On Sun, Sep 07, 2003 at 10:46:27PM -0000, mahdeltaphi wrote:
> The current implementation of the tesseract does not keep track of
> the orientations of the centre cubes of the 8 3x3x3 faces of the
> tesseract. Accounting for all permutations of pieces and their
> orientations, each rotation move on the tesseract seems to permit
> only even permutations. Consequently, in theory, it should be
> possible to have all the other pieces in place, and yet have the
> centre cubes of each of the 8 cubical faces in a variety of
> orientations. This was certainly the case of the original 3x3x3 cube
> (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). If
> the same result applied to the tesseract, marking the centre cubes
> would reduce the number of correct solutions from (6^8)/2 to 1.
This sounds plausible,
except there are 24 possible orientations of a cube (not 6)
so this number should be (24^8)/2 instead of (6^8)/2, I think.
Don
--
Don Hatch
hatch@plunk.org
http://www.plunk.org/~hatch/
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. 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. 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.)
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. 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.
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.
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.
So I guess my vote is that if this feature is included, it should be
optional and non-default.
-Jay
Jay Carlton wrote:
> [...] 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. 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.
you wouldn't need to "mark the center" in any special way. mark's
picture cube would "mark" (i.e. orient) all stickers equally and in my
mind is a reasonably elegant modification to the original puzzle. as an
aside, i may be remembering incorrectly but i think that rubik once made
their own official oriented cube that they ironically called "4D" even
though it had nothing to do with 4D; just had their logo on each face.
anyway, the perfect 4D analog to pasting a picture on each face of the
original cube would be to carve each 4D hyperface from a different block
of wood, marble, or some other 3D texture. very possible to do using 2D
textures applied to each face of each hypersticker, and you may not find
it unattractive, though i don't know if people would find it interesting
enough seeing as how the 3D picture cubes never really caught on.
-melinda
On Monday, September 08, "Jay Carlton"
>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
24. 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.
Thinking further, it is also clear that only the pieces which are
common to 3 or 4 cubical faces have their positions
absolutely fixed by their colours alone. The pieces that are common
to 2 cubical faces can also be correctly
positioned, colourwise, but be so in differing orientations.
This makes the full number of possible orientations even greater
than I first said (even allowing for the fact that I foolishly put 6
instead of 24 in my first message).
Below I offer some thoughts about moves which alter the orientations
of these pieces.
I am sure that it should be possible to mark the 1-face and 2-face
pieces in a manner which was not too ugly aesthetically, but which
nevertheless indicated their orientations uniquely. For example, I
simply used to cut a notch on one side of each centre face of my
3x3x3 cube, and a mathing notch on one edge piece for each 3x3 face.
The cube was in the correct orientation when all the notches matched.
I never saw a "good" 3x3x3 cube with pictures on the sides. I did
once have a Charles&Di wedding cube, but that was in appallingly
cheap and nasty plastic, and was next to impossible to use. Marking
a standard cube worked much better!
Moves similar to those which permute the centre faces of
the 3x3x3 cube will alter these pieces' orientations. For example,
the sequence:
Right 15 Left
Left 15 Right
Up 11 Left
Right 15 Right
Left 15 Left
Up 11 Twice
Right 15 Left
Left 15 Right
Up 11 Left
Right 15 Right
Left 15 Left
Up 11 Twice
[or R(15) L(15)^{-1} U(11) R(15)^{-1} L(15) U(11)^2 R(15) L(15)^{-1}
U(11) R(15)^{-1} L(15) U(11)^2]
gives a half-twist to the Top/Up centre piece, as well as giving a
matching half-twist to the Up/Bottom centre piece.
Additionally, I think that it gives a matching half-twist to the
centre piece of the Up face. In other words, it
gives a half-twist to the central spine of three pieces running
through the Up face from Top to Bottom.
Other moves which permute the orientations of the 3x3x3 centre faces
can be used similarly.
More interestingly, consider the sequence:
Up 11 Left
Down 11 Left
Right 15 Twice
Left 15 Twice
Back 15 Right
Right 5 Left
Front 11 Left
Right 15 Left (holding down the "2" button)
Front 11 Right
Right 5 Right
Back 15 Left
Left 15 Twice
Right 15 Twice
Down 11 Right
Up 11 Right
[ or U(11) D(11) R(15)^2 L(15)^2 B(15)^{-1} R(5) F(11) R(15,2) F(11)^
{-1} R(5)^{-1} B(15) L(15)^2 R(15)^2 D(11)^{-1}
U(11)^{-1} ]
Call this sequence Q.
