Message #593

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Something interesting and strange about permutations
Date: Sat, 27 Sep 2008 14:58:29 -0500

Wow, lots of good discussion and ideas here! I blink and I’ve fallen
behind! You guys have worked through everything so well, and so I only have
a few comments with which I will participate.

>>Let’s start
>>with a single reflection move. These can be
>>reflections about a point, line, or any space of
>>dimension lower than the puzzle itself.

>Not exactly. Any reflection will occur with respect
>to an (n-1)-dimensional reflection hyperplane. In
>that (n-1)-dimensional plane there can be embedded a
>hyperplane of lower dimension which one may wish to be
>included in the reflection plane. But a hyperplane of
>dimension less than n-1 does not uniquely determine a
>reflection. It does determine a family of possible
>reflection planes which include it.

Melinda’s thought of allowing reflections across any subspace of
intermediate dimension is interesting and worthy of further study! I had
only previously considered reflections across hyperplanes of dimension n-1
only as you describe, but the behavior of the other cases also appears to be
uniquely determined. Consider point
reflections<http://en.wikipedia.org/wiki/Point_reflection>,
which are easy to think about in any dimension (and only
orientation-reversing in odd dimensions).

However, I do have the feeling (no proof unfortunately) that the n-1
hyperplane reflections are the most fundamental operations. For example,
point reflections in 2D and 3D results in symmetries that can instead be
built up from reflections across lines and planes, respectively. But the
reverse isn’t true, e.g. in 2D you can’t make a reflection across a line by
composing point reflections (since it is just a 180 degree rotation).
Furthermore, you can’t express 2D rotations as a pair of point reflections,
so the "rotations as pairs of reflections" property clearly doesn’t extend
to all portions of this larger class of reflections.

Anyway, my intuition says if we limit ourselves to reflections across
hyperplanes as you’ve described, we can reach all mirror puzzle
permutations, just like we can reach all permutations in the rotation-only
puzzles when limiting rotations to being coordinate-axis aligned. But it
could be visually interesting to somehow support these other reflection
types, just as the edge and corner twists in MC4D bring a lot to that
puzzle.

Btw, one suggestion for naming the puzzle types we’ve laid out would follow
the language of topology: orientable or nonorientable, the latter being the
case where an odd number of reflection moves was allowed.

>>Even if all rotations can be expressed
>>as pairs of reflections, it might not follow that all
>>pairs of reflections can be expressed as rotations,

>But they can be. One thing that needs to be mentioned
>is that the reflection planes cannot be chosen
>arbitrarily. They must be such that the puzzle
>transforms onto itself in the same space. Given that
>restriction, it follows that the result of two
>reflections must be a non-reflecting reorientation.
>(What else could it be?)

Yep, this is correct. As another aside, without the "transform onto itself"
restriction, compositions of two reflections can result in rotations or
translations (or the identity), depending on whether the two reflection
planes intersect or not (distinct, parallel reflection planes result in
translations). We would limit our reflection planes to going through the
origin of a hyperface because we don’t want translations of faces to occur
during moves (also, think about what a single reflection would do to a
face if this was not the case), and with this limitation all pairs of
reflections will equivalently be a rotation for us.

One last thought. It does seem using reflections as a talking point for
discussing face "moves" has been very useful, since they are a more
fundamental motion than rotations. Coxeter
groups<http://en.wikipedia.org/wiki/Coxeter_group>,
which describe all the symmetries of regular polytopes among other
things, are "abstract groups that admit a formal description in terms of
mirror symmetries". I think it is natural we have been led in this
direction…

All the best,
Roice