Message #858

From: Chris Locke <>
Subject: Re: [MC4D] Fractal cubes
Date: Wed, 10 Mar 2010 16:58:58 +0900

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

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!


2010/3/10 Melinda Green <>

> 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 <<>
> > < <>>>
> >
> >
> >
> > 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 ( 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:
> > >
> > >
> > > There is there a comment indicating that this thing has a name -
> > > a 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.
> > >
> >
> >
> >
> >
> >