Talk:Triangle group

More references?

It would be nice to have a link to the proof of the classification. The current reference doesn't say much.

Missing one

Isn't this page missing the finite spherical group (1,1,1)? That is - a group made of two triangles where the three edges are along the equator?

There's also the group being one big triangle on a sphere (.5, .5, .5) where the edges have zero length and all three corners are 360 degree reflex angles.

Pmurray bigpond.com 01:22, 26 October 2006 (UTC)

Reverted changes

I am seeking feedback on the formatting that has been reverted. The page in its current form is unreadable to me. --Yecril (talk) 10:34, 1 October 2008 (UTC)

What about the article, in particular, makes it unreadable? I don't think there has been any widespread discussion of the math and bigmath templates, but personally I don't see the need for them. I agree with Gandalf that articles should not be converted to a new style without some sort of wider agreement that the conversion is desirable. — Carl (CBM · talk) 04:58, 2 October 2008 (UTC)

Reference doesnt exist march 2009

http://cs.smu.ca/faculty/dawson/images4.html remove if it is not back soon —Preceding unsigned comment added by 131.111.243.37 (talk) 15:56, 5 March 2009 (UTC)

fixed. JackSchmidt (talk) 16:04, 5 March 2009 (UTC)

Recent changes

The classical case is that of a discrete reflection group with fundamental domain a spherical, Euclidean, or hyperbolic triangle. Under this assumption, the group is a Coxeter group, we have a well known classification by the angles with all spherical and Euclidean cases explicitly described, and so on. That's what article was about until a few days ago. What is the source for "triangle groups" when the angles are just rational multiples of $\pi$? Is this OR? If not, should it be in this article? Currently we have a weird mix of standard facts and statements that are unreferenced at best, possibly wrong, and not integrated with the main case in any event. I am very much inclined to revert to the last stable version. Arcfrk (talk) 07:17, 17 April 2010 (UTC)

Sorry about that – just wanted to highlight the connection with Schwarz triangles.

As you correctly indicate, “triangle group” is used only to refer to non-overlapping tilings (Möbius triangles); I’ve corrected this and made a non-intrusive mention of Schwarz triangles and overlapping tilings at the bottom.

…and I’ve also added references (the article had none)!

—Nils von Barth (nbarth) (talk) 22:04, 17 April 2010 (UTC)

Tilings and figures

I generally like the idea of illustrating articles, but some of the figures made me scratch my head. Specifically, I don't see what idea do the "Euclidean" images convey. The dysdiakis — tetrakis stuff seems more appropriate at the tilings article, or at least, in a section of its own, rather than in the middle of classification. There is also a big dissonance between three black and blue spherical pictures and a rainbow tinted polyhedral picture. Arcfrk (talk) 02:03, 3 June 2010 (UTC)

Hi Arcfrk,

Sorry if it’s not clear – these tilings are the reflection domains (image of a fundamental domain) of the triangle groups on the universal cover – i.e., these are the natural geometries on which the triangle groups act. Among other things, they show immediately why these triangles correspond to spherical/Euclidean/hyperbolic.

The Euclidean tilings, while busy, show much data – the icons on the vertices shows the order of the symmetries at the vertex.

Perhaps some wording could be added to clarify?

I think it’s useful to include them in the enumeration so novices can immediately see the connection, but the pictures could probably use some introduction, rather than: “See?!”

For the dihedral picture, presumably one (say, Tom Ruen?, who created the other pictures) could create a consistent picture using the same software – I assume it’s straightforward.

—Nils von Barth (nbarth) (talk) 03:58, 3 June 2010 (UTC)

No need to apologize — you didn't create these pictures (do you know how to do it, by the way?). Yes, I understand their intent, to a certain degree, but they don't really fulfill their purpose. Pictures are there to help digest the algebra. If a picture requires an explanation, I think it's better not to use it all. It's OK not to have a dihedral picture, but note my comments about appropriateness of discussing tilings here. Arcfrk (talk) 11:44, 3 June 2010 (UTC)

The pictures are created in KaleidoTile by Jeff Weeks, apparently using the “curved style” – I’ve not used the software, but I’ve deduced that from the image description page.

Regarding the use of geometry to understand triangle groups – I’m approaching this from the point of view of geometric group theory (which I noticed hadn’t been linked or mentioned – I’ll add this), which suggests that a way to understand a group is via geometries on which it acts.

In this case the connection is very natural, as the groups arose as abstraction of symmetry groups of tiling, just as with Coxeter groups more generally. While one can discuss Coxeter groups without using polyhedra or tilings (and they can be defined very briefly axiomatically), the motivation and interest is very much from the geometry, and many properties of the groups cannot be understood without coming to grips with the geometry (or certainly are most easily understood in that way).

For example, why are the spherical triangle groups finite (a purely algebraic property)? Because the universal geometry on which they act is a tiling of the sphere, which is compact. Conversely the growth of the infinite groups is quadratic (if Euclidean) or hyperbolic (if hyperbolic).

