Message #50

From: rbreiten <rbreiten@yahoo.com>
Subject: Re: [MC4D] Orientations of the centre cubes …
Date: Sat, 01 Nov 2003 20:40:01 -0000

Sorry about resurrecting the thread; I didn’t check the group in a
while and missed out on this interesting discussion.

I am pretty familiar with the center face orientation supergroup in
three dimensions (occasionally I amuse myself by solving my 5^3 cube
corners and edges first, and leaving the 3x3 centers for last. Some
3-d analogs of the commutator Mark described in message 39 can be
useful here).

I’m a little sad that I didn’t think to consider the supergroup in
four dimensions; that the 2-color cubies contribute to it was a
little surprising at first. I expect there’s a codimension two issue
here.

Regarding the edge cubies in 4^3 and 5^3 cases Mark discusses below,
those constraints are ultimately algebraic, not physical. They
follow from the permutation rules of the cube. If a physical cube
were designed so that swapping an edge pair (in 4^3) without flipping
them or swapping the central edge (in 5^3) with a non-central edge
were possible, doing so seems to me exactly analogous to popping out
a corner and putting it back twisted.

rb

— In 4D_Cubing@yahoogroups.com, "mahdeltaphi" <mark.hennings@n…>
wrote:

> […]
>
> If we consider the 4x4x4 cube, it turns out the the 2-face cubies
> cannot have invisible orientation problems. If they
> are both correctly positioned as to colour alone, then they are
> either both correctly oriented or else they are both
> incorrectly oriented in a visible manner. The way in which the
4x4x4
> cube was constructed was to have a central
> sphere, with grooves along the octants. The 1-face cubies slid
along
> those grooves, carrying the 2- and 3-face cubies
> with them (just about - the cube tended to explode if roughly
> handled!). The fact that the 2- and 3-face cubies had
> to slide around a central sphere meant that they had curved backs,
> and the nature of those curves made it impossible
> for the 2-face cubies to exhibit invisible orientation problems.
> There were, however, invisible orientation problems
> for the 1-face cubies. The four such cubies for each 4x4 face of
the
> cube could be permuted. Once a cubie was in a
> particular position, however, its orientation was fixed (one corner
> of the cubie would always point to the centre of
> the 4x4 face). It was reasonably simple, then, to mark the 1-face
> cubies in such a manner to force a unique solution.
> Although the arguments given above for these limitations to the
> possible orientations are mechanical in nature, the
> device’s mechanism did permit all possible cube rotations, and so I
> would regard these limitations as theoretically
> justifiable as well.
>
> […]