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Date: Wed, 20 Aug 1997 17:22:07 -0400
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From: Dan Hoey
To: scotth@ichips.intel.com
Cc: Cube-Lovers@ai.mit.edu
In-Reply-To: <199708182132.OAA19914@ichips.intel.com>
Subject: Re: d-dimensional cube mechanisms
Scott Huddleston writes:
> Several years ago I worked out a solution to the d-cube 3^d, for d>3.
> This is most interesting combinatorially if you assume you're
> restricted to only rotating entire (d-1)-faces at a time, so that's
> what I assumed in my solution.
I thought that was most natural, but there are certainly at least two
others. In my article of 22 Dec 1993 in
ftp://ftp.ai.mit.edu/pub/cube-lovers/cube-mail-11 I mentioned how to
show they are different, at least in the 3^4 hypercube. I didn't go
into enough detail to determine "most interesting"--can you expand on
what you mean? By the way, do you know if this is the Kamack and
Keane model? I've never seen their paper, and I never figured out how
to read Charlie Dickman's document.)
> But when I thought about building a mechanism for the d-cube, I
> came to the surprising (to me) conclusion that any natural extension
> of the 3^3 mechanism to d dimensions would allow you to rotate any
> 2-face.
Wow! I can verify it for the most obvious natural extension I can
describe. We could form the 3^d Rubik's ball by taking the unit ball
B_d={z in R^d : |z|<=1} and some sufficiently small constant c, and
cutting the ball with the hyperplanes
P_i,s = {z in B_d : z_i = s c} for each index i in {1,...,d},
for each sign s in {-1,1}.
We arrange that all rotations are allowed that keep these pieces
inside the unit ball (perhaps enforcing this by attaching extensions
in the shape of a cube). It's then easy to see that a representative
2-face
{z in B_d : z_3 > c, z_4 > c, ..., z_d > c}
can be rotated by mapping
(z_1,z_2,z_3,...,zn) -> (z_1 cos th + z_2 sin th,
z_1 sin th + z_2 cos th,
z_3, z_4, ..., z_d).
That is a very surprising observation to me, too! Hands up anyone who
_isn't_ surprised!!
In fact, I think it may be worse (geometrically, if not
combinatorically). For instance, I expected that a cubical hyperface
of the 3^4 could only be turned by rotating the cube about an
orthogonal axis. But it looks to me like you could rotate the cube
around any axis you like. Maybe you have to eventually rotate it so
it occupies it's original space before you rotate a perpendicular
hyperface, but I'm still somewhat annoyed that you can put it in weird
orientations in between.
> I concluded that any mechanism that would restrict you to
> only rotating entire (d-1)-faces would require some sort of complex
> interlocking mechanism that would have to engage and disengage
> whenever a (d-1)-face was to be rotated.
> Has anyone else thought about this problem (d-cube mechanisms)
> enough to confirm or refute my conclusions?
What I haven't proven is that say, any decomposition of the ball into
3^d pieces that admit (d-1)-face rotations will also admit 2-face
rotations. If you've got that, it would pretty much support your
conclusion, but I don't know how to verify it off-hand.
Dan Hoey
Hoey@AIC.NRL.Navy.Mil