Message #1047

From: matthewsheerin <damienturtle@hotmail.co.uk>
Subject: [MC4D] Re: definition of a twist
Date: Mon, 19 Jul 2010 13:39:30 -0000

I think I see how these twists are sort of possible, since nothing overlaps during the rotations. I’m not sure whether a person in 4D with a physical n^4 would be able to perform these twists though. I think it would take someone who knows more theory about 4D then I do to answer that one. I also think it could be interesting to implement, especially since it might gave beginners an easier stepping stone into solving 4D puzzles.
To go into more detail about the A1 twists (I was waiting for clearer screenshots of the 5D cases, but more on those later), I’ll give a quick overview of parity restrictions. I’m sure these are in previous posts somewhere anyway, but just to mention them while they are relevant again. Permutation of 4C pieces must be even (a 90 degree face twist performs 2 4-cycles, and edge and corner twists can be constructed from these so parity must be the same for those too). Permutation of 2C and 3C pieces is linked, just like in 3D. Either both permutations are even, or both are odd. This is true (I’m sure) in all dimensions. 2C and 3C parities are linked, and pieces with more colours have even parity.
I’ve looked at the new 5D cases. Both are possible in the current interface, and I’ve already uploaded the log files. The first one I constructed the same way as B2. The second one built on the first one, and the setup moves for the commutators to make the position were a little tricky and took a little while to sort out, though I could tell it was possible.
While on the discussion of twists, I feel that it’s worth mentioning the old problem in MC4D. A face can be moved to most its possible positions in one move, but the three positions reached by two 90 degree twists require two moves. I’m sure it has been discussed before but a quick look didn’t find it.

Matt

— In 4D_Cubing@yahoogroups.com, Andrew James Gould <agould@…> wrote:
>
> Hello,
> Actually Klaus, I don’t see why they wouldn’t be possible on a 4D cube in a 4D space. That’s why I got into the conversations with Melinda and Roice. Any specific issues you see? In 3D, n = 3 so there’s only one choice when twisting anywhere from a 2 up to an n-1 dimensional section, but in upper dimensions, 2 < n-1 so there’s a choice. During these twists, no stickers nor cubies run into eachother nor do they overlap (I can provide detail here if that appears to be the issue).
>
> I wouldn’t want to mess up your leaderboards. I personally think your leaderboards are a more distinguishing list since only allowing n-1 dimensional twists is more restrictive. If the new twists were added, I think the way to go is separate leaderboards: only allowing n-1 dimensional twists vs. 2 up to n-1 dimensional twists.
>
> Matt, very neat that A1 is not possible, but that B1, A2, and B2 are. I didn’t imagine that being a possible set of outcomes, but it makes clear sense to me when you described an even number of A1 twists would be possible. (I’ll get caught up with the lingo–i.e. 4C pieces.) Sorry about the change in notation, it changed when I took the conversation from Melinda to Roice. In the 5D pics, the notation went: each new character (2, b, and ii) meant another restriction to -1/2 < variable < 1/2. I don’t know how the pic order got messed up, I clicked the MC4D pics first.
>
> Anyway, I uploaded the 5D pics again but without the faces that aren’t of mixed color after the twist–less obscuring and with labels: 5D_2D_2bii and 5D_3D_2b.
>
> –
> Andy