Message #4121

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
Date: Tue, 04 Sep 2018 12:40:39 -0700

Here’s the result of yesterday’s efforts (and a few hours this morning
making the video and writing the summary), as sent to the mailing list.
You probably shouldn’t watch the whole video - it’s pretty obscure.
Skim the message below, and sample a minute of the video, that’s my
suggestion.

https://youtu.be/e0aUuN0L1Dc

On 9/4/2018 12:00 PM, Marc Ringuette wrote:
> 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
>