# Message #134

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

Subject: Re: [MC4D] "Physical models" of Rubik’s Cube (tesseract, hypercube, etc.)

Date: Tue, 19 Apr 2005 09:32:11 -0500

On Wednesday, April 13, "Nathanael Berglund" <berglund@math.gatech.edu> wrote:

>If you’ve never taken apart a Rubik’s Cube, I

>recommend you find an old well worn one and try it.

How to do it: Rotate a face, say the top one, 45

degrees. You can then pry up the edge pieces of the

top face, popping them out without damaging anything.

(You can even do this on a new puzzle.)

>It’s great fun! If you do so you will notice that

>the pieces have some surfaces which fit together to

>form a sphere (O.K., so it’s not exactly a sphere.

I would say that the pieces have interior edges (one

dimensional) which lie on (or near) the surface of a

sphere. It is those edges behind which hook the

‘blocks’ extending inwards from the corner and edge

pieces to hold them in. Only small portions of the

sphere are realized, but that spherical shape is what

permits the move flexibility.

>The center piece consists of three mutually

>perpendicular cylinders to which the six center

>pieces attach. Seeing the cube taken apart, it

>becomes quite clear why the center pieces don’t ever

>change position. How these cylinders stay attached

>is a black box to me, since I don’t think I can take

>that part apart without breaking it (if anyone knows

>how this actually works, I would be interested in

>hearing about it).

I recall seeing a cube with stickers removed, and

there was a screw in the middle of each face piece.

(But not all cubes are made the same way.)

Inside each cylinder, there is a spring which pulls

the face piece towards the center, with the cylinder

stopping it at the right distance. During turning,

the face piece just slides rotationally on the end of

the cylinder. I think it would be difficult to make

it work without the bit of slop that those springs

allow for. I do not know which end of the spring has

the pivot; but I see no difficulty accomplishing such

a pivot, and I do not see why you would need more than

one cylinder. Some cubes make creaking noises when

you twist them because of the springs dragging inside.

>As a mathematician, of course I want to find an

>explicit formalization of this model.

I infer that the point of Nate’s message is that he

claims to have achieved this objective. However, I

must say that I remain confused about just what has

been achieved. As far as I can tell, Nate has come up

with a way of mathematically defining some 3-space

point sets whose shapes (for corners and edges)

correspond well to the shapes actually invented by

Rubik. Because they are based on the internal sphere,

they permit the required motions. But I think I must

still be missing something here, because I wind up

saying to myself, "So what?" Clearly a real physical

model can exist, because Rubik’s Cubes do work! The

fact that the shapes of the pieces of a working

physical realization can be captured via a

mathematical point-set description does not surprise

me.

>With this physical model, we know a posteri that the

>only possible moves are multiples of 90 degree

>rotations about the coordinate axes, or some

>combination of these.

I am not so sure about the "a posteriori" aspect of

this inference. It seems to me that it results more

from conscious intent and design.

>Of course a physical Rubik’s cube can be taken apart,

>but the theoretical one just constructed cannot be

>(or at least, that is my conjecture).

If I read Nate’s models correctly, it appears that his

model adequately reflects the blocks which extend

inwards (beyond the extent of the corresponding

cubical piece and into the interior ‘sphere’) from the

edge and corner pieces and which hook behind the edges

of adjacent pieces which lie on the sphere. I agree

that it cannot be taken apart. This is because Nate’s

‘theoretical’ face spindles do not have springs in

them.

It appears that a mathematical model has been achieved

which captures the property of this thing hanging

together. But this does not bear on solving the

puzzle. For my own purposes as a mathematician, it

seems to me that the simpler model - expressed as a

3x3x3 pile of 1x1x1 cubies, along with a description

of the motions permitted for certain subsets of those

cubies (the 3x3x1 slices) - captures the nature of the

puzzle better and is certainly easier to work with.

As a mathematician, I am anxious to _avoid_ details

related to the actual mechanics of a realization of

the object.

>We also know that these rotations form a group (I

>claim this is obvious from the fact that these are

>invertible functions applied from a set of states to

>itself, one of which leaves the state unchanged).

To me, it is obvious because it can be viewed as a set

of permutations on the 54 stickers. (Actually 48,

since the face cubies don’t move.) Permutation groups

are pretty well understood.

>HIGHER DIMENSIONAL "GEOMETRIC" GENERALIZATION:

>One will notice that this ‘physical model’ allows all

>of the moves one would expect for a Rubik’s

>Hypercube. However, also observe that any of the

>eight ‘cubical sides’ of this model can be subjected

>to all of the usual moves allowed for a 3-d Rubik’s

>Cube without causing any self-intersections either.

>Thus the natural generalization of the physical model

>leads to a quite different puzzle than the natural

>algebraic generalization of Rubik’s Cube.

The same conclusion could be drawn regarding the

puzzle as a 3x3x3x3 pile of hypercubies. Given a

3x3x3x1 pile corresponding to a hyperface, you can

ignore the dimension in which the pile is thin and

think of it as a 3x3x3 pile as with Rubik’s Cube -

ie., turning 3x3x1 subsets with respect to the rest of

the pile. My point is that Nate’s "physical model" is

not the only approach which could lead one to consider

moves which permute only a subset of the pieces

corresponding to a hyperface. Indeed, the same notion

falls out in a more easily graspable manner with the

pile of hypercubies approach.

