Message #859

From: Chris Locke <project.eutopia@gmail.com>
Subject: Re: [MC4D] Fractal cubes
Date: Wed, 10 Mar 2010 17:37:14 +0900

I actually found a good link that shows the coloring I was talking about:
http://www.mathematik.com/Menger/Menger2.html
You can see that even for the smaller void cubes, the insides of the cubies
are also colored based on which direction they are facing.

2010/3/10 Chris Locke <project.eutopia@gmail.com>

> I’ve attached an image for clarity.
>
> In the case I was talking of, a single twist won’t affect other parts of
> the puzzle, just the part that is twisted. So it’s like a normal puzzle,
> just with a crazy shape. In the image I attached, the green outlines
> examples of simple void cube twists that are permitted, and the teal
> outlines the larger twists that are permitted. Basically, the twisting
> works just like you expect it to if a physical level 2 Menger rubik cube
> existed. The mini level 1 cubes can be twisted just like a void cube along
> those axes, but in the other directions you have to move a larger 9x9x1 slab
> instead of just a 3x3x1 slice. This means that for the void cubes in the
> corners of the level 2 Menger cube, there is no way to just twist the cubies
> in that individual mini void cube; all twists are twists of 9x9x1 slabs.
> This process can be generalized up to any level as well. So for level 3
> Menger cubes, 3x3x1, 9x9x1, and 27x27x1 twists are possible.
>
> As for sticker coloring, if you look at the level 2 Menger cube, you will
> notice that the smaller 3x3x3 void cubes act as single cubies at that
> level. So if you would color the larger scale level 2 Menger cube simply
> like a void cube, the entire inside faces would be uncolored. In the
> picture I attached, I blacked out the top face of the front-bottom-middle
> void cube to show what that would look like. The only choice would be to
> color that blackened face blue, corresponding to the color directly above
> it.
>
> You’re definition of twisting where one twist propagates seems really
> interesting, but hard to visualize :D. Sounds like it could make for a
> pretty interesting puzzle. I’d imagine it might be similar to how in
> Roice’s new MagicTile program the infinite tilings are repeated so a twist
> of one face twists all equivalent faces. Only in this case it’s not tiling,
> but ‘fractaling’ :D. I’ll have to think about that a bit more later! Seems
> like it would be one heck of a difficult puzzle anyway!
>
> Chris
>
> 2010/3/10 Melinda Green <melinda@superliminal.com>
>
>
>>
>> We may not be talking about the same basic geometry, but if we are then
>> I think that I see the problem. When I say that a twist on a level 1
>> cube (I.E. a Void cube) will affect all other cubes, even of larger
>> levels, meant for twists on larger level cubes to be in proportion to
>> those cubes. So in your 2-level version, twisting a 3x3x1 slice will not
>> only make the same twist on all 19 other level 1 cubes, but will also
>> twist the corresponding 9x9x3 face of the level 2 cube. Rather than
>> mixing up the puzzle beyond all hope (for say a 4 level cube), this
>> design mixes them not at all and seems to leave us with a single void
>> cube to solve.
>>
>> Regarding realizations of possible puzzles, I only mean computer
>> implementations.
>>
>> Regarding the coloring of inside stickers, I didn’t completely follow
>> what you proposed, but it doesn’t seem like an important issue because I
>> don’t think that the inside stickers can contribute to the puzzle. I
>> think they should automatically be correct when the rest of the puzzle
>> is complete. I’d therefore probably color them with the same color as
>> the outside stickers that face in the same direction. That way when the
>> whole puzzle is solved it will appear to be more "complete", but maybe
>> that was what you were saying.
>>
>> So how to define a Rubik’s Menger Cube that’s harder than a Void cube
>> yet not impossible? Here’s another idea: What if a twist on any level 1
>> cube twists all other cubes of all levels that are currently playing the
>> same role? IOW, if you twists a level-1 cube that is part of the FUR
>> cubie, then all other cubes that are also FUR elements of other cubes
>> will twist, and *only* those cubes. That might make all cubes into
>> individually solvable void cubes yet hopelessly interconnected with each
>> other. I don’t know but I sense that there may be a natural, elegant,
>> and hard-but-not-impossible-puzzle definition lurking here somewhere.
>>
>> -Melinda
>>
>>
>> Chris Locke wrote:
>> >
>> >
>> > Well, since the void cube is a level 1 Menger sponge, it would always
>> > be possible to go simply to just level 2 for now and have a finite
>> > puzzle in that sense. I don’t know if it would be possible to
>> > construct a physical one without having it rounded in shape like the
>> > V-Cubes, but on a computer it would be no problem. How would it
>> > work? Let’s say you take the front upper middle edge cube (almost
>> > equivalent to a void cube). You would be able to do L, M, and R
>> > twists on this as much as you want, but if you try, say, a U twist,
>> > you will end up twisting the entire upper layer. This would move the
>> > top 8 cubies of the front upper middle cube and move it to the top 8
>> > cubies of one of the side upper middle cubes, and similarly for other
>> > twists like it. This would give the cube a crazy level of
