# Message #549

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: Re: [MC4D] Something interesting and strange about permutations

Date: Mon, 11 Aug 2008 17:16:57 -0500

On Sunday, August 10, "Melinda Green" <melinda@superliminal.com> wrote:

>MC2D may not be a proper analogy but it is not a

>misnomer because a square is definitely a 2D cube.

I would agree that, in the context of multidimensional

puzzles, a square for MC2D is every bit as much a cube

(2-cube, in this case) as is the tesseract (4-cube) in

the MC4D case. So, in saying "misnomer", with

emphasis on "cube", I think Nelson got a bit carried

away. However, and more importantly, he correctly

observed that, unless you allow reflecting twists, the

‘puzzle’ does not twist at all. (So Nelson’s

"misnomer" could probably be taken appropriately with

respect to "magic".)

The reference to "equivalent 2D puzzle" regarding MC2D

on the Superliminal MC4D page is misleading. In the

description for MC2D, it needs to be made clear that,

to permit it to be at all interesting, mirroring

twists are allowed and that this does violate the

analogy with MC4D and the regular 3D puzzle. Melinda,

if you fix the description, this would also be a good

point to mention that MC3D also allows the mirroring

twists extension in the context of the 3D puzzle. In

either case (and in higher dimensions as well), it

could be argued that reflections can be thought of as

being achieved by embedding the n-dimensional puzzle

in (n+1)-dimensional space, so that reflection can be

achieved by performing a 180 degree rotation for which

the extra spatial axis is in the plane of rotation.

>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.

I don’t think that there is an important distinction

to be made here. The (n-2)-dimensional subspace

orthogonal to the plane of rotation is also called the

"fixed space" for the rotation. In 3D, it is just a

line. In 4D, the fixed space for a rotation is a 2D

subspace. Defining a rotation in terms of its fixed

space or its plane of rotation are essentially

equivalent, since the two subspaces are always related

by orthogonality. When Nelson wrote, "The axis of

rotation of a figure in N-space will always be

composed of a segment of N - 2 dimensions.", his "axis

of rotation" would more appropriately be referred to

as the "fixed space for the rotation" and his

"segment" would more appropriately be referred to as

"subspace" or "hyperplane".

>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.

I think this is poor advice. It is often very useful

to be able think about a rotation in terms of its

fixed space. Indeed, the reasoning for what to use

for a rotation often involves thinking about the

aspects of state that you do NOT want to change.

I.e., it is a constraint on the fixed space that

may motivate the choice of rotation plane.

>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.

There does not have to be just one choice; and, in the

presence of multiple possibilities, there is no need

to evaluate any choice as "best". E.g., the 3D puzzle

is interesting whether or not you allow reflecting

twists. Why try to exclude a choice? Regarding

mirroring, only one choice makes sense in the 2D case;

but, for a higher dimension, in addition to the

non-mirroring twists normally considered, one can also

consider a variation which permits reflections. (You

could argue that making this sort of choice is

analogous to whether or not one regards the

orientation of face stickers to be relevant for the 3D

puzzle. By making a different choice, you create a

somewhat different 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.

Where does this "8 states!" come from? Orientation of

a corner 2-cubie depends only on its position.

However, the 4 corner 2-cubies can be permuted in all

24 different ways. In what sense can 3 different

permutations all be regarded as the same state? I had

pointed out the apparent discrepancy in Melinda’s

analysis in somewhat greater detail a couple years

ago:

http://games.groups.yahoo.com/group/4D_Cubing/message/330

Since Melinda is now repeating the dubious claim, I

wonder if she ever saw my old message replying to

hers. In that old message, I also touched on some of

the other issues which have arisen again in the

current discussion as well as some other issues which

have not rearisen (yet).

Regards,

David V.