# Message #4117

From: Andrew Farkas <ajfarkas12@gmail.com>

Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid

Date: Mon, 03 Sep 2018 17:26:22 -0400

Hey all!

I’m *very* tentative to call the twisty-stacky a 2x2x2x1. I’ve already

half-written and deleted multiple emails trying to work through the

behavior of a 4D cuboid, but I could never come to a concrete conclusion.

Here is one line of thinking I ran down:

Some observations about the NxNx1, with the Z axis as the short axis:

- The puzzle has 2D cuts along the XZ and YZ planes.

- 180-degree moves are possible in two planes: XZ and YZ.
- Moves are not possible in the XY plane, because there are no cuts

in that plane. - In a 3x3x3 emulating a 3x3x1, pieces in the E layer never mingle

with pieces in the U and D layers.

- 180-degree moves are possible in two planes: XZ and YZ.

The 2^4 has 2D cuts alo– well no, they would be 3D cuts along each of the

… 12? 24 possible 3D slices? But how does that correspond to which moves

are allowed?

Let me try another one:

To use a 2x2x2 like a 2x2x1, use only U/D-invariant moves in a plane

containing the U/D (short) axis.

This looks like something we can work with

. To turn an N^4 into an (N^3)x1, use only I/O-invariant moves in a plane

containing the I/O axis. For the (virtual) 2^4 and 3^4, that means moves

like Ru2, Rf2, and its symmetries on the other faces. Doing these on a

virtual puzzle certainly seems believable. But what about a move like Ro?

It’s not in a plane containing the I/O axis, but is there any sure property

of (N^M)x1 puzzles that it broke?

The more I think about this, the more confused I become.

- Andy

On Mon, Sep 3, 2018 at 2:36 PM Marc Ringuette ringuette@solarmirror.com

[4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:

>

>

> CORRECTION – I mixed up some axes, so the best part of my last message

> ended up quite scrambled. I got all confused because the corner axes

> of the regular and mini puzzle are named differently (R-L versus I-O).

> Here’s a better version of the comparison.

>

> For the 2x2x2x2: the short exchange of the U-D and corner (R-L) axes

> is Iy Oy’, while the pure rotation, FOro, is Iy Oy’ Rx2 Bz2 Uy2 Rx2..

> The short half-exchange is Iy.

> For the 2x2x2x1: the short exchange of the U-D and corner (I-O) axes

> is M y M, while the pure rotation, FRro, is M y M R2 F2 R2 z2. The

> short half-exchange is M U M.

>

> Note that I’m using a tweaked version of Luna’s gyro for the 2x2x2x1,

> FRro, where I put the y rotation in between the restacks. This version

> draws out the parallels between the 2x2x2x1 FRro and my favorite gyro

> for the 2x2x2x2, ROIL FOro. 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.

>

> Whatever the details, the parallel between these is still sweeeet.

>

> –Marc

>

> On 9/3/2018 10:22 AM, Marc Ringuette ringuette@solarmirror.com

> [4D_Cubing] wrote:

> > Here’s a different gyro for the 2x2x2x1, FUro, that corresponds

> > (CORRECTION, NOT REALLY) to the mini version of my favorite gyro, the

> > ROIL version of the FUro (CORRECTION, FOro) 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 (CORRECTION,

> > U-D and L-R) axes is Iy Oy’, while the pure rotation, FUro

> > (CORRECTION, FOro), 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.

>

>

>

"Machines take me by surprise with great frequency." - Alan Turing