Message #3643

From: Christopher Locke <project.eutopia@gmail.com>
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
Date: Sat, 11 Feb 2017 19:00:13 -0800

To continue a bit of my previous line of thought. I considered how this
would work in 3D case: http://imgur.com/a/sUqEh

Essentially my conclusion is that in the 3D case being reduced to two 2D
squares, there is no rigid orientation of stickers that allow U, D, and
R moves to work using clean rotations of the physical 2D squares. You
can get it to allow 90 degree twists of U, D, and 180 degree twists of
F, B, R, L simply, but there is some necessary "cubie twisting" that is
required to implement F, B, R, L twists (e.g. to do a R twist, you need
to rotate the middle 4 square cubies around by 90 degrees, while also
twisting each cubie to mix up x- and z-face stickers). This is
analogous to the 4D case Melinda created, where 90 degree twists that
mix up red/blue (w-face stickers) with other stickers requires
simultaneous twisting of the cubies themselves too.

It might be possible if the stickers themselves were slightly
free-floating within the cubie, and somehow the internal orientation of
the stickers could be moved during twists. I illustrated this idea with
the 2D representation of the 3D case in the second image. I analyzed the
R twist, by breaking it down into the rigid 90 degree rotation, and the
necessary internal twist of stickers. If you split each square cubie
into 120 degree sticker "sectors", then when you do the R rotation, you
need to rotate the internal stickers also by +/- 75 degrees to keep the
stickers in the correct location.

These considerations imply to me that it is impossible to make a simple
2D representation of the 3D puzzle as two squares that has the same
simple 90 degree rotations possible as the original 3D puzzle. This can
only be done by having some very clever stickers that are "floating"
within each cubie and that rotate correctly to keep stickers in the
correct position during 90 degree twists. This could possibly be done
by having slightly rounded stickers that can roll over each other in
some nice way to allow both U or D twists, and the central R twist.
Note that since the internal twists are alternating between 75 and -75
degrees, a 180 degree twist is okay still, just like on the physical 4D
puzzle.

Best regards,
Chris