# Message #133

From: Nathanael Berglund <berglund@math.gatech.edu>

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

Date: Tue, 19 Apr 2005 01:27:45 -0400

Melinda Green wrote:

> hello nathanael,

>

> welcome to the MC4D group!!

>

> you will be interested to know that don and i originally had a

> protracted debate about the "best" 4D generalization of the 3D puzzle.

> my initial feeling was that it was all about 90 degree rotations and

> that the proper generalization would only allow twists about hyperface

> axies passing through center/center cubies. don’s position was that all

> rotations about hyperface cubies should be allowed, if only because you

> can get to any of these through some, possibly painful, sequence of 90

> rotations anyway. eventually i convince myself that his ui does make

> perfect sense if we look at the N dimensional puzzle by defining a twist

> like this: 1) detach the face to be twisted from the rest of the model.

> (note that this includes some stickers from other faces.) 2) apply any

> rotation of that face that results in its original orientation. 3)

> replace the face. with this definition, a twist on a 3D cube detaches a

> 3x3 slice of cubies which can only be turned into 4 unique positions.

> the fact that these are all rotations about a single axis is just a

> coincidence that happens in the special case of 3 dimensions. in 4D, the

> way to think of the analogous process is is to take a 3x3x3 face of

> stickers along with the slice of stickers on each adjoining face, and

> pull the entire cubic assembly out of an imaginary box. you should then

> be allowed to rotate this 3D cube into any orientation that fits back

> into that box, and then slide it back in. this definition generalizes

> into N dimensions even if the program’s UI does not. i have no idea how

> to create a similarly holistic 5D puzzle interface that a human could

> reasonably solve.

>

> -melinda

>

> Nathanael Berglund wrote:

>

> >A few years back I thought of a possible "physical model"

> >for higher dimensional Rubik’s Cubes.

> >

>

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>

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. Here’s why I say this: Suppose we define a

particular direction as ‘up’, and we wish to rotate the ‘top’

hyperface. If we look at the original partition of the hypercubes into

81 identical hypercube pieces, then it is clear that any rotation which

is orthogonal (perpendicular) to the ‘up’ direction can be applied to

the top hyperface without causing any self-intersections of pieces. To

an observer inside the top hyperface, this means any rotation about the

origin is allowable. Now if we change the model to the more realistic

interlocking pieces which slide along a hypersphere, note that in each

3-d cross section orthogonal to the ‘up’ direction (that is touched by

at least one of the ‘top’ pieces), we are just rotating a sphere within

a cube remove the sphere. So clearly this will allow any 3-d rotation.

In fact we could twist and twirl the 3-d hyperface all we wanted, as

long it we put it back in the aligned position again at the end. 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. 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, 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.)

– Nate