Message #54

From: rbreiten <rbreiten@yahoo.com>
Subject: Re: [MC4D] Orientations of the centre cubes
Date: Fri, 21 Nov 2003 23:52:26 -0000

— In 4D_Cubing@yahoogroups.com, "ojcit" <oscar@i…> wrote:

> By making that change, you’re causing a fundamental change in the
> underlying rules of the puzzle, perhaps almost as radical as
> extending from 3 to 4 dimensions. The fact that a center face on
> the original cube can be rotated is a necessary flaw, but one that
> I believe the state counting already takes into account. As the
> system configurations are defined traditionally, there is still
> only one solution. It’s a similar problem to the one that comes
> up in inverse trigonometry all the time (e.g. arcsin(1) = pi/2 +
> k*pi, for any integer k. Although the k=0 solution is more
> pleasing, it’s no more valid than any of the others.) So I guess
> my point is that if you want to differentiate the orientations of
> the "fixed" faces, you’re altering, not merely clarifying, the
> rules. In the original cube, the correct orientation of any piece
> is defined by its neighbors, not by the configuration it comes
> from in the factory.

I find this argument rather bizarre. The underlying rules of the
cube are that you can rotate its faces and scramble it up. The goal
of the cube as a puzzle is to find a method of restoring it to its
original configuration by following those rules. It is a fact that
the orientation of a 1-color cubie on a standard physical cube exists
but is not easily apparent. One may define "original configuration"
to include, or not, this extra information (whether tracked by notes
on paper or marks on the cube). In my opinion the phrase "necessary
flaw" fails to refer. The trig analogy I also find unhelpful. We
aren’t choosing a canonical element from a set of solutions, we’re
considering information inherent in the definition of an object that
has been modded away in a particular model of that object. It seems
strange to me that this could be viewed as somehow arbitrary or evil.

> Also, from an aesthetics standpoint, one of the most pleasing
> aspects a solved 3x3x3 cube is the fact that each face is a solid
> color, with 9 identical squares. I think the elegance would suffer
> if you mark the center square.

This I think is a reasonable position, but I don’t have any strong
affinity toward it. My cubes are all abstract mathematical objects
somewhere up in Platonic Heaven. Some of them happen to be
incompletely modeled by physical objects (or computer programs). I
guess I could argue, if forced, that this incompleteness is an
aesthetic detraction or a necessary flaw, but as Mark says, the
quotient groups are interesting enough….