—Nils von Barth (nbarth) (talk) 19:43, 3 June 2010 (UTC)

Yes, I generated the images from KaleidoTile, any triangle group (p,q,r) generation. I made the alternating colored triangles to show the triangle domains. Unfortunately it couldn't make prismatic domains spherical (p 2 2). I substituted a different image I made myself for (3 2 2). Tom Ruen (talk) 23:16, 3 June 2010 (UTC)

Here's a comparable table used in Wythoff symbol: Tom Ruen (talk) 23:27, 3 June 2010 (UTC)

Dihedral spherical		Spherical
D2h	D3h	Td	Oh	Ih
*222	*322	*332	*432	*532
(2 2 2)	(3 2 2)	( 3 3 2)	(4 3 2)	(5 3 2)

Thanks Tom – you rock!

—Nils von Barth (nbarth) (talk) 23:50, 3 June 2010 (UTC)

Reference for "overlapping triangles" ?

There is a section called "Overlapping Triangles" which essentially explains that "There are also tilings by overlapping triangles, which correspond to Schwarz triangles whose angles are non-integer rational multiples of π."

But no reference is given, other to the article Schwarz triangle, which merely defines overlapping triangles tersely and also gives no reference.

Can anyone suggest references for overlapping Schwarz triangles? These might be good to include in the article. Many thanks.Daqu (talk) 07:16, 11 April 2011 (UTC)

Coxeter's (Regular Polytopes (book), table 3, p296) is the only reference I know, and it lists them, but he doesn't call them "overlapping triangles". He defines density on p94, as the order of overlap for a given triangle group. Tom Ruen (talk) 07:47, 11 April 2011 (UTC)

Many thanks, Tom! If it seems appropriate I'll try to put a reference superscript in the "Overlapping triangles" section to that Coxeter book.Daqu (talk) 00:49, 12 April 2011 (UTC)

But wait! Such a triangle is defined merely as having angles that are non-integer rational multiples of pi. A list of them would be very long.Daqu (talk) 00:51, 12 April 2011 (UTC)

The list of triangles, copied from Coxeter's book is here Schwarz_triangle#Schwarz_triangles_for_the_sphere_by_density. Tom Ruen (talk) 00:57, 12 April 2011 (UTC)

Yes, as soon as I saw your post I looked it up in my copy of Coxeter. But if overlapping is allowed, then can't we just start with any triangle with angles A·π, B·π, C·π, with A, B, C rational (necessarily such that A+B+C > 1)  ?

If not, then what is the definition of a Schwarz triangle in the overlapping case?Daqu (talk) 18:00, 12 April 2011 (UTC)

I'm not sure how the full list is derived, but all of the overlapping forms have a resulting system of one of the nonoverlapping Mobius triangles (3 3 2), (4 3 2), (5 3 2). So basically the rational divisors eventually repeat by overlaps as multiples of whole angles (2*π), but other angles won't ever repeat. Tom Ruen (talk) 20:26, 12 April 2011 (UTC)

I'm not sure if you're suggesting the same thing as this: Maybe the "overlapping" Schwarz triangles -- when repeatedly reflected about their sides -- are the ones that end up covering the sphere a precise integer > 1 number of times (away from vertices). If so, then any other spherical triangle with angles A·π, B·π, C·π, with A, B, C rational cannot. (In my previous comment I inadvertently omitted the rational condition, so I've now revised it for clarity.)Daqu (talk) 21:54, 12 April 2011 (UTC)

Probably? :) Tom Ruen (talk) 22:10, 12 April 2011 (UTC)

Definitely! Just found a very good reference in §6.8 of Coxeter (p. 112). Also, the 3-dimensional version of the problem is discussed in §14.8 (p. 280). These are too numerous for Coxeter to have classified in full, but I'd guess someone's done it with a computer by now. (Incidentally, some of the 3D cases he does mention have strange densities, like 191.)Daqu (talk) 20:49, 13 April 2011 (UTC)

Just a few at Goursat_tetrahedron. Coxeter used the triangle groups to enumerate the uniform star polyhedrons. Apparently the search wasn't so simple for star uniform polychorons. You can see some of the problems in the star polyhedra, as composite/nonsimple Wythoff symbols. In 4D there's more degeneracies that have to be accounted. BUT that's on the polytope side, but you might be right - no reason not to think all the rational tetrahedra have been enumerated by someone by exhaustive search. Tom Ruen (talk)

p.s. I added the nonlinear 3-sphere graphs from Coxeter at Goursat_tetrahedron#3-sphere_.28finite.29_solutions, Regular Polytopes, p 283, and put out a query of there's any easy wider lists to reference! Tom Ruen (talk) 02:59, 14 April 2011 (UTC)

"rotational" triangle groups?

I noticed the term (rotational) triangle group was used without being defined in the history and applications sections of the page. I have a vague sense that this means some index 2 subgroup(??) of the full triangle group (certainly the modular group is an index 2 subgroup of the (2,3,∞) triangle group), but I'm not confident enough to edit, nor have I had a chance to check out the sources. Akriasas (talk) 21:05, 28 August 2011 (UTC)

Lost

I'm lost, after reading the definition. If a and b are reflections on resp. the sides BC and AC, application of b results in the image bB of B. Then abB is a point far outside the triangle and repeating ab gives ababB still further away. Any further image, in my opinion, never will be the original B. What is it I don't understand. Madyno (talk) 22:15, 23 November 2021 (UTC)