Message #38

From: David Vanderschel <DvdS@Austin.RR.com>
Subject: Re: Orientations of the centre cubes
Date: Tue, 09 Sep 2003 03:42:31 -0500

On Monday, September 08, "Jay Carlton" <oscar@its.caltech.edu> wrote:
>mahdeltaphi wrote:

>>(I used to mark the centre square of each 3x3 face to indicate in
>>which direction it should be pointing, and doing so reduced the
>>number of possible solutions down to 1 from a total of (4^6)/2).

>By making that change, you’re causing a fundamental
>change in the underlying rules of the puzzle, perhaps
>almost as radical as extending from 3 to 4
>dimensions.

This change is not fundamental at all. The
possibility of paying attention to the orientation of
the face cubies was recognized very early on with
Rubik’s Cube. Indeed, see

http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Alan_Bawden__[no_subject]_(2).html

which I believe was the third message ever posted to
the cube-lovers list back in 1980. Folks immediately
began referring to it as "the extended problem". If I
recall correctly, few folks got excited about it.

>The fact that a center face on the original cube can
>be rotated is a necessary flaw, but one that I
>believe the state counting already takes into
>account.

There is no reason to regard it as a flaw, and it is
possible to take it into account if one chooses to.

>As the system configurations are defined
>traditionally, there is still only one solution.
>It’s a similar problem to the one that comes up in
>inverse trigonometry all the time (e.g. arcsin(1) =
>pi/2 + k*pi, for any integer k. Although the k=0
>solution is more pleasing, it’s no more valid than
>any of the others.)

I dispute the relevance of this analogy. The issue
with Rubik’s Cube is whether or not the additional
information about the orientation of each face cubie
is presented or not. When it is presented and paid
attention to, there are more configurations possible
and the number of orbits in the group goes from 12 to

1. Jay’s attempted analogy is just a particular case
   of the very common situation (like x**2 = 1) in which
   an equation admits multiple solutions. So what?

>So I guess my point is that if you want to
>differentiate the orientations of the "fixed" faces,
>you’re altering, not merely clarifying, the rules.

I would say that you are creating a different puzzle -
one which happens to have additional information to
deal with. This new puzzle is so closely related to
the other that I would hesitate to say that the
"rules" have changed. Manipulation of the puzzle
remains the same. The objective is just slightly more
elaborate.

>In the original cube, the correct orientation of any
>piece is defined by its neighbors, not by the
>configuration it comes from in the factory.

Huh? The way I understand it, the correct position
and orientation (in a solved cube) for any cubie is
determined by the set of stickers on it. But this is
also consistent with the way it comes from the
factory. I do not understand anything about relation
to neighbors. It does happen that, in a solved cube,
all stickers in the same 2D plane are the same color.
But this is a statement about the stickers, not the
cubies. Recall that a 1-color cubie can never ‘leave’
the face of the whole cube in which it starts. The
positions of the 1-color cubies are normally regarded
as determining the orientation of the whole cube.

>Also, from an aesthetics standpoint, one of the most
>pleasing aspects a solved 3x3x3 cube is the fact that
>each face is a solid color, with 9 identical
>squares. I think the elegance would suffer if you
>mark the center square.

On the contrary. Many cubes were made with various
sorts of pictures on each 3x3 face (six pictures
total). (You could even order customized cubes with
your choice of pictures.) The folks who made such
cubes obviously regarded them as aesthetically
pleasing. Working such cubes required paying
attention to the orientation of the face cubies.

>Lastly, the center cubes in the 4D system are fairly
>difficult to see as it is, and if they were marked
>with seven different colors each, it would just be
>painful.

It is important to realize that what MC4D presents are
the sets (27 in each) of hyper-stickers on each 3x3x3
face of the big hyper-cube. The hyper-stickers are
themselves 3D cubes embedded in 4-space (which
involves both position and orientation in 4-space).
For a 1-color hyper-cubie, there is no theoretical
reason why the orientation of each of the 6 faces of
the cube could not be presented. A seventh color
would distinguish on which face of the hyper-cube
these hyper-stickers are stuck. Yes, it would be
messy in both implementation and use. I would want to
see, on each face of the center hyper-sticker for the
big hyper-cube face, the color associated with the
direction in which the face of the hyper-sticker
faces, and the color associated with the direction
opposite to that in which the face of the
hyper-sticker faces. (The problem is that I can only
‘see’ 3 faces of a cube at a time, and I don’t want to
continuously have to reorient the entire puzzle to
look at the other sides of the hyper-stickers.) I
also want to see the color associated with the
direction in which the face faces. One possiblity
would be to first draw each face with the color
corresponding to the direction in which it faces, then
to draw on each hyper-sticker face a big centered dot
for the color corresponding to the opposite facing
direction, and finally with a smaller centered dot
indicating the face color (same for all six
hyper-sticker faces). My suggestion leads to a
picture which would be hard to draw and difficult to
interpret. There is hardly enough space. How about?:
If the user clicks on such a center hyper-sticker
(presented as now with only a single color), such an
enlarged and elaborated version of it would be
rendered over on the side.

>So I guess my vote is that if this feature is
>included, it should be optional and non-default.

Clearly a valid point, as it is a different puzzle.

It is my opinion that tracking the orientation of the
1-color hyper-cubies is not interesting enough to
justify complicating the puzzle or the program. The
puzzle is already interesting enough and hard enough
without this complication. However, the subject of
possible orientations of the 1-color hyper-cubies is
of moderate interest from a theoretical point of view.

The analogous problem with the 3D cube is actually
more interesting from the practical point of view
because it was not difficult to portray the extra
information. Indeed, for pictures on the faces,
attending to the extended problem becomes essential.
For the 4D cube, it is difficult to imagine how such
information could enter in a realistic manner. (As
one example, I can imagine hyper-stickers which are
transparent 3D cubes with some distinguishable
symmetry-lacking objects embedded in them.) All this
reassures me that the preferable view of
hyper-stickers is that they have uniform color
throughout their 3D volumes and that different
orientations of a hyper-sticker are not
distinguishable.

I have a program for 4D cubing which does make it easy
to keep track of orientation for the 1-color
hyper-cubies. I have not yet resolved all the
questions I have managed to ask myself regarding what
possibilities exist theoretically. I do know that the
possibility of unobvious orientation variation is not
unique to the 1-color hyper-cubies. In the 4D
analogue, it can also occur with the 2-color
hyper-cubies.

Regards,
David V.