Message #845

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Introducing "MagicTile"
Date: Thu, 04 Feb 2010 21:19:13 -0600

What an great flow of ideas. I love it!

About solving towards a single cell, I think the program can help answer
this by using the setting to only show the fundamental set of tiles. I’m
not perfectly confident, but my intuition says it is possible to avoid
multiple disjoint unsolved cells at the end. My reasoning is that even if
the topology of the object as a whole (fundamental and orbit regions glued
up) is not simply connected, the fundamental region alone is simply
connected (as long as you never leave the boundary of it, so that it behaves
topologically as a disk). Looking at only the fundamental set of Klein’s
Quartic, it is easy to pick a solve order leaving only one cell at the end,
by using sort of a "solve wave" to push unsolved cells from one end to the
other. This would be my strategy to "keep the edge a simple closed curve",
as Melinda nicely described things.

If I require that I solve cells in a connected fashion, where each cell
solved must be adjacent to the prior one, it appears I lose the ability to
end on a single cell. As an aside, I wonder if it is possible to have
fundamental sets of tiles which are not simply connected, which could
invalidate my logic above.

Anyway, you’ve all convince me that it is the topology that is important
factor in all this (though it still seems the symmetries and topology are
intertwined). After taking another look at the 12-colored octagonal, also
genus 3, I see it actually does behave the same.

Regarding which puzzle(s) may be "just right", I’ve only done a few so far,
but last night I did go through a solve of the 9-colored torus puzzle and
can say I found it really enjoyable - perfect for a solve in one sitting for
sure. And the repeating units caught me a couple times, which was fun. It
may be inevitable that the 3x3x3 will always be the most perfect puzzle in
my mind though :)

One small comment related to difficulty… The state-space of the
3-coloreds is small enough that they will often solve themselves if I just
randomly click.

Yesterday, I learned about some additional coloring possibilities on the
hexagonal puzzles which I think will be good, balanced puzzles when it comes
to the tradeoffs being discussed, e.g. a 7-colored (This was pointed out to
me by someone from the twistypuzzles forums). Applying the tiling algorithm
which produces those to the hyperbolic puzzles will probably allow some
further balanced variants there as well. The 6-12 color range feels nice to
me, both in terms of coupling and tedium.

Btw, a while back I realized repeating units can be applied to spherical
puzzles too, which will end up throwing even a few more into the mix. A
Megaminx with a 6-colored fundamental region and opposite sides being orbits
would topologically act as the
non-orientable<http://en.wikipedia.org/wiki/Orientable> real
projective plane <http://en.wikipedia.org/wiki/Real_projective_plane> (I
verified by playing with color settings). Due to the non-orientability of
that surface, orbits would rotate in an opposite sense to the fundamental
cells! One could do a non-orientable hexagonal version as well,
topologically a Klein bottle <http://en.wikipedia.org/wiki/Klein_bottle>. I
definitely want to support these variants at some point, which will only
involve minor code changes to deal with reversing the twisting.

Re: Alexander… I bet doing a ctrl+click view rotation without animation
wouldn’t be too bad, and something like that could likely be the first pass.
Also, I’ve been doing the same head twisting and so am sharing the desire
for rotation control of the projected puzzles too, which will be trivial to
add. And load/save is at the top of the list. I hope to address the
majority of the requests I’ve received so far, though I expect it will be at
least a month before I’ll be able to provide any updates.
Regarding publicizing, I have been sending out an email or two each evening
to try to get the word out. Mathpuzzle <http://www.mathpuzzle.com/> gave
MagicTile a really nice shout out this morning. I want to do a blog post
too (both about MC4D and this), though that will probably just get the word
out to my mom and siblings :)

Re: Chris… so sorry about the loss of work :( I will definitely switch
this resetting behavior. It caught me the other night too, at which point I
noted the poor behavior in my list. I was lucky enough to not be half way
into a large puzzle. Also, on the 4D front, I do think it’d be cool to have
stereographically projected versions of the puzzles in MC4D. I even opened an
issue for it<http://code.google.com/p/magiccube4d/issues/detail?id=14&q=stereographic>a
while back :D

Have a good night all,
Roice

P.S. I did move the donate button up (thanks for the suggestion on that
too, Chris and Alexander)

On 2/4/10, Melinda Green <melinda@superliminal.com> wrote:
>
>
> Chris Locke wrote:
> >
> >
> > 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.
>
> I think that this phenomenon is the same one that makes the megaminx
> easier to solve than the cube because there’s more "room" on the surface
> of the puzzles with more faces in which to squirrel away some parts
> while working on others. The smaller, puzzles are "tighter" and cause
> every action to affect just about everything else. I have a funny
> feeling that there is some sort of natural difficulty metric in which
> the original 3^3 Rubik’s cube will turn out to be the hardest of all the
> similar puzzles in all dimensions, but I can’t quite see how to define
> that metric. I only feel it.
>
> > 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.
>
> You’d need a genus 4 or higher surface to end up with 3 or more
> disconnected unsolved faces.
>
> I’m not sure if you are right that one should be able to follow a
> modified layer-by-layer method on any of these puzzles and always end up
> with a single unsolved face. The key would be to keep track of the
> "outer edge" of your solved patch as you grow it. Never let one part of
> that edge connect with any other part. In other words, you make sure to
> keep that edge a simple closed curve. Any time a part of it wants to
> connect with another part, just leave that area and work on some other
> part of the border that has room to spread. The question is whether the
> initially growing edge will eventually shrink back down to a single face.
>
> At first I thought yes, but as I started to write the above, I’m now not
> so sure. This is reminding me a lot of the short Wikipedia article that
> Roice cited on simply connected
> <http://en.wikipedia.org/wiki/Simply_connected> spaces. Imagine trying
> the above solving strategy with a {4,4} puzzle defined on the surface of
> a torus. I think you’ll end up with a seam that you can’t get around. I
> just can’t quite see it in my head without some good pictures.
>
> > […] Less colors is too hard, and more colors takes too long.
>
> Each color is a single face. Each time that you see a red face, it’s
> always the same face just seen from a different perspective. Kind of
> like how gravitational lensing can let you see multiple images of the
> same distant galaxy in a single photograph. I agree that tighter puzzles
> take more brain work and less tedium but when they become extremely
> tight, they seem to get easier again in a way as you can begin to get
> your head around the whole thing. In addition to wanting to know which
> puzzle is hardest for it’s size, I also want to know is which puzzle out
> of all the twisty puzzles different people think has the challenge that
> is "just right" for them.
>
> -Melinda
>
>
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