Message #860

From: David Vanderschel <>
Subject: Re: [MC4D] Fractal cubes
Date: Wed, 10 Mar 2010 18:20:12 -0600

Melinda wrote:
> I’m aware of Menger cubes, Serpenski gaskets, Cantor dust,
> etc. …

Yet you had originally written, "I have no idea what’s
behind it."; so I figured some explanation would be welcome.

I described the construction as growing at each step because
the presentation makes a point of coloring the lowest level
cubes. For a ‘real’ sponge, taken to the limit, there are
no visible ‘cubies’ to color.

> Does anyone else have ideas for better ways to make this
> thing real?

Did you see my suggestion to treat it as a variation on the
order-9 3-puzzle? I meant that in the sense of allowing
twists on the slices that have holes in them. (No problem
for a simulation.) Chris had a slightly different take on
an order-9 variation.

> Regarding the coloring of inside stickers, I didn’t
> completely follow what you [Chris] 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.

This is not obvious to me. Do you have proof, reasoning,
reference? I thought this was the interesting thing about
the order-9 variation.

> 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.

I suppose something along those lines is what it would take
to call the puzzle "fractal". But I can’t think of any
useful way to make it work either.

Speaking of "fractal", it should be noted that these various
space-filling curves were known before Mandelbrot appeared
on the scene. They are so deliberately self-similar that I
have never regarded them as particularly interesting in the
larger fractal context.

In that coloring that Chris pointed us to, it may noted that
it is equivalent to a pile of identically decorated cubies
in which every cubie has stickers on all six faces with the
starting position having the same color facing in each of
the six possible directions. Interestingly, even the basic
order-3 3-puzzle can be viewed this way. With this view,
what makes the cubies distinguishable is that the _visible_
set of stickers on each cubie is unique and unchanging.

David V.