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 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.

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