Message #841

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Re: Introducing "MagicTile"
Date: Wed, 03 Feb 2010 23:09:27 -0800

I think that the reason you end up with two bottoms is due to the
surface topology and not the polygonal symmetries. My page on infinite
regular polyhedra
<http://www.superliminal.com/geometry/infinite/infinite.htm> describes
the topologies of these sorts of animals. Scroll down there and look at
the second black & white diagram showing a genus 3 IRP. In an infinitely
tiled construction you simply connect lots of those shapes in a cubic
lattice, but the more interesting case is when you have just one copy
which connects back onto itself as indicated by the arrows. You can do
that within a repeating *finite* space with the topology of a 3-torus
which is the 3D equivalent of the finite repeating space you may
remember from the old "Asteroids" video game. The modern game "Portal"
uses 3D spaces in this way.

To visualize the reason for the two bottoms of the Klein’s Quartic
puzzle is even simpler: Just connect up the open mouths of a single copy
of the figure with simple tubes that follow the arrows. The {7,3} or its
{3,7} duel can be drawn on that surface and the twisty puzzle version
can be solved there. You can smoothly bend the shape around all you like
without changing the topology. You can even have the handles pass
through each other without interacting. Just imagine unfolding it until
you have the equivalent of a hollow ball with three tubular "handles".
Now imagine holding it by one handle and letting the rest dangle
downward. If you start your solution at the top of that top handle and
solve by crawling your way ever outward from there, you can see how your
solution can spread around the body of the ball and then down the other
two handles until finishing at the bottom of each of them. You
essentially end at two different local minima.

Here is a cross-eyed stereo photo of part of the infinite {3,7}
<http://www.superliminal.com/geometry/3_7st.jpg>. You can easily imagine
how each of the triangles can be carved up into the twisty puzzle
version. I think that I’ll rebuild that puzzle sometime such that it is
colored to show the heptagonal patches of Klein’s Quartic. I probably
won’t get to that right away but I’ll post photos when I’ve done that.
Unfortunately I don’t have 24 differently colored sets of triangles, but
hopefully I will find a nicely symmetric 4-coloring. If so I expect that
it will be rather handsome.

This stuff was a huge interest of mine for a long time and it’s fun to
think about it again. I never would have dreamed that Rubik’s cubes
would connect back to this stuff in any way, but there it is. Thanks for
plugging these two interests of mine together Roice. It gives me a
really nice feeling of completeness.

-Melinda

Roice Nelson wrote:
>
>
> 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.
>
>
>
>
>