# Message #855

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Fractal cubes

Date: Mon, 08 Mar 2010 21:55:45 -0800

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.

>