>My personal experience seem to be that the physical

>generalization is somewhat easier to solve, due to

>the increased number of allowable moves, combined

>with the fact that once 2/3 of the Rubik’s Hypercube

>is solved, the problem reduces to merely solving a

>3-d Rubik’s Cube.

Indeed, it is much easier. Ironically though, the

corresponding group is much larger - which some

might regard as corresponding to greater complexity.

Actually, the group is not as much larger as you might

think. It turns out that the smaller group already

includes the move that turns a center 3x3x1x1 slice of

a 3x3x3x1 pile. Such a move is not even difficult to

discover. Turning an external 3x3x1x1 slice of a

3x3x3x1 pile is the type of permutation that Nate

added.

Nate, have you convinced yourself that your 4D model

cannot fall apart? I have difficulty thinking about

how the hooks work in the 4D case, and I cannot

convince myself. I am not saying that it is not

true - just that it is not obvious to me. I am

particularly concerned about how the hook on a

2-sticker piece works.

>I should mention that I have also thought of a 4-d

>model that would allow only the moves usually allowed

>in the Rubik’s 4-Cube, but it is much more

>complicated geometrically. At this point I have not

>put down specific numbers for it, but have managed to

>convince myself of it’s existence.

I would be very surprised if such a model exists. The

reason is that a valid model must support turning a

center 3x3x1x1 slice of a 3x3x3x1 pile. (Not in a

single move - but without affecting anything else.

Ie., the group admits this permutation.) So the

constraint in the smaller group is that there is some

sense in which opposite external slices of an affected

3x3x3x1 pile must maintain the same orientation

relative to one another. Given that the center slice

between them _can_ turn and that this is true for each

of the three relevant axes, it strikes me that that

constraint would be difficult to achieve in a

‘physical’ model - even granting the existence of 4D

construction materials. If it turns out that it is

possible, I will be very impressed.

In response to Melinda, Nate wrote:

>Indeed, from the point of view of algebra, both of

>your models yield the same set of possible states. In

>fact, I believe with my "physical model" all the

>rotations in MagicCube4D are possible, and no

>detaching of hyperfaces is necessary.

That’s a good point. I think "detachment" is

primarily just a way of speaking. Just looking at

MC4D’s animated projection into 3D is helpful for me

to convince myself that rotating a hyperface about

axes that are not aligned with coordinate axes does

not lead to pieces intersecting. (It takes some

imagination.)

>The elegance of MagicCube4D is that it contains a

>direct move (i.e. geodesic, inertial rotation, or

>2-dimensional rotation, whichever you like to call

>it) to each of the 23 (24 if you count the null

>rotation) rotations.

It has struck me as extremely elegant that the 23

reorientations of a cube can be obtained by 2D

rotations about the axes which pass through corner,

edge, and face positions. There are 4 such axes

through corners admitting 2 new twist positions each.

There are 6 such axes through edge-middles admitting 1

new twist position each. There are 3 such axes

through the face-centers, admitting 3 new positions

each. 4x2+6x1+3x3 = 23. Clearly the reorientations

are all different. (So that was actually a proof by

enumeration, since we know that there cannot be more

than 24, counting the identity transformation.) I

doubt that this elegance is just a happy accident, but

I have never come up with an intuitive argument about

why it _should_ be possible to represent all the

reorientations in this way. I wrote a program which,

given a reorientation transformation represented as a

3x3 matrix, directly computes which single axis to

twist about. It is not trivial.

>Any more rotations than this would not gain the user

>anything, and any less would make some moves

>unnecessarily burdensome. For Rubik’s 5-cubes and

>higher, non-inertial rotations become possible (since

>the hyperfaces are now 4-cubes). However, one could

>make an argument against allowing such rotations in a

>UI, since as a consequence of the Shur Decomposition

>Theorem any such rotation can be written as the

>product of orthogonal 2-dimensional rotations, which

>being orthogonal must commute,

There is something here that I do not understand at

all: Even 90 degree rotations of a 3-cube about two

different coordinate axes (clearly orthogonal) do not

commute.

>and so conceptually these rotations would be

>independent from each other.

>This would bring down the 192 rotations possible for

>a 4-cube shaped side to a smaller, perhaps more

>reasonable number (I’m not quite sure what that

>number is. If anyone has a way of calculating it, I

>would be interested to know about it.)

But Nate has already mentioned that, if 90 degree

turns about any two axes are permitted, then those two

moves generate the entire group of 192 such rotations

of the 4-cube. Since I like 90 degree coordinate-

axis-aligned twists, I don’t see much opportunity for

a method of reducing this which would make sense.

But, as long as the coordinate-axis-aligned twists are

in the UI, then all 192 reorientations can still be

generated. The impact is on twist-count. I think it

is still cleaner to regard any such reorientation as

being a single twist, no matter how one specifies it.

(Where 192 comes from: A reorienting transformation

must map each of the 4 axis-aligned unit vectors onto

itself or another, and there are 24 ways to permute

them. For each mapping of a basis vector onto a basis

vector, there may be a sign change. There are 2^4

ways of assigning signs. When you multiply it out,

you get 384; but half of those are mirror-image

transformations which are not permitted.)

Regards,

David V.