Talk:Schläfli symbol

message: 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)
thread_title:  TEST CHART for Extended Schläfli symbols for polychora 

message: 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)
replies: text: 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)
thread_title:  Uniform polytope move 

message: The Schläfli symbol was independently discovered by Hope and by Gosset.  Schläfli and Hope wrote it as eg (5,3,3), regarding it as the outcome of (5,3) + ,3  [ie 3 dodecahedra at an edge].  Gosset writes |5|3|3|, regarding it as the application of |5 to |3|3|. [Coxeter, 1973, Regular Polytopes, p144].  D M Y Sommerville (1929) makes passing references late in the book "the geometry of N dimensions".  Coxeter may have also discovered it himself as well.
The curtail (ie {3 [over] 4} for Cuboctahedron), is due to Coxeter, comes from a back-formation of using the Coxeter-Dynkin graphs in lining methods.  It is not apparent from the manner that Schläfli et al did this.  
The use of the t_ operator is a direct representation of Stott's e_ operator, except that e_0 and t_0 behave differently. (e_0 is assumed unmark the marked, while t_0 is assumed to marked the unmarked).  Stott's operators were applied to the base regular figure, which is named under a different scheme.
Coxeter's names for marking the branches of say 2_21 is very complex (see "Wythoff Construction" in Twelve Essays (1968)), because the alignment of the graph is difficult.
It should be noted that the Schläfli symobl is a kind of continued fraction, from which one can calculate a value of both a figure and the vertex-figure, and thence the diameter of the represented figure.  Neither Coxeter's curtail, nor the t_ notation can be used in this way.
--Wendy.krieger (talk) 13:27, 25 December 2008 (UTC)
thread_title:  Schläfli Symbol 

message: Nice article - a lot of useful information presented very clearly. But a bit repetitive in my view, with the Description section trying to do too much and therefore a bit jerky to read.
Some suggestions, therefore. 
First, we could define the Schlafli formally for the general case at the start, i.e. (as you did)
:The Schläfli symbol is defined recursively as {p1, p2, ..., pn − 1} if the facets have Schläfli symbol {p1,p2, ..., pn − 2} and the vertex figures have Schläfli symbol {p2,p3, ..., pn − 1}. 
Then, you could go on to give examples for 2, 3, 4 dimensions. I think that would improve the structure and coherency of the article, and make it less repetitive. But maybe you feel that doing this would make it harder for novices to catch on?
Also, I think less mainstream topics such as symmetry groups and star pols should be at the end.
Also, this and many, many polytope articles should mention the differences for abstract polytopes - but that is a major undertaking requiring much time (and knowledege!!!).
Good stuff - hope my comments are helpful. SteveWoolf (talk) 09:29, 4 July 2010 (UTC)
thread_title:  Suggestions 

message: I have some question. Is it necessary that the symbol must not have space after comma? I should know because Thai Wikipedia currently uses a bot for cleaning up commas. --Octra Bond (talk) 14:13, 17 January 2011 (UTC)
replies: text: No, that's mainly for compactness or to prevent the symbol from being wrapped, but same effect can be done with nbsp {3, 3, 3, 3}, just ugly to edit. Tom Ruen (talk) 20:40, 17 January 2011 (UTC)
replies: text:  Easier with the nowrap template: }. —Tamfang (talk) 00:47, 24 June 2011 (UTC)
thread_title:  Space by comma 

message: Facet (geometry) describes only polyhedra and is mostly about gem stones. I have updated the link to Facet (mathematics) but I'm not sure how clear it is that the second definition given on that page is the intended one. 
I have also removed the incorrect statement that "There are z facets around each vertex." I am not sure how this can be accurately re-stated. I do think the attempt further up "A regular 4-polytope with r {p,q} regular polyhedral cells around each edge is represented by {p,q,r}, and so on." could be clearer - but I see the problem!  — Preceding unsigned comment added by Tesspub (talk • contribs) 08:15, 16 July 2011 (UTC)
replies: text: "around each ridge"? —Tamfang (talk) 03:12, 24 July 2011 (UTC)
thread_title:  Facets 

