Talk:Schläfli symbol

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 t_0,1{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, George Olshevsky.

Tom Ruen 23:48, 11 February 2006 (UTC)

If you use George Olshevsky's uniform polyhedra, George Olshevsky. we have: Polyhedra:

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

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 tessellations 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)

I no longer have contact with either —Tamfang (talk) 22:55, 8 July 2023 (UTC)

Don Hatch does indeed have an 'extended' schläfli symbol for his figures, but it is not very general. Its novelty is that it goes in a 'hatch-loop', marking nodes. I translate it fairly literally in the PG. It's a kind of dynkin-symbol for mobius groups.

John Conway et al, seemed to have cracked the code for writing the symbols for all uniform polyhedra (in S,E, H). This is in one of the forthcomming books by Breugel, Conway, and Goodman-Strauss. However, i don't think this is the one he showed Coxeter on his 95th birthday, because my recollections is that Coxeter was then dead.

In any case, the article gives 'orbifolds', which is a Conway-Thurston notation for groups. I suggested the notation was worth decorating (in order to show polytopes), and gave some examples: eg 3 * 2 = pyritohedral group, and "3 / * / 2 %" is the icosahedron in that symmetry. However they had a different method, in the form "3 * 2 <1,3> (2)". The particular figure has no wythoff-symmetry because 3*2 is non-wythoffian.

I am also rather confused with his namings, as well. Both 3*2 and <1,3>(2) were called 'orbifolds', even though the second has no form with the first. It can occur in any group p * q. I gloss the former as orbifold and the latter as archifold, and he may have also adopted these. So he may have called the thing as a kind of Extended Schläfli Symbol, even though it has little to do with that symbol. --Wendy.krieger 07:38, 26 September 2007 (UTC)

None the same, i think there is very little to go on from the second reference, except that Conway is probably playing with some kind of decoration to the orbifolds.--Wendy.krieger 07:38, 26 September 2007 (UTC)

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 t_0,1{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)

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)

The notation t_0,1{3,5} is a composite of the modified Stott-notation and the Schläfli notation. Coxeter used it in 1935 paper 'Wythoff's construction of uniform polytopes'. We should now eplain what it means.

t_0,1 corresponds to Stott's e₀e₁ operator, although she usually suppoesd e₀ was always present. So e₁.Cube is an edge-expanded cube. Schläfli's notation gives meaning to the numerals for the subscripts.

The particular notation gets to describe what is needed for the assorted papers it appears in, without too much explanation. However, it is not ready for heavy general applications, which is why those who went further, such as Bowers and myself, found need to devise a more GP notation. Bower's notations is rather limited to uniform polytopes, whereas mine handles a much richer field of enquiry. Wendy.krieger 08:00, 22 September 2007 (UTC)

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)

Schläfli was the first of many to independently discover that regular polytopes can be designated by a series of othermargin-index (eg 4,3 = squares, 3 at a vertex, 4,3,3 cubes (4,3), 3 at an edge. Thorald Gosset devised it in the recriprocal form, ie cube = squares on |3|, {3,4,3} = |3 applied to |4|3|, and so forth. Coxeter also discovered them independently. The closeness between the Schläfli symbol to the Coxeter-Dynkin graph is quite an annoyance, because people tend to think they're somehow related.

Wythoff's main contribution to the scheme is an independent thread, to show that one can take Mrs Stott's figures, and construct these as motifs of the mirror-group [5,3,3], although neither he nor Mrs Stott had this notation.

Coxeter, de Witt and Dynkin each derived the "Coxeter-Dynkin" symbol independently, to represent a class of Lie groups. Coxeter understood also that the Lie groups directly related to the mirror-groups, but needed the push from reading Wythoff's paper to make the connection of marked nodes. Wendy.krieger 07:25, 22 September 2007 (UTC)

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)

Coxeter largely extended the Schläfli notation, beginning right back in his 1935 paper "Wythoff's Construction for Uniform polytopes", reprinted in "Twelve Essays" [Dover, 1999]. The actual notation is a composite of the Stott-notation and the Schläfli Notation, slightly modified.

