Message #40

From: ojcit <oscar@its.caltech.edu>
Subject: Re: Orientations of the centre cubes
Date: Tue, 09 Sep 2003 09:54:34 -0000

> This change is not fundamental at all.

"Fundamental" isn’t the best word for what I think I was trying to
get at. I shouldn’t have reacted without reading the whole thread.
All I was trying to say was that assigning a stringent orientation
requirement is a change to the goal of the game, whereas extending
the cube to four dimensions is a generalization of the same game.

I’m 0-1 with math analogies so far, but I’m feeling lucky so here’s
another one: If I’m defining an algebra that works on, say,
matrices, I have to define, among other things, a notion of
equality. I can define the equality test for any nxn matrix
system. If I am accustomed to using matrices in R3x3, and I switch
to R4x4, I expect my equality concept to survive the migration.
What I meant when I said that the orientation stipulation was
a "fundamental" change was that it would be like redefining the
concept of equality for my matrix system, and that change requires a
whole different algebra.

> Jay’s attempted analogy is just a particular case
> of the very common situation (like x**2 = 1) in which
> an equation admits multiple solutions. So what?

What I was trying to illustrate was a simple example of the same
solution state being described in multiple ways, leading to an
invertibility problem. It really doesn’t have much relevance, but
the fact that it’s a common problem doesn’t make it an trivial one
to deal with. That said, everyone here seems more than equipped to
deal with it, but I find it fascinating. It’s the first time I’ve
gotten to think about complex analysis and stuff in a while, so I
got kinda excited. I wonder if the center sticker is a branch
point…

> >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.
>
> Huh?

By neighbor, I meant adjacent sticker, but now that I think about
it, I’m wrong. For the center and edge cubies of each face,
matching colors on only two adjacent stickers is sufficient to
ensure that the cubie is in the proper orientation, but for the
corners, you have to look at the diagonal neighbors. Since
neighborhoods don’t stay put, I’ll refrain from using them as a
state descriptor in the future.

Also, you’re very right about the need for orientation of the center
cubie if you’re going to map a surface on to the faces, but, like
you said, it’s a different puzzle.