>> > interconnectedness between all the mini void cubes and would make for
>> > one crazy puzzle to solve. This method could also be extended to any
>> > level of depth desired, except beyond level 2 the number of cubies
>> > just becomes waaay too big for any sane person to deal with :D
>> >
>> > As for coloring of the inside stickers (the void cube is black inside
>> > but the Menger sponge has to have inside colors at least above level 1
>> > but it’s no more difficult to have for all cubies) just have the same
>> > color as the whole cube. So you would still have 6 colors, but if you
>> > look at one of the edge pieces of a mini void cube, instead of being 2
>> > colored, it would be 4 colored. But because of the way the coloring
>> > works, the 4 coloredness of the cubie is equivalent to the normal 2
>> > colored edge pieces.
>> >
>> > Chris
>> >
>> > 2010/3/9 Melinda Green <melinda@superliminal.com<melinda%40superliminal.com>
>> > <mailto:melinda@superliminal.com <melinda%40superliminal.com>>>
>>
>> >
>> >
>> >
>> > David,
>> >
>> > I’m aware of Menger cubes, Serpenski gaskets, Cantor dust, etc. Their
>> > constructions are quite simple. Normally quite boringly simple, but I
>> > find this coloration of a Menger cube really quite evocative. I don’t
>> > have any more context for the image. I just get Google alerts on the
>> > term "Buddhabrot" which lately includes a lot of postings such as
>> > this
>> > one by a person who used that term as their account name. Quite
>> > flattering really and I often like what they come up with. Yes, the
>> > image appears to be a random Rubik colored Menger cube and not any
>> > sort
>> > of actual puzzle. It just makes me wonder how it could best be
>> > made real.
>> >
>> > My current thought is to treat every 20-cube figure identically,
>> > regardless of scale. So a twist on a 20 "atom" cube would cause a
>> > twist
>> > of every other 20 atom cube as well as every 400 atom cube and so
>> > on up
>> > the chain to some maximum scale cube. Scrambling such a beast
>> > would mix
>> > the colors so completely that at high levels it will just appear as a
>> > single grainy mix of all the colors. You’d need to zoom in to the
>> > atomic
>> > level in order to work on it. In some ways, the fundamental puzzle
>> > would
>> > not be terribly interesting because I think that it would just be
>> > normal
>> > void cube (http://en.wikipedia.org/wiki/Void_Cube) because as you
>> > point
>> > out, the cubies would not mix outside their respective cubes at any
>> > level. Still, it’d be fascinating to watch it being solved.
>> >
>> > Does anyone else have ideas for better ways to make this thing
>> > real? The
>> > only constraint that I urge is that a twist on any scale should have
>> > identical affects on all scales, but just how that might work is
>> > an open
>> > question.
>> >
>> > -Melinda
>> >
>> >
>> >
>> > David Vanderschel wrote:
>> > > This is just a typical fractal generation with a highly
>> > > regular algorithm. (I am trying to distinguish it from the
>> > > more interesting fractals (think of coastlines) that exhibit
>> > > randomness.) Instead of going smaller on each iteration,
>> > > the pattern becomes larger in this case. The basic starting
>> > > pattern is a pile of 20 cubes, corresponding to a 3x3x3
>> > > stack with the central ‘cross’ (7 cubes) removed. That
>> > > stack, with the holes in it, can be treated as a cube
>> > > itself. So 20 such cubical piles can be piled together in
>> > > the analogous fashion to create the next generation - a pile
>> > > of 400 little cubes. Etc.
>> > >
>> > > The coloring looks random to me. (I can imagine interesting
>> > > looking non-random colorings, some of which could improve
>> > > one’s ability to see the picture correctly.)
>> > >
>> > > As an abstract thing, the 20-cube pile could be ‘worked’
>> > > like a 3D puzzle. (I have a recollection that there is a
>> > > commercial physical version of such a puzzle.) The only
>> > > catch is that, in the absence of face-center pieces, you
>> > > have to use some other method to assign the face colors.
>> > > However, this is already a familiar problem with the even
>> > > order puzzles. If you use the analogous motions to ‘work’
>> > > a 400-cube pile, you see that little cubes can never move
>> > > from one 20-cube pile to another; so that does not lead to
>> > > an especially interesting puzzle. OTOH, the 400-cube
>> > > pile could be regarded as a variation on the order-9
>> > > 3-puzzle; and this one is interesting, as we can see
>> > > cubies that are not in external slices. I.e., we can begin
>> > > to concern ourselves with the permutation and orientation
>> > > of interior cubies that are normally invisible to us.
>> > >
>> > > Melinda, how did you encounter this? Surely there must
>> > > be some context that would provide a little more info about
>> > > its significance (or lack thereof).
>> > >
>> > > I managed to find some context:
>> > > http://dosenjp.tumblr.com/post/430499178/via-dothereject
>> > >
>> > > There is there a comment indicating that this thing has a name -
>> > > a Menger sponge:
>> > > http://en.wikipedia.org/wiki/Menger_sponge
>> > > If you found my explanation of its ‘construction’ too terse,
>> > > there is a much
>> > > more elaborate version on the wiki.
>> > >
>> > > So the image was basically a coloring of a Menger sponge in the
>> > > manner
>> > > of a Rubik’s Cube.
>> > >
>> >
>> >
>> >
>> >
>> >
>>
>>
>
>