Hyperbolic Hopf Fibrations The Hopf Fibration of S^3 is amazing and beautiful. Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson. nilesjohnson.net/hopf.html To understand it better (and fibrations in general), I recommend this talk by Niles too. www.youtube.com/watch?v=QXDQsmL-8Us It turns out there is an analogue of the Hopf fibration for H^3. In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case. There is a 2-parameter family of them, plus one additional fibration that does not fit the family. As with S^3, fibers in the H^3 cases are geodesics. They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration. I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this. www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf I'm posting three early pictures of H^3 fibrations below, labeled
The conjecture makes sense. Since the state is completely symmetrical, any move is equivalent (in the quarter-turn metric) to whichever move starts the solving process, and in the half-turn metric at least does not increase the distance. So the superflip is at least a local maximum.
ReplyDeleteThat the superflip is the only center in these cases also makes sense, since changing the position of any pieces can't be in the center, since you have arbitrary 3-cycles, and any rotation would have to be applied to every single piece of the same type in the same way to deal with position-permutations, but "the same way" is only meaningful for Z/2Z; the other orientation groups have automorphisms that can be realized via twisting (in the Rubik's sense, but also the bundle topology sense)
Thank you for the insights Layra Idarani!
ReplyDeleteI guess instead of "quarter-turn" and "half-turn" we need new descriptors. I guess "quarter-turn" means "minimal nontrivial turn on a single (hyper)-layer" and "half-turn" means "whatever turn on a single (hyper)-layer".
ReplyDeleteI'm assuming in the 3^4 case that we're looking at moves that along axes that go through a face of a cubical hyper-face, since in the case of at least some 4d Rubik's cube programs you have rotations around axes that go through the edges or corners of a hyper-face as single moves. If those are considered single moves, then my argument for the superflip being maximally scrambled falls apart since there are non-equivalent moves available.
Layra Idarani, yeah, we could think of two metrics, one allowing only minimal moves and the other allowing any reorientation of a face as a single move.
ReplyDeleteThat's right on the 3^4. The entire group can be constructed from minimal 90 degree twists, which can be composed into 120 degree rotations and two kinds of 180 degree rotations.
I suspect the superflips are maximally scrambled under either metric, but I have to confess I don't even know if that is true for Rubik's cube (and it must be known in that case).
The Superflip on a 3^3 is globally maximally scrambled in the half-turn (arbitrary-turn) metric, but not in the quarter-turn (minimal-turn) metric. The Superflip+four centers-swapped-with-opposites is globally maximally scrambled in the quarter-turn (minimal-turn) metric.
ReplyDeleteSurprising!
ReplyDeleteSo, for details, the 20 steps (half-turn metric) algorithm that I know for the superflip starts with a quarter-turn and ends with a half-turn (or vice-versa) and includes 8 half-turns in total. There is a 24 quarter-turn move algorithm, but I don't know what it is.
ReplyDeleteHere is a 24 move quarter-turn sequence (counting central slice twists as one). The video is showing how to use the superflip for a magic trick. https://youtu.be/LByJjGg-Kh8
ReplyDelete