Message #863

From: Chris Locke <project.eutopia@gmail.com>
Subject: Re: [MC4D] Fractal cubes
Date: Thu, 11 Mar 2010 18:58:33 +0900

I’ve had a bit more time to think about the fractal-twisting idea and here
are my thoughts on it (and please let me know if I’m misrepresenting your
ideas here, Melinda):

Since if you have a single twist make a corresponding 3x3x1, twist, a 9x9x3
twist, a 27x27x9 twist, and so on, the twisting propagates up the Menger
levels all the way to the top level. So if you have a level 1 puzzle, you
will have the sticker you clicked twist in one way; if you have a level 2
puzzle it will twist in two ways, and so on. In this way, the puzzle is
seems like the level you work with will play a fundamental role in the way
twisting works. This isn’t necessarily a bad thing, just an observation.
Now, Melinda has proposed what seem like two potential ways that twisting
will work in this fashion. The first where a twist on a single 20 cube
"atom" will also twist every other 20 cube "atom", and the equivalent twists
will also apply to each 400 cube atom, and so on. This will mean that each
twist is truly global in every sense of the word. As for how the puzzle
will actually function, my intuition from playing with some of the highly
interconnected puzzles in MagicTile, is that due to the globalness of the
moves and the potential size of the puzzle, would make for one of the most
mind-bendingly difficult puzzles to solve. There’s also the more distant
(but the more I think about it increasing) present possibility that this
high level of interconnected-ness might reduce the puzzle in complexity
(like it might collapse into a single void cube). Without a working model
to work with, I can’t say for sure.

The second form of twisting is a play on the same theme, but with a little
bit less everything-is-twisting global-ness. In this case the sticker you
click affects which 3^n x 3^n x 3^(n-1) twists are done. So for instance,
on the level 2 puzzle, clicking one of the front face stickers won’t cause
every mini 20 cube atom to twist, but just the one containing the sticker
you clicked. It will also cause the entire front 9x9x3 face to twist. So
if you left-clicked a sticker on the front-upper-middle atom, then that
3x3x1 face will undergo a counter-clockwise twist, and the twisting of the
whole front 9x9x3 will cause that atom to also itself move to the
front-left-middle atom’s spot. If the puzzle was level 3, then the atom you
clicked would twist, move within the level 2 atom, and then move within the
level 3 atom as well for a total of 3 moves. Now, with this case of
twisting, you still get some globalness to each twist, but because you
aren’t affecting essentially the whole puzzle at once, I believe that the
possibility it would reduce to a single void cube in terms of complexity
vanishes. This method of twisting I could see becoming a standard if this
puzzle was implemented. Although I think for human reasons the size might
be limited to level 3 (although I’ve been surprised with what some people
have conquered in the past…. 7^5 and 120-cell anyone? :D).

Now, I think that the method of twisting I proposed would have some merit as
well. It might not be exploiting the fractalness of the shape, but it does
exploit the weirdness of the shape by taking the obvious physical
interpretation of what twisting would do. Furthermore, because the 9x9x1
twist moves around pieces that can be moved by 3x3x1 twists also, there is
still a large amount of global mixing and inter-connectedness going on. In
a sense, actually, this method of twisting is the most generalizable because
the fractal twists can be imitated through selective choices of the twists
done here. For the level 2 puzzle, you could implement the second method of
fractal twisting by doing a single 3x3x1 twist, then three 9x9x1 twists.
Naturally, though, there is a certain elegance to the fractal twisting that
makes it quite appealing (and potentially ground-breaking in the
puzzle-scene), but I think both kinds of twisting would result in
interesting puzzles. My proposed example might be more mundane in it’s
definition, but by very nature of the geometry in question it becomes quite
interesting in its own right. I guess it’s like the new additionally
puzzles of MagicCube4D in the sense that there’s no new deep innovation in
them, but they draw their intrigue from their interesting shapes, which
themselves give rise to new and unseen difficulties.

….. I wonder if there’s a way we can make this idea a reality ^_^

Chris

2010/3/11 Melinda Green <melinda@superliminal.com>

>
>
> Chris & David,
>
> Both of your suggestions allow for twists on elements such as a 9x9x1
> slice which are not part of any sort of standard cube of any scale.
> Also, twists in both of your designs have only local effects. I don’t
> want to put them down because I’m actually very happy to see some folks
> giving this problem some thought. While we might come up with a hard yet
> solvable puzzle this way, these aspects just don’t get me excited. As
> you both point out, these purely local operations are not very
> fractal-like and therefore don’t exploit the fundamental nature of this
> geometry.
>
> Yes, I was thinking of interactions very much like Roice’s Magic Tile in
> which portions of the geometry are not just copies of other portions but
> in actual fact *are* the same bits of geometry. I was hoping to do the
> same thing but with in scale as well as in position, but maybe that’s
> not the best approach. At least it didn’t seem to be generating anything
> terribly new.
>
> Well here is yet another possible design that I would like to offer: A
> click on a given sticker will affect all twistable 3^n x 3^n x 3^(n-1)
> faces that contain that sticker. IOW, clicking on a sticker will twist
> the face of its level 1 void cube as well as the level 2 cube that
> contains it, and then level 3 cube that contains that one, and so on up
> to some maximum (probably ending right there). If we don’t allow twists
> on "inner" stickers, then not all faces in every such chain will be
> twistable. I’m fine with whichever design makes more sense. This gives a
> puzzle that is sort of a combination of local and fractal.
>
> Thoughts?
> -Melinda
>
>