Message #4113

From: Marc Ringuette <>
Subject: 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
Date: Mon, 03 Sep 2018 10:22:56 -0700

Hi gang,

It’s becoming clearer to me that the mini-puzzle does indeed correspond
to a 4D virtual puzzle, still not precisely defined, that is some flavor
of 2x2x2x1 cuboid.  As you’ll see by my discussion of gyro rotations
below, the connection is looking pretty deep.    In fact, I started
messing around with the MC4D 2^4 hypercube puzzle (thinking of various
colorings, projections, and allowed moves) in the hopes that I could
figure out what 4D puzzle the mini-puzzle emulates.  Can anyone see what
it is yet?  I’m not quite there.

Gambling that this correspondence will pay off … I hereby name the
mini-puzzle the 2x2x2x1  (or depending on context, "Melinda’s 2x2x2x1"
or "the physical 2x2x2x1", but please never "Marc’s 2x2x2x1" because I
prefer to stay under the radar, name-wise).

My latest bit of insight was prompted by Luna’s discussion of the face
swap versus the pure face swap:

> [ I will call M S the face swap.]   The pure face swap is M S R2 F2 R2
> Y’.        ~Luna

The pure face swap that Luna describes looks just like a gyro rotation
for the 2x2x2x1 !  (Actually, I might propose that a Z2 rotation be
added to the end of his pure face swap, making it  M S R2 F2 R2 Y’ Z2 ,
so that both the F-B and the L-R face pairs are completely unchanged.  
This is the 4 dimensional "FR rotation", holding the F-B and R-L axes
fixed while rotating the other two. So, this modified gyro for the
2x2x2x1 would sensibly be called FRro.

Here’s a different gyro for the 2x2x2x1, FUro, that corresponds to the
mini version of my favorite gyro, the ROIL version of the FUro gyro. 
The connection is deep enough that there are even three versions of each
rotation:  the short one on the physical puzzle, the longer one with
cleanup moves to correspond to a one-click MC4D rotation, and the short
half-exchange, that re-aligns only half of the puzzle corners.

For the 2x2x2x2:    the short exchange of the L-R and I-O axes is Iy
Oy’, while the pure rotation, FUro, is Iy Oy’ Rx2 BO2 UO2 Rx2.   The
short half-exchange is Iy.
For the 2x2x2x1:    the short exchange of the L-R and I-O axes is E X E,
while the pure rotation, FUro, is  E X E F2 U2 F2 Y2.  The short
half-exchange is E R E.