Message #1833

From: schuma <>
Subject: Re: State graph of MC2D
Date: Sun, 31 Jul 2011 08:21:31 -0000

Hi Melinda,

I don’t know why the links fail. I am using the Yahoo group files and
copied the URL from the files. It worked when I tested them. But now
they don’t work any more. Anyway, all the four files have always been
in the Yahoo group folder:

There are four files:

color.gif is for MC2D, where the nodes and edges are color coded. It’s
a GIF animation.

labels.gif is for MC2D, where the nodes are labeled by matrices. It’s
also a GIF animation.

Torus_2x2x2.png is for the 2x2x2 with 180 deg turns, where only one
large hexagon is shown and the opposite sides are identified.

Periodic_2x2x2.png is for the 2x2x2 with 180 deg turns, where several
copies of large hexagons are tiled.

About the outcome of identifying opposite sides of a hexagon, I cannot
agree with you. Topologically, it should still be a regular torus.
Please see the section "hexagonal torus" in this page:


— In, Melinda Green <melinda@…> wrote:
> Nan,
> I only get 404 errors for most of the image links you gave. The one that
> I *could* see is your amazing hex graph (the first one below).
> Topologically it’s not a simple one-holed torus, but rather a torus-like
> surface with three holes
> <>.
> What makes this much more interesting to me is that it has a higher
> genus which is perhaps the most fundamental thing about topology in
> general. It’s amazing to me how this topic feeds-back into MagicTile and
> Roice’s recent support for IRPs. I will definitely have to attempt to
> sketch the type graph if someone doesn’t beat me to it. This has been
> one crazyamazing day for me in several ways. Thank you for your
> wonderful new result, Nan!
> -Melinda
> On 7/30/2011 11:42 PM, schuma wrote:
> > Melinda, thank you for providing the context about our meeting, and the context about why the MC2D could be interesting.
> >
> > Out of curiousity, I analyzed another simple puzzle and drew its state graph in a geometrically symmetric manner. The simple puzzle is:
> >
> > *** 2x2x2 with only 180-degree turns ***
> >
> > I assume the LBD corner is always solved. Only U2, R2, F2 are the valid moves. Just like MC2D, there are also 24 states. When I asked Mathematica to draw the graph, it didn’t provide a symmetric arrangement for the 24 nodes. Then I manually drew the graph and found a symmetric arrangement:
> > <>
> > Here is how to look at the above graph. Inside of the large hexagon, there are 24 nodes where thin black lines meet. These 24 nodes represent the 24 states of the 180-degree-turning 2x2x2 puzzle. Every node has three edges, representing U2, R2, F2. The opposite sides of the large hexagon are identified. So topologically it’s a torus. It’s good to think of it as a torus because it’s easy to see why all the nodes are equivalent.
> >
> > One can also make copies of the large hexagon and tile them on a plane like a honeycomb:
> > <>
> > In that way every state (out of 24 states) can find a representitive in each hexagon cell of the honeycomb.
> >
> > Why are there so many small hexagons? It’s because a 6-move algorithm [U2,R2]x3 always bring you back to the original state. So do [U2,F2]x3 and [F2,R2]x3. Therefore from each starting point there are three hexagonal loops. Note that [R2,U2]x3 is another such algorithm. But it’s simply the inverse of [U2,R2]x3. So these two algorithms are nothing but going along the same loop in different directions. Using this property, one can easily label the thin lines by U2, R2, and F2.
> >
> > I’m not good at identifying which states belongs to a type (an equivalency class with respect to recoloring and reorientation). It would be interesting to find out whether there’s a bottleneck in the graph of types.
> >
> > Nan
> >
> > — In, Melinda Green<melinda@> wrote:
> >> Thank you Nan for posting this to the group. For context, Nan, Brandon
> >> and I all met for a marathon discussion and brainstorming session a
> >> little while ago and had a fabulous time. The topic of MC2D came up in
> >
> >
> >
> > ————————————
> >
> > Yahoo! Groups Links
> >
> >
> >
> >