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 Post subject: Unique parity situations with Anisotropic/ Gear CubePosted: Fri Oct 08, 2010 2:40 am

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Some time ago, I buckled down and ordered the Anisotropic mod from Shapeways, along with an extra white Gear Cube as well as some "Fisher-Style" Cubesmith stickers. They arrived. For the Anisotropic edges, I used some old Cubesmith 4x4x4 stock to fill in the blank areas.

And I must say, it is a beautiful puzzle.

Well, I finally took the plunge. I scrambled the Anisotropic Cube. One feature that I have observed about this cube, is that the eight corners always have fixed orientation with respect to the freely-turning layer, as do the Anisotropic pieces always align themselves onto the same plane. The Anisotopic Cube can probably be solved as a subset of the Rubik's Cube, were it not for those pesky little gears which seem to defy the natural logic of twisty puzzles. After a fair amount of manipulation, I came to the shocking conclusion that it is possible to only twist one gear while the remaining seven stay in perfect alignment with their corresponding 3x3x3 edges. Have a look...
Attachment:

Anisotropic Gear parity - No Monkey Business.JPG [ 135.84 KiB | Viewed 1936 times ]

There is no monkey business going on here: Despite not photographing the backside of this cube, I can assure you that none of the other gears have been twisted with respect to their edges, nor are any of edges in a "flipped" state. All of the red or black either faces up or down. Naturally, I thought this to be a contradiction, but upon closer inspection, I learned why: Every time you do an "LR" or "FB" move, each of the four gears along the center slice parallel to the receive a twist of -5/6. This is equivalent to a +1/6 (clockwise) turn for each of the four gears compared to their mated edge, hence every quarter turn of the opposite faces yields a +4 overall gear parity. But unlike it's Gear Cube sibling, on the Anisotropic Cube, the edges are not strictly confined to groups of four and are allowed to be freely scrambled. Because the least common denominator of 4 and 6 is 2, it is only necessary that the total twist metric of all gears merely be even. And since "even" can include a twist of +2/6 or -2/6 (+ or - 1/3), it is fully possible to twist only one gear out of alignment

This also lead into some insight as to why the "flipped edge" parity shows up on a standard Gear Cube with "Fisher Style" stickers: For a set of gear to be rotated 180 degrees implies that their adjacent layers must be rotated to an odd turn metric. A half-twist is equal to 3/6 and 3 is an odd number. This "odd metric also dictates that a slice is one quarter turn out of alignment, which makes it possible for an edge to rest on the color of an adjacent face while the gear reflects either the matching or opposite color. Because each set of four edges operates in tandem, if one set is flipped, then another set must also be flipped. And while the gears appear to be correct in this "pseudo-solved" state, they actually have an odd twist-metric to them. There are three ways to select an en even number of items out of a superset of three, three groupings of two plus the solved state (no grouping), hence if you solve your Gear Cube without regards to the edge orientation, there is a 3/4 chance of hitting this parity with the "Fisher-style" sticker set

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 Post subject: Re: Unique parity situations with Anisotropic/ Gear CubePosted: Fri Oct 08, 2010 2:51 pm

Joined: Thu Sep 24, 2009 12:21 pm
Location: Chichester, England
Wow. I can't even see how this is possible. I think I'll be scared when I get mine...

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 Post subject: Re: Unique parity situations with Anisotropic/ Gear CubePosted: Mon Oct 11, 2010 6:29 pm

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
SEBUVER wrote:
Wow. I can't even see how this is possible. I think I'll be scared when I get mine...
It's due to a parity mismatch. Every time you rotate the outside layers (LR or FB) you perform a total of four gear twists. It requires 6 twists total to get a cog back to it's solved state, therefore by shuffling the edges around and then retwisting the cogs, virtually any combination of even gear parity can be achieved. The cog in the picture is rotated 1/3 revolution, which is a 2/6 turn and therefore even parity. However rotating a single cog by flipping it 1/2 turn (3/6), despite preserving the overall cubic shape, would be totally impossible, just like flipping one edge on a standard 3x3x3 is impossible.

I have no idea of what type or how long an algorithm would be necessary to individually rotate a single cog 1/3 turn whilst leaving the rest of the puzzle in tact, but it would probably not be an easy position to solve.

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 Post subject: Re: Unique parity situations with Anisotropic/ Gear CubePosted: Mon Mar 07, 2011 2:18 am

Joined: Sat Feb 12, 2011 8:32 pm
I've figured out the gear cube almost completely by myself until I ran into this problem. After trying my previous methods before and repeating steps over and over, I finally gave up with trying to deal with this parity.

I looked it up and its actually pretty simple to solve. It involves the algorithm:
(R2 U) x2 (R2' or R2) x4 (U' R2') x2

R2' or R2 is situational which relies on the orientation of your twisted edge. You have to repeat this algorithm three times, so it gets a little monotonous.

I got this information from redkb on youtube. His video tutorial showing how to fix the edge parity is here:

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