There_s a really nice bit in +Roice Nel

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Tim HuttonTim Hutton - 2015-11-18 15:25:43+0000 - Updated: 2015-11-18 21:04:51+0000
There's a really nice bit in +Roice Nelson's video at 1:33 where it shows a Rubik's Cube on the Klein Quartic. It looks like a {7,3} tiling but if you look carefully you can see that for each dial that is twisted lots of other dials twist at the same time - the ones with the same colors. That's because they are actually the same dials! This is really helpful for understanding what +John Baez meant by gluing the matching numbers together, here:

So presumably, Roice, it would be possible to make the same Rubik's cube video on the 3D embedding of the Klein Quartic, as shown at the top of John's post? That could be nice to see.

Also, I really want to see an animation of how the {7,3} or {3,7} tiling folds up into the 3D embedding. I can't get my head around how that is even possible! +Jos Leys I've been enjoying your recent animations, could you make one for the Klein Quartic? (Oh, I just saw you were already asked for that here: Excellent!)

A demonstration video of MagicTile, which abstracts the faces of the original Rubik's Cube as a special case of more general tilings to create a huge number ...

Reshared by: Dave Gordon
Roice Nelson - 2015-11-18 19:28:54+0000 - Updated: 2015-11-18 22:43:19+0000
Hi +Tim Hutton , yes!  I've even spent quite a bit of time thinking about (and coding towards) the goal of rending these as embeddings instead of their universal cover on the hyperbolic plane.  So I have a lot (probably too much) to say about it.  Here's a little streaming:

* If you pasted the faces of the {7,3} quartic tiling onto Greg's picture at the top of John's post, it would be highly contorted.  The approach I've been hoping for is to paste the faces onto what are called Lawson surfaces.  See for some nice pictures of them.  You can construct one with arbitrarity genus, and they have nice properties, like being minimal surfaces in S^3.  It's debatable, but I also think it'd be nicer for the embedding to be a surface, rather than have edges like in Greg's picture.

* Lawson or not, what I need is a map from the hyperbolic plane to the embedded surface.  My experiments so far have been with using relaxation techniques on a mesh to achieve this.  I haven't been successful.

* +Henry Segerman and +Saul Schleimer have since made a really nice embedded model of the quartic that would be a good surface to use, and they've shared their code with me.  This seems like the most promising approach for the quartic in the short term, though it would be nice to find a method that would work for other tilings in general too.  Check out their model here:

If you search Henry's stream, there are a number of additional posts about their surface:

* There is some literature on this topic, known as "regular maps".  There are multiple papers by Jarke J. van Wijk, for example: Those techniques could be leveraged to render embeddings.  The surfaces aren't quite as nice as the ones above, from the perspective of curvature anyway.

* I thought I'd mention that some tilings can partially roll up into "infinite regular polyhedra" living in R^3, including the {3,7}, and these are supported already by MagicTile. Here's a picture:

However, this is not the Klein Quartic.  The identification of colors doesn't follow the same pattern.  Also, these are infinite structures, repeating in a euclidean lattice over all of R^3. A finite embedding would be nice too.

Anyway, if you have access to a Windows computer, I recommend downloading the latest MagicTile and experimenting with the quartic (and other tilings) for intuition about the identifications.  The program lets you drag around the hyperbolic plane, so you really can get quite a good feel for it!

Martin Andersson - 2015-11-18 22:48:01+0000
Ping +Malin Christersson​, I thought you might enjoy this.
Malin Christersson - 2015-11-18 23:13:43+0000

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