Message #1297

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] 8Colors solved
Date: Tue, 28 Dec 2010 11:02:17 -0600

A single twirled 4C piece was very surprising to me! At first blush, I
would expect things to behave exactly as described in section 6d of
the Keane/Kamack
paper <http://udel.edu/~tomkeane/RubikTesseract.pdf>.

We’ve seen single twirled corners on the 5D puzzle, and the allowance for
that has to do with being able to mirror pieces through four dimensions
(using the 5th dimension), as described by
Don<http://games.groups.yahoo.com/group/4D_Cubing/message/243>.
So my initial guess was that the 8-Colors puzzle must therefore have a
non-orientable <http://en.wikipedia.org/wiki/Orientability> topology, which
would allow mirrored pieces to show up, and that a combination of
reflections could give the observed result. But the topology is orientable.
(For any interested, to check to see if the puzzle was non-orientable, I
traced paths from a cell to its copies, then looked to see if any of the 4C
pieces in the copy appeared mirrored relative to the parent cell.)

At a loss for an explanation, what I tried next was to count the number of
4C pieces connected to a given cell (fourteen), and in doing so, I think I
stumbled onto the cause of this. If you look at how these 4C pieces move
during a 2C twist of this cell, there are two sets of 6 that change
positions in 6-cycles. But the remaining two 4C pieces change positions in
2-cycles, with their orientations twirling a third after each two cycle.
This seems to be the root of the unusual behavior, though I still haven’t
nailed down exactly which part of Keane/Kamack’s analysis no longer applies.

Why are there two 4C pieces that have this unique behavior during each 2C
twist? I think the cause of that has to do with the fact that the periodic
painting on the 8-colored puzzle is
chiral<http://en.wikipedia.org/wiki/Chirality_(mathematics)>.
You can get a sense of this by studying the pattern of colors on a pristine
puzzle from within one of the cells (such that you can easily see the color
pattern on the hexagonal grid). Contrast with the 12-Colors puzzle, where
the periodic painting is not chiral. In the latter, you have eighteen 4C
pieces connected to every face, which breaks down into 3 sets that each
permute positions in 6-cycles, so there are no uniquely behaving pieces.
I’m placing my bet that a single twirled corner on the 12-Colors puzzle is
impossible.

I would of course appreciate further insights or corrections to my thoughts
about this unusual behavior! One last comment on the topic though - the
single twirled piece is noteworthy because, unlike the other kinds of
impossible-looking states usually encountered, this is not due to an
even/odd (binary) phenomenon, but instead a trinary one. So "parity" is
probably not a word that should even enter the discussion :D

I want to add to the chorus and also say* thank you to Andrey* very much for
the new goodies in MHT633. I really appreciate the Poincare Ball view in
particular. It’s very nice, and I consider it the closest possible visual
analogue to MC4D in terms of display and controls. I like that you’ve
matched the controls so well, though to be as close as possible, I would
recommend switching the directions in the left-click dragging (so that the
dragging pulls the front of the ball model, rather than the rear). I’d also
like to echo others thoughts on how nice the animated twisting is for
understanding.

Here is some *very minor* feedback for things I ran across while playing
with the latest version (some of it probably not even worth fixing):

• The ctrl-left-click to set a new center feature looks to no longer work.

• If you left-click to twist while it is still animating the previous
  twist, both twists are instantaneously applied.

• Ctrl-z to undo does not always undo relative to the same clicked sticker
  when animating, but instead always undoes relative to some fundamental
  location. This only looked noticeable when you are viewing the puzzle from
  within a cell, and the end result of the state is the same regardless, so I
  don’t think it is worth spending time on. In fact, it was a nice insight
  for me for further possibilities in MagicTile (I didn’t realize things fit
  together so well in a hexagonal periodic painting, so I see now there are
  some additional slice-masked twists I could allow).

Thanks again Andrey! MHT633 is the ultimate Rubik analogue!

All the best,
Roice

On Fri, Dec 24, 2010 at 3:29 PM, Andrey <andreyastrelin@yahoo.com> wrote:

> Yes, 3-cycle on 4C is not a bug, it’s really possible situation. But may
> way to resolve it was too long (more than 1500 twists?).
>
> Andrey
>