Message #843

From: Chris Locke <project.eutopia@gmail.com>
Subject: Re: [MC4D] Re: Introducing "MagicTile"
Date: Thu, 04 Feb 2010 16:51:04 +0900

Congratulations Roice on another jaw-dropping program! The community of
beyond physically realizable Rubik cubers I’m sure is both shocked and
pleased by the recent flurry of new puzzles. First we get tons new cool 4D
shapes to play with, and now hyperbolic puzzles?? What’s next, 4D
equivalents of these hyperbolic puzzles? :D

First thing I would note is that you should move the "donate" button to a
more visible location. As you can see by Alexander’s post it’s not easy to
find. I didn’t even know you had it on the page until recently when I
actually scrolled all the way to the bottom!

As for the puzzles themselves. I find that from my limited playing around
with the hexagonal and heptagonal tilings, that low color counts completely
baffle me, but as the number of colors increases, the amount of coupling
between the twists decreases, which makes it much easier to work with. The
3 and 4 color hexagonal tilings are just weird to try and solve :P. But in
the cases with many colors, it’s feels like it’s just a matter of patient to
reduce it down to a last layer. From there, Megamix LL
algorithms/commutators can be generalized to finish them up.

As for the double bottoms thing, Melinda gave a pretty good explanation as
to how it’s a result of the topology. In the case of the 3 holed torus
(Klein’s quartic) it’s pretty easy to see how spreading out from one
position can leave with with a case of two disconnected unsolved faces.
Furthermore, I don’t think it’s possible to end up with 3 disconnected faces
in this case, and also if you understand the topology from the tiling, it
should also always be possible to solve it such that you end up with just
one unsolved face. But that in itself is a challenge too :D.

While we’re on the topic of suggestions: can you make it so that when you
change the performance setting, the puzzle doesn’t reset? I was playing
with the Klein’s quartic and halfway through my solve I tried that setting
and it resetted it :(. So I decided to solve the 9 color hexagonal tiling
instead as I didn’t want to work on the same puzzle twice ^^. Less colors
is too hard, and more colors takes too long.

Thanks again for this sweet program!

Chris

2010/2/4 <alexander.sage@jacks.sdstate.edu>

>
>
> 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<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.
>
>
>
>
>