Q is made up of three parts. The first part is the sequence
U(11) D(11) R(15)^2 L(15)^2 B(15)^{-1} R(5) F(11)
which moves all the pieces of the Front face away from the Front
face, placing them where the second part of Q,
namely the move
R(15,2)
does not affect them. The remaining part of Q is the inverse of the
first part. Thus the only part of the Front face
that is affected by Q is the centre cube, which has been replaced by
the centre cube from the Up face. Of course, the
rest of the tesseract has been messed up!
Applying a rotation to the Front face will not affect the relative
(within the Front face) positions of the components of that
face, and will not affect the way in which the rest of the tesseract
has been muddled up. Undoing Q then returns the
centre of the the Up face to the middle of the Up face (but with an
altered orientation, due to the Front rotation),
and undoes all the other moves that have been performed on that
cube. The result is that the Front face has been
restored (with its centre face in its original orientation, but the
rest of that face rotated). Applying an
appropriate rotation to the Front face should restore the tesseract
to its original colour status, but the Front and
Up centre cubes will have been rotated!
Thus the sequence
Q F(11) Q^{-1} F(11)^{-1}
should be one such move.
I would be very interested in an implementation of the 4D cube which
enabled me to think about moves like this more easily!
Mark
> This change is not fundamental at all.
"Fundamental" isn't the best word for what I think I was trying to
get at. I shouldn't have reacted without reading the whole thread.
All I was trying to say was that assigning a stringent orientation
requirement is a change to the goal of the game, whereas extending
the cube to four dimensions is a generalization of the same game.
I'm 0-1 with math analogies so far, but I'm feeling lucky so here's
another one: If I'm defining an algebra that works on, say,
matrices, I have to define, among other things, a notion of
equality. I can define the equality test for any nxn matrix
system. If I am accustomed to using matrices in R3x3, and I switch
to R4x4, I expect my equality concept to survive the migration.
What I meant when I said that the orientation stipulation was
a "fundamental" change was that it would be like redefining the
concept of equality for my matrix system, and that change requires a
whole different algebra.
> 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?
What I was trying to illustrate was a simple example of the same
solution state being described in multiple ways, leading to an
invertibility problem. It really doesn't have much relevance, but
the fact that it's a common problem doesn't make it an trivial one
to deal with. That said, everyone here seems more than equipped to
deal with it, but I find it fascinating. It's the first time I've
gotten to think about complex analysis and stuff in a while, so I
got kinda excited. I wonder if the center sticker is a branch
point...
> >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?
By neighbor, I meant adjacent sticker, but now that I think about
it, I'm wrong. For the center and edge cubies of each face,
matching colors on only two adjacent stickers is sufficient to
ensure that the cubie is in the proper orientation, but for the
corners, you have to look at the diagonal neighbors. Since
neighborhoods don't stay put, I'll refrain from using them as a
state descriptor in the future.
Also, you're very right about the need for orientation of the center
cubie if you're going to map a surface on to the faces, but, like
you said, it's a different puzzle.
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 (46)/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. 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. 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.) 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. 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.
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.
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.
So I guess my vote is that if this feature is included, it should be
optional and non-default.
-Jay
--- In 4D_Cubing@yahoogroups.com, "ojcit"
> 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. 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. 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.) 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. 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.
I find this argument rather bizarre. The underlying rules of the
cube are that you can rotate its faces and scramble it up. The goal
of the cube as a puzzle is to find a method of restoring it to its
original configuration by following those rules. It is a fact that
the orientation of a 1-color cubie on a standard physical cube exists
but is not easily apparent. One may define "original configuration"
to include, or not, this extra information (whether tracked by notes
on paper or marks on the cube). In my opinion the phrase "necessary
flaw" fails to refer. The trig analogy I also find unhelpful. We
aren't choosing a canonical element from a set of solutions, we're
considering information inherent in the definition of an object that
has been modded away in a particular model of that object. It seems
strange to me that this could be viewed as somehow arbitrary or evil.
> 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.