message: Text currently reads:
 The symbol { } means a digon or line segment.
I think this is wrong.  A digon is the (degenerate) 2D object {2}, while the line segment is a (non-degenerate) 1D object, the unique regular 1-polytope.  A digon is bounded by two segments, which coincide if Euclidean.  Conflating digon with segment is analogous to saying that {4} is not only a square but also a square dihedron.
I'm less certain about whether the other uses of digon in that section are correct, so I'll wait to see what other people think.  If no objections, I'll make just the one change in a few days.  Joule36e5 (talk) 07:36, 15 June 2015 (UTC)
replies: text: I made the change.  Now I'm also wondering if it's worth covering the trivial 1-polytope {} in the main text, and using this, instead of polygons, to bottom out the recursive definition.  The polygon {p} would be the polytope with line segments {} as facets, p of them surrounding each peak, but the peak is the empty set (dimension -1) at this level.  Joule36e5 (talk) 00:46, 23 June 2015 (UTC)

text: I agree { } is a line segment, a 1-polytope with 2 vertices, and {2} is a digon, a 2-polytope with 2 vertices, 2 edges. Tom Ruen (talk) 00:52, 23 June 2015 (UTC)
thread_title:  The empty symbol 

message: https://en.wikipedia.org/w/index.php?title=Schl%C3%A4fli_symbol&diff=next&oldid=695690717
(A tesseract has 3 cubes around every edge, as stated . Undid revision 695690717 by Bohocmasni (talk))
Hey, that's not true!
It has 4 cubes. See the pictures, or calculate it. 
It has 8 3-cube hyperfaces zusammen, and, every 3-cube includes 8 vertices. This is 8*8 = 64 different (cube,vertex) pair that the cube includes the vertex.
64 can be calculated an other way: it has 16 vertices, and X cubes around. This is X*16 = 64 (cube,vertex) pair. So, X equals 4.
This definition is simply wrong. I think somebody just tried to generalise the definition can be found here: http://mathworld.wolfram.com/SchlaefliSymbol.html
For the good recursive definition, see the 'Higher dimensions'.
edit: minor changes
Bohocmasni (talk) 16:21, 18 December 2015 (UTC)
replies: text: A tesseract, {4,3,3}, has 4 cubes, {4,3}, around a vertex (in a {3,3} tetrahedron vertex figure) but around an edge, its three cubes. The mathworld article only describes the polyhedron case {p,q}. Four cubes around an edge makes for a cubic honeycomb in 3-dimensions. Tom Ruen (talk) 16:22, 18 December 2015 (UTC)
replies: text:  Yeah, in my mind 'edge' was interpreted as vertex. In the general recursive definition, does 'edge' means n-2 face? Bohocmasni (talk) 16:47, 18 December 2015 (UTC)
replies: text:  Close. Ridge (geometry) means n-2-face, and facet means n-1-face. Tom Ruen (talk) 17:12, 18 December 2015 (UTC)

text:  So what does 'and so on' means in the definition exactly? I don't feel its generalisation in higher dimensions self-evident. Bohocmasni (talk) 17:43, 18 December 2015 (UTC)

text: See  "The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes." Tom Ruen (talk) 16:27, 18 December 2015 (UTC)
replies: text:  Ok. I see. Thank you. Bohocmasni (talk) 16:32, 18 December 2015 (UTC)
thread_title:  Wrong definition 

message: I have edited what I believe to have been a mistake in the statements of the general case of the Schläfli symbol for an n-dim regular convex polytope.
As correctly stated, {p,q} is a polyhedron (3-polytope) with {q} faces (facet) around each vertex (0-face). {p,q,r} is a 4-polytope with r {p,q} cells (facet) around each edge (1-face).
The general case is that a {p,q,...,y,z} has z {p,q,...,y} facets around each peak ((n-3)-face), not each ridge as previously stated.
--Davyker (talk) 23:10, 19 May 2016 (UTC)
thread_title:  Edit of the general case for an n-dim RCP 

message: The explanation of what "snub" is isn't enough. It should have pictures to show what it means.Rixn99 (talk) 08:46, 12 December 2018 (UTC)
thread_title:  Snub 

message: ??? A Loaf Of Bread has returned (talk) 11:59, 21 August 2022 (UTC)
thread_title:  is there some sort of app that allows you to play with these? 

message: Axial polytopes containing vertices on parallel offset hyperplanes can be represented by the || operator. A uniform prism is {n}||{n} and antiprism {n}||r{n}.
how can we use this for polychoral prisms, antiprism, cupolae, rotundas and bicupolae?
tetrahedral cupola: {3,3}||rr{3,3}
cubic: {4,3}||rr{4,3}  and so on...
but what about other like 5-cell cupola or rotunda? 185.160.226.33 (talk) 13:52, 7 October 2022 (UTC)
thread_title:  Schläfli ( || )