. Tom Ruen 05:50, 16 September 2006 (UTC)

Okay, I finished cleaning up this article, reducing examples, referencing other articles which do a better job, and adding a unified new section for prismatic forms. WELL, it is better I hope, at least as a sort of example overview of the uses for the Schläfli symbol and related notations. I've also read the vertex configuration notations can be considered Schläfli symbols, but didn't reference that here. I need to consult some more sources! Tom Ruen 05:30, 5 October 2006 (UTC)
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