This I think is a reasonable position, but I don't have any strong
affinity toward it. My cubes are all abstract mathematical objects
somewhere up in Platonic Heaven. Some of them happen to be
incompletely modeled by physical objects (or computer programs). I
guess I could argue, if forced, that this incompleteness is an
aesthetic detraction or a necessary flaw, but as Mark says, the
quotient groups are interesting enough....
rb
On Friday, November 21, "rbreiten"
>--- In 4D_Cubing@yahoogroups.com, "ojcit"
>> By making that change, you're causing a fundamental
>> change ...
>I find this argument rather bizarre. ...
So did I. Oddly, Jay had posted the same comment back
in September:
http://groups.yahoo.com/group/4D_Cubing/message/36
At that time, I disputed his argument (in a manner
similar to the way rb did on this second occasion):
http://groups.yahoo.com/group/4D_Cubing/message/38
Why Jay posted an unaltered version of his earlier
questionable observation mystifies me. Perhaps he
blundered with respect to what file he intended to
post in his latest message.
Regards,
David V.
--- In 4D_Cubing@yahoogroups.com, David Vanderschel
> So did I. Oddly, Jay had posted the same comment back
> in September:
It seemed so familiar that I checked to make sure I wasn't replying
to an accidental double post.
Anyway, let u f r d b l refer to clockwise quarter turns of up,
front, right, down, back, and left faces of 4^3 cube as usual, and
let U F R D B L refer to clockwise quarter turns of the layer
adjacent to the face with lowercase name (so uU == dD plus a spatial
reorientation). Let u' and u2 refer to anticlockwise quarter turn
and half turn of the u face, etc.
To rotate just the up face of the internal cube a half turn, perform
U2.f2r2.U2.rl'du'fb'r2f'bd'ur'lf2.U2.r2.U.r'f2r2fr'f'r'f2ldrd'l'.U2.d'
f2d2fd'f'd'f2uldl'u'.U.f2
The first U2 is our half turn of internal u face along with some
naughty stuff in the shell, and the rest unscrambles all the naughty
stuff without permanently affecting the internal cube.
The first longish subsequence rotates the center four 1-color squares
of the f and r faces a half turn while leaving the rest of the cube
fixed. When conjugated by U2, this restores the 1-color cubies to
their original positions. The only tricky point is that exponent
counting seems to show that f and r faces end up rotated a half turn
total, but the conjugation by U2 kind of distributes these half turns
to the l and b faces in a way that makes everything work out (try it
on paper, or if you can bear to look at it, mark up your 4^3). The
last U2 is really U'U', and the last two longish subsequences are
edge pair flips (the same one in the bottom face and two in the
middle layer), that when conjugated by U and U' respectively, restore
the middle edges that were disrupted by the first U2.
There is still a bug in my sequence for rotating two adjacent faces
of the internal cube a quarter turn each (wrote it down late late
last night). I'll work it out tonight and post it tomorrow.
I'm testing these at work via
http://www.nrr.co.uk/rubik/cuboid/?planes=444
which has the nice property of taking keyboard input (left my cube at
home). It uses shift for anticlockwise and ctrl for inner layer.
But I do need to get *some* work done today.
rb
--- In 4D_Cubing@yahoogroups.com, "rbreiten"
> There is still a bug in my sequence for rotating two adjacent faces
> of the internal cube a quarter turn each (wrote it down late late
> last night). I'll work it out tonight and post it tomorrow.
Gah, I just forgot to write down a d2.
Here is a sequence which shows that our group G contains all even
permutations of the internal cube's corners. I argued before that G
is a subgroup of this group, so it is in fact this group.
U.r2l2.D.rl'u2d2rl'f2rl'u2d2rl'b2.D'.r2.D2.fbrf'b'r2fbrf'b'r2.D2.l2
R.u2d2.L.rl'u2d2rl'f2rl'u2d2rl'b2.L'.u2.L2.fbuf'b'u2fbuf'b'u2.L2.d2
f2.D2.l u .r'b2r2br'b'r'b2luru'l'.u'l'.D.b'l2b2lb'l'b'l2fubu'f'.D
L2.d'r'.r'b2r2br'b'r'b2luru'l'.r d .L.l'u2l2ul'u'l'u2rblb'r'.L.f2
I used one U and one R for the "real" turns, and D and L in multiples
of 4 to restore stuff. I fixed up the 1-color cubies from the U
before doing R to make it a bit less tedious.
The first two longish subsequences are my other 3^3 supergroup
buddies that go along with the first longish subsequence in
yesterday's U2 sequence. The last two lines are conjugated pair
flips to restore the edges as before.
Thanks for an interesting problem!
rb