Message #840

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Introducing "MagicTile"
Date: Wed, 03 Feb 2010 00:03:53 -0600

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<http://math.ucr.edu/home/baez/klein.html>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<http://en.wikipedia.org/wiki/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<http://www.amazon.com/gp/product/0521004195?ie=UTF8&tag=gravit-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0521004195>,
but you can download a free version
online<http://www.msri.org/publications/books/Book35/>(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.
>
>