# Message #4113

From: Marc Ringuette <ringuette@solarmirror.com>

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.

Sweeeeet.

–Marc