Message #51

From: mahdeltaphi <mark.hennings@ntlworld.com>
Subject: Re: [MC4D] Orientations of the centre cubes …
Date: Mon, 03 Nov 2003 00:21:35 -0000

I fully agree that the fundamental constraints I discussed are
algebraic rather than mechanical. That the cubes can be constructed
physically merely (! - apologies to the engineers who worked out how
to do it) represents the fact that the algebra can be implemented in
a concrete form. However, the mechanical constraints in the 4^3 case
are such as to provide a ready heuristic proof, without the need to
set out the full algebraic proof of the problem.

My main interest in raising this thread in the first instance was
that the 3^4 cube represented a case where the cube could not be
represented physically, but only through computer simulation, and
the MC4D visualisation of the cube did not allow for ready
consideration of the problems involved.

DV has sent me an alternative visualisation of the 3^4 cube which
does permit such consideration - I just have not had time to get to
grips with the problem recently!

Mark

— In 4D_Cubing@yahoogroups.com, "rbreiten" <rbreiten@y…> wrote:
> 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.
> >
> > […]