# 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