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...
Anisotropic Gear parity - No Monkey Business.JPG [ 135.84 KiB | Viewed 1939 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
My Creepy 3D Rubik's Cube Video
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.