# Message #853

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: Re: [MC4D] Fractal cubes

Date: Mon, 08 Mar 2010 01:14:14 -0600

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).

Regards,

David V.

—– Original Message —–

From: "Melinda Green" <melinda@superliminal.com>

To: "MagicCube4D" <4D_Cubing@yahoogroups.com>

Sent: Sunday, March 07, 2010 9:59 PM

Subject: [MC4D] Fractal cubes

Have a look at the following image. It brings together two of my

favorite subjects: Fractals and Rubik’s cubes.

http://proofmathisbeautiful.tumblr.com/photo/1280/431394685/1/tumblr_kyv2avNAWg1qa3dmv

I have no idea what’s behind it. Has someone actually built a

fractal

twisty puzzle? If not, would that be possible? I expect that a

working

fractal cube would feel very much like a 3D version of Roice’s

Magic

Tile in which each area is repeated across space but at multiple

scales

in addition to translations and rotations.

Fascinated,

-Melinda