Talk:Orbifold notation

Credit
I'm Heidi Burgiel, one of Conway's co-authors on a book that discusses this topic in detail (to appear).

As I understand the history, Conway did invent an orbifold notation but discarded it after learning the notation described here. The notation that he has so dilligently publicised was invented by William Thurston. Conway credits the first use of such a notation to Murray MacBeath.

While many may know this as "Conway's Orbifold Notation", Conway is not solely responsible for it.


 * Hi Heidi. I didn't add this article myself, but glad for your information. I first learned about it in a "Geometry and the Imagination" class in 1991.
 * I just added a link to the class notes. (Looks like the notes reference "Conway's names", crediting him for naming the forms more than the "star-cap" notation.)
 * Feel free to edit the article to include better credits. Otherwise thanks for the notes here, and maybe someone else will want to expand this as well. Tom Ruen 23:54, 10 July 2006 (UTC)

Coxeter notation for 3,3,3:
Coxeter used a Delta sign for this, but i don't know how to make it, eg Δ or &Delta;. So the orbifold *333 is Δ and 333 is Δ+ Wendy.krieger (talk) 00:55, 18 January 2009 (UTC)

Understandability
This page needs to be better oriented towards the lay-people, however mathematically inclined. As it is, it seems fairly unreadable to the non-experts. Zoltan 20:24, 2 August 2009 (UTC) —Preceding unsigned comment added by Lzkelley (talk • contribs)

Inconsistent notation
What's the correct symbol for "miracle"? This article renders it as a letter "x" in various inconsistent fonts, but I've also seen a times sign. Ishboyfay (talk) 17:49, 30 October 2009 (UTC)


 * Conway used x in his oldest notes I have copied from 1991, AND I checked in the 2008 published book symmetries of things, and x there too. Tom Ruen (talk) 20:39, 30 October 2009 (UTC)

Clean up
This article badly needs a clean up. The definition of orbifold notation reads very much as it does in Conway's "Quaternions and Octonions" and it is as incomprehensible here as there. I speak as someone with a degree in mathematics who took group theory in my final year. If I can't work out what the notation means, then I suspect its going to be hard for anyone. Sure there are lots of examples and I might be able to deduce what it means, but that's not really the right way to write an article.

The most obvious problem with the definition is it leaves a great deal unsaid, eg:


 * What about numbers neither to the left or right of an asterisk? They clearly occur, but the definition makes no mention of them.
 * What about the strings "**" and "*x"? Again, there is no explanation of them.

So the definition seems (to me) to be woefully incomplete. It would be great if someone who knows what it means could complete it.

The second problem is I have no idea what (say) 55 means (or *55). What in conventional group theory terms does this mean? That I have a generator of order 5, two generators? What? Why is a pentagon *55 but the flag of Hong Kong 55? What exactly is being said here?

At the moment the definition is (i) incomplete and (ii) not very clear. Francis Davey (talk) 22:21, 23 July 2012 (UTC)


 * Maybe I can add some more details to help. I don't think the notation makes sense in 2D symmetry, like *nn, nn, while for 3D, the orbifold is a lune with two order n rotation or reflection points, one at each "north and south poles". Tom Ruen (talk) 21:26, 14 January 2013 (UTC)

Non-triangular symmetry
I was curious about the hyperbolic groups starting at *2223 with non-triangular domains, and constructed a few as subgroups of *642 by mirror removal, with hand-recoloring the fundamental domains. Maybe we can get some nicer generations of these sometime! Tom Ruen (talk) 00:29, 14 January 2013 (UTC)
 * Hyperbolic 642 subgroups-small.png

Hyperbolic groups
I found one list of hyperbolic groups that is larger than given by Conway in "The Symmetries of things", in a paper "Two Dimensional symmetry Mutations" by Daniel Huson,, showing how fundamental domains in one symmetry can be remapped into other similar symmetry structures. Table 3 gives a count of "equivariant equivalence classes of tiling-transitive", but I'm just looking at the symbols, and some only exist in hyperbolic. Tom Ruen (talk)


 * I added the table to the article, but looking for more sources as well. Tom Ruen (talk) 20:47, 17 January 2013 (UTC)

Here's a nonreflective example, starting from {6,4} tiling as mirrors (*222222), removing alternate mirrors around the hexagon, and replacing them with 2-fold gyration points and adding 3-fold gyration points in the hexagon centers, making a (*23) orbifold. So this symmetry corresponds clearly to the (22*) orbifold of the Euclidean plane, which would start from the *2222 symmetry by the same process. For clarity I colored the gyration points and mirrors similarly. The gray/yellow background represent mirrored domains, and the magenta is a single fundamental domain of the orbifold. I'll keep looking for published examples! Tom Ruen (talk) 02:12, 24 January 2013 (UTC)
 * 23star orbifold example.png