Message #123
From: Nathanael Berglund <berglund@math.gatech.edu>
Subject: "Physical models" of Rubik’s Cube (tesseract, hypercube, etc.)
Date: Wed, 13 Apr 2005 23:56:39 0400
A few years back I thought of a possible "physical model"
for higher dimensional Rubik’s Cubes. The phrase "physical
model" must of course be taken with a grain of salt since
there’s the question of whether higher dimensional materials
could exist, or if they would even follow similar physical
laws. My model worked as follows:
The 3d Rubik’s Cube, if you take it apart, consists of 27
pieces, carefully machined so as to fit together snugly and
allow the slices to rotate without the whole thing falling
apart. If you’ve never taken apart a Rubik’s Cube, I
recommend you find an old well worn one and try it. 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. Spherical surfaces are
not fun to try to machine, so other surfaces are used
instead, but since a sphere would work just as well, let’s
just assume it’s a sphere.) 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). However,
I can imagine at least one way of doing this that would
work. Imagine 3 rods of different radii attached to each
face piece, the middle rod having the smallest radius. Then
by having the internal piece wrap cylinders around each of
these rods, they can be prevented from being pulled out.
As a mathematician, of course I want to find an explicit
formalization of this model. I do this as follows:
PRELIMINARY DEFINITIONS:
Consider the cube [36, 36]x[36, 36]x[36, 36]
(where by A x B x C we mean the set of points x such that
x_1 is in A, x_2 is in B, and x_3 is in C. We wish to
partition a subset of this cube into 27 nonoverlapping
‘pieces’.

Let M denote the set of all 24 matrix rotations that
rotate this cube onto itself. (Generated by a 120 degree
rotation about [1, 1, 1], and a 90 degree rotation about one
of the coordinate axes). 
Let S denote the open ball of radius 24 about [0, 0, 0],
given by { x : sqrt(x_1^2 + x_2^2 + x_3^2) < 24 } and S~
denote the complement of this ball. 
Let D(r, x_i) denote a disk about the origin of radius
r with normal along x_iaxis (for example D(12, x_1) =
{ x : x_1 = 0, sqrt(x_2^2 + x_3^2) < 12}). 
If we write D(r1, r2, x_i), we will mean
D(r2, x_i)\D(r1, x_i), or in other words, the annulus formed
by the points outside of D(r1, x_i) but inside of
D(r2, x_i). 
Finally, we will often take icl(K) := int(cl(K)) =
the interior of the closure of K, in order to ensure that a
piece is a connected set, and also that it does not
intersect any other pieces.
EXPLICIT DECOMPOSITION:
internal piece: union_{A in M} A(icl(K))
where the set K of 3d points is given by:
K = ((0, 6) x D( 6, x_1)) union
((6, 9) x D(4, 6, x_1)) union
((9, 12) x D(2, 6, x_1))
face pieces: For a given A in M, A(icl(K))
where K = (( 6, 9) x D(4, x_1)) union
(( 9, 12) x D(2, x_1)) union
((12, 24) x D(6, x_1)) union
(((12, 36) x (12, 12) x (12, 12)) intersect
S~)
(of course, some entries of M may give the same face piece,
so there are really only six distinct face pieces, but of
course we already know that)
edge pieces: For a given A in M, A(icl(K))
where K = (((12, 36) x (12, 36) x (12, 12)) intersect S~)
union
((( 6, 24) x ( 6, 24) x ( 6, 6)) intersect S )
corner pieces: For a given A in M, A(icl(K))
where K = (((12, 36) x (12, 36) x (12, 36)) intersect S~)
union
((( 6, 24) x ( 6, 24) x ( 6, 24)) intersect S )
POSSIBLE MOvES:
By a possible move, we mean a continuous path of rigid
motions (rotations/translations), applied to some
subcollection of the pieces, such that at no point during
the move do two pieces ever intersect, and at the end of the
‘move’, the pieces occupy the same collection of sets in
space as they did originally (with the pieces possibly
permuted). 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. 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). 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).
HIGHER DIMENSIONAL "GEOMETRIC" GENERALIZATION:
It becomes interesting when we attempt to generalize this
model to 4d space. Similarly to before consider the
hypercube [36, 36]^4 (i.e. [36, 36] cross itself four
times). Similarly to before, let M denote the set of all
matrix rotations that rotate this hypercube onto itself.
Let S denote the open 4d ball of radius 24 about the
origin, and S~ denote the complement of this 4ball.
Let D(r, x_i) denote a 3d ball about the origin of radius
r with normal along x_iaxis (for example D(12, x_1) =
{ x : x_1 = 0, sqrt(x_2^2 + x_3^2 + x_4^2) < 12}). And
also as before, let D(r1, r2, x_i) = D(r2, x_i)\D(r1, x_i)
(the ‘hollow ball’ between the radii r1 and r2). Then we
have:
EXPLICIT DECOMPOSITION:
internal piece: union_{A in M} A(icl(K))
where K = ((0, 6) x D( 6, x_1)) union
((6, 9) x D(4, 6, x_1)) union
((9, 12) x D(2, 6, x_1))
cube pieces: For a given A in M, A(icl(K))
where K = (( 6, 9) x D(4, x_1)) union
(( 9, 12) x D(2, x_1)) union
((12, 24) x D(6, x_1)) union
(((12, 36) x (12, 12) x (12, 12) x (12, 12))
intersect S~)
face pieces: For a given A in M, A(icl(K))
where K = (((12, 36) x (12, 36) x (12, 12) x (12, 12))
intersect S~)
union
((( 6, 24) x ( 6, 24) x ( 6, 6) x ( 6, 6))
intersect S )
edge pieces: For a given A in M, A(icl(K))
where K = (((12, 36) x (12, 36) x (12, 36) x (12, 12))
intersect S~)
union
((( 6, 24) x ( 6, 24) x ( 6, 24) x ( 6, 6))
intersect S )
corner pieces: For a given A in M, A(icl(K))
where K = (((12, 36) x (12, 36) x (12, 36) x (12, 36))
intersect S~)
union
((( 6, 24) x ( 6, 24) x ( 6, 24) x ( 6, 24))
intersect S )
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 3d Rubik’s Cube without causing any selfintersections
either. Thus the natural generalization of the physical
model leads to a quite different puzzle than the natural
algebraic generalization of Rubik’s Cube. 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 3d Rubik’s Cube.
I should mention that I have also thought of a 4d model
that would allow only the moves usually allowed in the
Rubik’s 4Cube, 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.
Now, if only someone would invent 4dimensional modeling
clay!
– Nathanael Berglund