Message #548

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Re: Something interesting and strange about permutations
Date: Sun, 10 Aug 2008 19:41:45 -0700

MC2D may not be a proper analogy but it is not a misnomer because a
square is definitely a 2D cube. Or if you want to be completely strict,
only the 3D version is a cube but all of these puzzles are based on the
"measure polytope" in various dimensions.

Regarding rotations, I really don’t think that it is helpful to try to
think of N-D rotations as involving rotation axes. The fact that 3D
rotations are easily visualized as happening *about* an axis is really
just a quirk of three dimensions. A better way to think of rotations is
that they always occur *within* a 2D plane. In other words, while an
object moves under the influence of any single rotation in any number of
dimensions, any point of that object will move in a circular arc within
a single 2D plane. In 3 dimensions there will be a single rotation axis
that cuts through the centers of rotation of all those parallel planes
but in 4 dimensions there can be more than one axis that does that, so
try to forget about axes and just look for the planes of rotation.

Now as to the legitimacy of classifying MC2D with the other puzzles, it
all depends upon how we want to define these puzzles. We can choose to
allow mirror operations or not, and we can allow twists involving higher
dimensions or not. I don’t have a strong opinion on the best choice, and
I’m perfectly happy if there does appear to be a best choice which does
not allow a valid 2D puzzle. For me, the most interesting thing about
MC2D as implemented is that one can easily sketch the entire state graph
for the puzzle (8 states!) and thereby begin to get an idea of what the
topology of other similar puzzles might look like.

-Melinda

spel_werdz_rite wrote:
> Well, there’s a simple way to think of this that I believe I brought
> up to Roice about a few months ago. The axis of rotation of an a
> figure in N-space will always be composed of a segment of N - 2
> dimensions. To put this in visual terms, look at a cube (or any 3D
> solid) and rotate it. You can imagine a line about which the figure
> rotates. When you rotate a piece on MC4D, you will always see 3
> stickers that stay in place (maybe not orientation) through which a
> line could pass. The same holds true for any 2D solid. Rotate a dollar
> bill, or a face on a Rubik’s cube. There exists a unique point, as
> there did a unique line in 3D through which any part of matter in it
> stays stationary. Anybody familiar with vectors can imply this as curl
> *F* = *0*. Even if you look at a face on MC5D, you will see
> a 3x3 array of stickers that do not move. A 4D face’s movement has a
> 2D plane of rotation. But in the case of rotating a line, this would
> imply -1D axis, which, of course, does not exist! You may think of
> twisting it through 2-Space as done in MC2D, but this is analogous to
> turning a shirt inside out. This process is not allowed in out concept
> of "twisty puzzles." Thus, MC2D is really a non-existent puzzle, even
> a misnomer! Magic /Cube/ 2D!
> -Nelson G.