Message #1053

From: matthewsheerin <damienturtle@hotmail.co.uk>
Subject: [MC4D] Re: definition of a twist
Date: Tue, 20 Jul 2010 21:50:15 -0000

Interesting construction, which seems to show that these twists are indeed possible. Following on from this idea, I think I can clarify for the other twists too. The construction you gave describes (unless I have got mixed up somewhere) and A1 twist: essentially taking one 3x3x3 face and turning one face. It follows that if you can rotate one face clockwise and the opposite face anticlockwise, you have the essence of turning a middle layer of the 3x3x3 face. In other words, if all outside layers can rotate by themselves, the inner ones can too, which gives a B1 twist. Since a B2 twist can be constructed from two B1 twists and two normal face twists, and none of these seem to overlap, it stand to reason that they can somehow be performed at the same time, which gives a B2 twist. I’m still not 100% convinced personally, only because I have difficulty thinking of a 4D hand interacting with a physical 4D cube. A question I can’t help but think of, is one of mechanism. If a 4D person made a 4D cube, which of these twists would be standard? Would these extra twists require a more sophisticated mechanism, or would they be inherently possible? I know that this is slightly off the point, not necessary to have answered, and I have no idea how to answer it anyway, but I decided to put the question out here to see what reaction it gets.

In response to Melinda: just to say that I would also consider the 37 twists as 1. I would consider moving any face (and/or any layer(s) parallel to that face) from any valid position of that face, to any other valid position of that face. To clarify, if more than one layer is moved, then all the layers must move in the same way, ie. doing R L on a 3x3x3 would be 2 twists since layers are going in different directions. Obviously this is an ambiguous issue as it is possible to make a valid argument for more than one system, but that is the most natural definition for me.

Matt

— In 4D_Cubing@yahoogroups.com, "Andrey" <andreyastrelin@…> wrote:
>
> If we consider two sides of 4d cube then we can rotate each of them around the axis that goes in direction of another side (click of two stickers of the same 2C in MC4D). In both cases block 3x3x1x1 will be rotated around its axis but it will to that as a part of different 3^3 sides - so it can freely twist relative to both of them! Looks like it’s really possible to twist this 2D facet in any construction of 3^4. It’s more difficult to imagine the rotation of middle layer of 3^3, but probably it’s possible as well.
> But I think that without such twists puzzle it a little more interesting )))
>
> Andrey
>