Message #3642

From: Christopher Locke <project.eutopia@gmail.com>
Subject: Re: [MC4D] Re: Physical 4D puzzle achieved
Date: Sat, 11 Feb 2017 01:53:08 -0800

I figured out some more possible twists. See my attached images.

The first is a labelling of the 2x2x2x2 cube in MagicCube4D. The -w
face is central, surrounded by the +/- x, y, z faces with +w hidden.
The second shows the axis definitions of the physical cube. The
leftmost 8 cubies those part of the -w face, and the rightmost part of
the +w face.

If we label a simple 90 degree twist by first specifying the face, then
the axis to rotate about (e.g. -w:+z rotates the central -w block around
the +z axis), then we can investigate some twists.

• Any twist of the form +/-w:+/-(x,y,z) is done on the physical puzzle
  by rotating the +/-w block (left or right 2x2x2 cube) about the
  respective x,y,z axis. This is the easiest twist to make. Essentially
  you can detach and freely rotate the left or right blocks any way you
  desire, and it is a valid twist.

• Any twist of the form +/-(x,y,z):+/-w can also be done relatively
  easily. For instance, the +z:+w twist involves doing U’ on left block
  and U on right block. +y:+w is R on left block and L’ on right block.
  All possibilities involve twisting one block with some normal Rubik’s
  cube rotation, and the mirrored twist on the other block.

• Putting these two moves together, and you are allowed to do any
  rotation of the left and right blocks, or do a single 90 degree regular
  Rubik’s cube twist to both left and right simultaneously (do not need to
  do the exact mirror, because w:(x,y,z) twists can put left/right blocks
  in any orientation). So from this point of view, it is like solving two
  2^3 cubes by doing any even number of 90 degree twists to each block, or
  an odd number to each block (can always do then undo a move on one side,
  so it is the parity of the number of twists that must remain
  unchanged). This is the state of the puzzle without being able to mix
  up red/blue (+/-w face) stickers with the other stickers.

To make the puzzle more 4D, you need to be able to mix up w and (x,y,z)
stickers, and this can only be done using twists that involve only x,y,z
axes. However, if you can figure out how one such twist works, then
reorientations of the physical puzzle (rotate the left/right physical
block any way, while doing the mirror rotation to the other block) can
put it into an orientation to get any twist you desire. Let’s focus on
the +x:+z twist.

This twist moves the frontmost 8 pieces of the physical puzzle to the
left, cyclically (so those on the left are brought all the way to the
right). One 90 degree +x:+z twist of the 4D puzzle involves moving
these physical pieces to the left, while also rotating each cubie 180
degrees. The 180 degree twist is about a diagonal line in the "x-z"
line, as drawn roughly at the bottom of the 2nd figure. The bottom
figure shows the rotation for one of the cubies in the top half. The
cubies in the bottom have the same 180 degree rotation, but mirrored
along the x-y plane that splits top/bottom. This mixes up w-face and
y-face stickers.

All such "difficult" twists I think are accomplished by moving the
respective 8 physical cubies around cyclically, while also doing some
180 degree twist of the individual cubies. I imagine these would be
fairly annoying to do on the physical cube, so it would be worth
investigating if some of those "inversion" moves Melinda showed can be
combined with simpler moves in a commutator/conjugation to give an
equivalent x,y,z twist.

Best regards,
Chris