Message #4120

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
Date: Tue, 04 Sep 2018 12:00:55 -0700

Hi gang!

I had a lively day yesterday exploring the nature of the 2x2x2x1 mini
puzzle, also known as the twisty stacky 2^3.   Thanks to Luna
particularly for a bunch of help with it!   Thanks also to Andy, Dylan,
and Joel.

"Cuboid, not so much."   After studying it for a couple of days, our
consensus is that this mini puzzle should probably NOT be called a
"2x2x2x1 cuboid".  That would be more misleading than helpful.  The 4D
version of it has the dimensions of the 2x2x2x1 cuboid, but the stacking
moves and/or the Gyro move, that provide access to the full 12
orientation states of the 4D pieces, do not seem to correspond to 4D
rotations that would apply naturally to generalized cuboid puzzles.  In
fact, I’ve stopped calling the Gyro for the puzzle "the FR rotation" or
even a rotation at all.   It moves and reorients pieces, and it holds
the F-B and R-L axes fixed, but unless we discover that the Gyro has a
natural geometric interpretation as a 4D rotation, I’ll quit calling it
a rotation altogether.

As Luna says about the mini puzzle, "I think it’s just a subset of the
2^4. A puzzle that’s obvious with our physical puzzle, but ultimately
random. It’s still interesting though."

I’ve made a reasonably nice emulation of the mini puzzle inside the MC4D
2^4 hypercube puzzle using a big pile of macros.   Hopefully that’ll
help us clarify further what the mini-puzzle is in relation to the full
puzzle and to other subsets of it.  Here’s a 10 minute side-by-side video.

  32   Side by side demo for 2x2x2x1 mini puzzle    10m14s
https://youtu.be/e0aUuN0L1Dc

I’ve attached my MC4D macros file to this mail.   Anybody with MC4D can
play with the virtual mini-puzzle.   I’ve scrambled and solved the
puzzle using the macro buttons, and it’s not too horrible to use.    The
handles for the macros are my usual ones, with the mnemonic "IFUR FUR"
(find the IFUR sticker, and click its F U R subfaces).   Just go into
macro creation mode using those handles, then click freely on the
buttons.   Sliding the "Eye W Scale" knob to the right will make the I-O
axis skinnier as in my video.

All later macro buttons can be composed from sequences of the first 7
(the twists and the Gyro).  Non-obvious algorithms used:
   Gyro(phys) = M S R2 F2 R2 y’ z2    (from Luna)
   Gyro(virt macro using 2^4 moves) = RFro UF2 IF2 UF2   (from Luna)
   M U M’ (virt macro using prev macros) = x Gyro L2 F2 L x’ Gyro B2 L2
x   (from Marc)
   Monoflip RUF by x2 =  y’ S R S’ R U2 R’ U’ R U’ R’ U2 S R’ S y R U
R’ U R U2 R’ U2  (from Marc, applies to both real & virt 2x2x2x1)

Probably we’ll end up considering the 2x2x2x1 mini puzzle to be a fun
oddity:  a somewhat arbitrarily defined subset puzzle that’s quick and
easy to try if you already have Melinda’s 2x2x2x2.

Cheers
Marc