The unmodified symbol eg {3,3} corresponds to a mark in the first position. This is true for Schläfli's notation (which handles only the regular figures), and for Stott's notation, (where operators correspond to inserting nodes to a graph with the first node already marked. The modification by Coxeter, is that when the t- operator is used, the first node is assumed to be unmarked.

The use of "double-decker" symbols is Coxeter's attempt to render the branching groups like 2_21, into something that looks like a Schläfli symbol. But the magic of the formula is rather strained by the process.

The use of the letters like t-, r, etc, is probably due to NW Johnson. Note that Coxeter never got as far as running the notation into lining text, as eg x3x5o3o. When polytopes became prime-time, this became more of an issue: We see the Bowers diagrams (which render the thing into small symbols.

The notation on the article page is pure Coxeter-style modifications. Wendy.krieger 07:37, 22 September 2007 (UTC)

Hi Wendy. Thanks for helping with more of the history of usage. Mostly I'm content to abandon usage of the vertical Schläfli symbols invented by Coxeter, or mixing r/t prefixes (by Johnson). The t-subscript-Schläfli notation is useful for uniform polytopes with linear Coxeter-Dynkin diagrams, less clear when used with branching/cyclic CD graphs.

Perhaps this article should be reduced exclusively to expressing what Schläfli used for regular polytopes? But what should the t-subscript notation be called? Perhaps Coxeter-Schläfli symbol? I wonder what Norman would suggest? Maybe it's sufficient to just call these "Extended Schläfli symbols", as done, if there's no printed sources to reference a name? Tom Ruen 00:21, 27 September 2007 (UTC)

OH, it would be nice to include a table column of comparisons to a Stott notation, especially if given back to a specific publication. I don't have any books from her or giving information about her notations.

I also have a copy of Elte's 1912 The semiregular polytopes of the hyperspaces book, not easy to read all of his notation, although some like tC₁₂₀ is clear. Tom Ruen 00:32, 27 September 2007 (UTC)

I'm not having read Stott directly. The essential arguments come to me from Conway, and Coxeter fills out the flesh. None the same, the notation replaces t_a,b,c by e_ae_be_c, with some differneces for subscript 0.

I am not really sure what to call the particular notation. It's a horrible one, which i abandoned as soon as i saw it. The thing has very little to do with Schläfli at all! Still, this is the way the names fall :( --Wendy.krieger 08:59, 27 September 2007 (UTC)

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)

Schläfli Symbol

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)

Suggestions

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, ..., p_n − 1} if the facets have Schläfli symbol {p₁,p₂, ..., p_n − 2} and the vertex figures have Schläfli symbol {p₂,p₃, ..., p_n − 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 knowledge!!!).

Good stuff - hope my comments are helpful. SteveWoolf (talk) 09:29, 4 July 2010 (UTC)

---

I would greatly appreciate a section on Schläfli Symbols for antiprisms and crossed antiprisms like the shapes below:

https://upload.wikimedia.org/wikipedia/commons/4/41/Pentagonal_Antiprism_%28Dodecahedron%29.svg

https://en.wikipedia.org/wiki/Pentagrammic_crossed-antiprism#/media/File:Pentagrammic_crossed_antiprism.png

https://upload.wikimedia.org/wikipedia/commons/4/40/Antiprism_12-7.png

as I am doing research on them and am not sure how to notate them. — Preceding unsigned comment added by 70.36.193.221 (talk) 06:06, 25 July 2024 (UTC)

in other words  ? Antiprism suggests s{2,2n}. —Tamfang (talk) 22:14, 26 July 2024 (UTC)

