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