Message #842

From: alexander.sage@jacks.sdstate.edu
Subject: RE: [MC4D] Re: Introducing "MagicTile"
Date: Thu, 04 Feb 2010 01:22:56 -0600

I played with the hyperbolic puzzles more, and realized that this is certainly a well written program. I love this idea of playing with twisty puzzles that lack the spatial ability to exist, including both these hyperbolic polyhedra, and the polychora. This is a really ingenious idea, one that I would probably never thought of in my life, if not for you.

I’m glad to hear that {4,4} is coming along nicely.

I first noticed the double bottoms on the hex tiling, when I was using 9 colors. needless to say, I was rather suprised. I think that the only regular euclidian polyhedron that has anything like double-bottoms is the tetrahedron, but that is obviously an exception.

Having used the program more, I have a few more words of advice. A save function would be very nice. Right now, even the professor’s cube is really hard to do in one sitting. You said that it would be difficult to change the hyperbolic viewpoint, but we really don’t need to see animation for it. It might be hard, but even if the change was instant, that feature would be very helpful on some of the bigger puzzles. One of the other features I would really like to see is a simple rotation, x-y plane. Sometimes it’s nice not to have to turn your head sideways to try to figure out if an algoritm is going to have your desired effect. (turning one’s head is faster than figuring it out spatially.)

Two more suggestions. (If you haven’t already,) Put a paypal box up on your website. Your programs are very deserving of some compensation. Second, get more people interested in this. I know that it’s hard, having tried myself (with no good results), but if you’re smart enough to envision these programs, you probably can think of something. ;)

To: 4D_Cubing@yahoogroups.com
From: roice3@gmail.com
Date: Wed, 3 Feb 2010 00:03:53 -0600
Subject: Re: [MC4D] Re: Introducing "MagicTile"

I found the "two bottoms" observation extremely interesting :D I tried to figure out the why of this last night by rereading John Baez’s article on Klein’s Quartic, but didn’t have much luck finding the insight I was looking for (though the two cells in the last layer is mentioned there, and the article is full of other neat information). It felt like the behavior should be related to the topology and the fact that a 3-holed-torus (genus 3, which is the topology of the puzzle) is not simply connected. But the 12-colored octagonal puzzle is also genus 3, and it doesn’t behave the same.

I found some more info this evening, and it turns out there is an entire book on Klein’s Quartic! Amazon has it, but you can download a free version online (however, the pictures seem to be missing). In the first section by Thurston, he notes "The infinite hyperbolic honeycomb is divided into 3 kinds of groups of 8 cells each, where each group is composed of a heptagon together with its 7 neighbors.". Together, these groups account for the 24 cells, and after labeling the 3 groups red/green/white, he writes:

It is interesting to watch what happens when you rotate the pattern by a 1/7 revolution about the central tile: red groups go to red groups, green groups go to green groups and white groups go to white groups. The person in the center of a green group rotates by 2/7 revolution, and the person in the center of a red group rotates by 4/7 revolution. The interpretation on the surface is that the 24 cells are grouped into 8 affinity groups of 3 each. The symmetries of the surface always take affinity groups to affinity groups. This is analogous to the dodecahedron, whose twelve pentagonal faces are divided into 6 affinity groups of 2 each, consisting of pairs of opposite faces.
So I think it has more to do with the symmetries of the object than the topology (though perhaps there is some interrelation). I think what Nelson found was one of these 8 affinity groups. Btw, by editing colors, you should be able to use the program to more easily see the reg/green/white groups described above - I’ll have to try this.

Also, I did note to myself last night that it is possible to solve the {7,3} cells in an order such that you’d be left with one cell at the end instead of two, but you wouldn’t be working "layer-by-layer" in that case.

Very cool discovery of this unusual behavior Nelson!

Roice

On 2/1/10, spel_werdz_rite wrote:
In a follow up with Roice, I’d like to share some more interesting details with the Klein’s Quartic puzzle.

The strategy to solving it was very much similar to how one would solve a Megaminx (my method at least). Edges, then sides, then edges, working all the way down to the bottom of the puzzle. Doing this method lead to me a very interesting discovery that, surprisingly, not even Roice new about. It turns out that Klein’s Quartic has two "bottoms." By which I mean if you follow this method of inserting pieces downward until you reach the bottom of the puzzle, you will end up at 2 different faces. At this location, solving became a bit of a new task, but still not much of a challenge. The first step was making sure the remaining 2C and 3C pieces were on their corresponding face and oriented correctly. After that, I borrowed many techniques I used for the Megaminx. However, due to some obvious differences, the end took a lot of guesswork. In the end, the puzzle took about 2.5 hours (factoring in my "hey let’s get distracted a lot" variables).

My final thoughts. Very fun. It was a true joy to play a technical 3D puzzle that technically couldn’t exist in the 3D world.