Message #856

From: Chris Locke <>
Subject: Re: [MC4D] Fractal cubes
Date: Tue, 09 Mar 2010 15:12:53 +0900

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.


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.
> >