Fractional entries

There's no explanation at all for where, why and how you use Schläfli symbols with fractional entries, like {5/2,3}. All it says it that it is used for some stellated polyhedra. All the information on the page on stellation about dividing by the number of steps and stuff should be included here and explained properly. —Preceding unsigned comment added by 120.145.12.233 (talk) 08:53, 14 December 2010 (UTC)

fractions refer to the polygrams. A symbol like {p/d} refers to what happens when the edges connect the vertices of {p} taken in steps of {d}. For example, {12/5} would be drawn on a clockface, connecting each hour to one five later (ie 12 - 5 - 10 - 3 etc). The {5/2} is the pentagram, and the {6/2} is the hexagram or 'star of david'. These might be applied like regular polygons, giving rise to things like {5/2, 5}, the result of stellating the pentagons of the dodecahedron, to create peaks on each of the faces. Wendy.krieger (talk) 09:19, 23 June 2011 (UTC)

This section needs expansion. While forms such as {5/2} and {6/2} are well explained, the general case {p/s} is not defined for all rational 0 ≤ s ≤ p. Some questions:

• {5/2} and {5/3} should produce exactly the same plane figure, but drawn in reverse order of intermediate vertices. Does this effect have a use, for example in theory or applications when the polygon is considered to have directed edges?
• If so, how does drawing direction, which changes as s crosses the threshold value p/2, affect the sign of the (variously defined) area of the star polygon {n/s}?
• Does drawing in the opposite direction produce the opposite face of the polygon (considered as an object with two "sides", that is, surfaces, rather than just a plane figure)?
• More generally, does the theory of polytopes also cover signed areas and opposite surfaces, and if so, how does the Schläfli symbol reflect this?
• Are {5/5} and {5/0} defined and both equal to a single point?
• Does {6/3} produce an asterisk, and is the class of asterisks {2n/n} considered to be a class of star polygons of zero area?

yoyo (talk) 15:25, 23 June 2016 (UTC)

I'm not aware of directionality ever being relevant or meaningful, but perhaps someone more knowledgeable can chime in on those first four points. {5/5} or {5/0} would logically be not a single point, but five copies of the monogon {1}, each of which is a single point connected to itself by an edge. {6/3} would indeed be an asterisk, consisting of three copies of the digon {2}, with area zero. Joule36e5 (talk) 10:17, 7 July 2016 (UTC)

One standard is to remove to the common factor, so {6/3} becomes a compound 3{2/1} with equally spaced orientations. You can see some at List_of_regular_polytopes_and_compounds#Two_dimensional_compounds. Another interpretation for star polygons is multiwound, overlapping edges, so {6/3} look like one edge overwritten 6 times! Tom Ruen (talk) 16:19, 7 July 2016 (UTC)

The relation t{p/q} = {2p/q} only makes sense if the second interpretation is followed, which is why I am convinced that Grünbaum is right and it's the correct one. So {6/0} = {6/6} are degenerate points, {6/1} is a hexagon, {6/5} is a retrograde hexagon, {6/2} is a double triangle (not a Star of David), and {6/3} is a triple edge. The directionality can matter: see for example the small rhombihexahedron, which has a vertex configuration typically written 4.8.4/3.8/7: the first square and octagon proceed forward around the vertex, the second two go backward. Double sharp (talk) 11:47, 28 March 2020 (UTC)

Space by comma

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)

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)

Easier with the nowrap template: {3, 3, 3, 3}. —Tamfang (talk) 00:47, 24 June 2011 (UTC)

Facets

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)

"around each ridge"? —Tamfang (talk) 03:12, 24 July 2011 (UTC)

The empty symbol

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)

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)

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)

Wrong definition

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)

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)

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)

Close. Ridge (geometry) means n-2-face, and facet means n-1-face. Tom Ruen (talk) 17:12, 18 December 2015 (UTC)

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)

See [4] "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)

Ok. I see. Thank you. Bohocmasni (talk) 16:32, 18 December 2015 (UTC)

Edit of the general case for an n-dim RCP

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)

Snub

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)

is there some sort of app that allows you to play with these?

??? A Loaf Of Bread has returned (talk) 11:59, 21 August 2022 (UTC)

Schläfli ( || )

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)