>Working on a uniformly coloured 3x3x3 cube or tesseract is not
>solving the full symmetry group. The subgroup of the full symmetry
>group which fixes the colours, but ignores orientations, is a normal
>subgroup of the full group, and the quotient group of the full
>symmetry group by this normal subgroup is the group that is being
>studied when working with a uniformly coloured cube/tesseract.
>Although nothing like as big as the full symmetry group, the colour-
>preserving subgroup is nonetheless respectably large, and probably
>deserves some consideration.
David's comment:
>I think there may be a slight ambiguity in the above
>which I would like to understand properly. There are
>two sorts of relevant orientation issues for a given
>hyper-cubie - that which is forced by the stickers on
>it (the obvious orientation issue) and that for the
>axes for which the hyper-cubie has no stickers (the
>unobvious one applying to 1- and 2-color
>hyper-cubies). So there are two levels at which one
>may "ignore" orientations.
Yes, there are two types of orientation, which could be classed
as "visible" and "invisible" (on a uniformly coloured tesseract). A
cubie sharing 2,3 or 4 faces can be in the "correct position" (its
2,3 or 4 colours match the colours of the centre cubies of the faces
to which it belongs, but in an incorrect orientation in that
position. This sort of orientation problem is, of course, a standard
component of solving the tesseract as it currently is. We could call
these "visible orientation" issues. Like David, when solving the
tesseract I first place the cubies in the correct positions, and
then fix their visible orientations (well, I normally do this first
for the 2-face cubies, then the 3-face cubies, and finally for the 4-
face cubies - it is much easier to solve the 2-face cubies if you do
not have to worry about what you are doing to the 3- and 4-face
cubies, and similarly it is simpler to solve the 3-face cubies if
you only have to worry about not mucking up the 2-face cubies, and
do not yet care about the 4-face ones - solving the whole position
plus visible orientation problem for each level of cubie before
proceeding to the next stage makes it easier to see what you are
doing!).
The second type of orientation, the "invisible" type, relates only
to those cubies which share 1 or 2 faces of the tesseract. These
cubies can be in the right position (the 1-face cubies have to be!),
and (in the case of the 2-face cubies) can be in the right "visible"
orientation with respect to the faces to which they belong. However,
they can still be in a number of different orientations themselves.
It is currently impossible to see whether, for example, a 2-face
cubie has been given a quarter twist along the axis running through
the centre cubies of the faces to which it belongs or not.
Similarly, it is possible for the centre cubie of a face of the
tesseract to be rotated, without affecting any other part of that
face.
When I previously referred to "orientations", I meant "invisible
orientations". Thus a colour-preserving permutation of the tesseract
is which which puts all the cubies in their correct positions and in
their correct visible orientations. If a colour-preserving
orientation of the cube is performed then a uniformly coloured
tesseract, which started by looking solved, will still look solved.
Start with a clean cube, perform a colour-preserving permutation
(such as Q F(11) Q^{-1} F(11)^{-1}), and the programme will give you
a congratulatory beep!
There is, as David suggests, yet another subgroup of the full
permutation group, namely the one that comprises all orientation-
adjusting permutations, both visible and invisible. How you choose
to describe it is up to you. We could let G be the full permutation
group of the tesseract, and then let C be the (normal) colour-
preserving subgroup of "invisible orientation" permutations. The
quotient group G/C would then be the permutation group of the
uniformly coloured tesseract. We could also consider the normal
subgroup P of G consisting of those permutations of the tesseract
which fix the positions of all cubies, but can alter both visible
and invisible orientations. The colour-preserving subgroup C is of
course itself a normal subgroup of P. We could then consider the
group G/P, which is the symmetry group of the positions of the
cubies alone. Alternatively, we could consider the quotient group
P/C, which is the permutation group of the visible orientations of
the uniformly coloured tesseract. By the Third Isomorphism Theorem,
P/C is a normal subgroup of G/C, and the quotient group (G/C)/(P/C)
is isomorphic to the group G/P. We would interpret (G/C)/(P/C) as
the positional symmetry group of the uniformly coloured tesseract.
Reassuringly, the Third Isomorphism Theorem tells us that it does
not matter whether we start with a uniformly coloured tesseract or a
fully marked one - if we are only interested in positional issues,
it does not matter if we choose to disregard orientations in two
stages (first the invisible, then the visible) or all in one go!
Although respectably large, the position-fixing permutation group P
is still much smaller than the full symmetry group G and so, as
experience shows when solving the tesseract, dealing with the
positional permutation group G/P takes up the lion's share of
solution time!
Mark
mahdeltaphi wrote:
> David's comment:
> [...] So there are two levels at which one
> >may "ignore" orientations.
>
> Yes, there are two types of orientation, which could be classed
> as "visible" and "invisible" (on a uniformly coloured tesseract).
i don't know if you included this but i suspect there may be a 3rd type
of ambiguity found in puzzles with more than 3 cubies along an edge. for
example when there is more than one "middle" cubie along an edge they
can all be correctly oriented yet placed in interchangeable positions.
this would also show up with a 4^3 picture cube, or more generally, an
N^D textured cube for all N > 3. so even positions can be ambiguous, not
just orientations. this should apply to more than just edge pieces too
and in fact should apply to all cubies that are not corners or face
center-center (0 colored) cubies when N is odd.
-melinda
>i don't know if you included this but i suspect there may be a 3rd
type
>of ambiguity found in puzzles with more than 3 cubies along an edge.
for
>example when there is more than one "middle" cubie along an edge they
>can all be correctly oriented yet placed in interchangeable
positions.
Certainly, as the cube gets bigger, there are more orientation
problems. The fact that the 3x3x3x3 cube is uniformly
coloured makes it sensible to talk about visible and invisible
orientation issues, but other classifications could be
possible.
If we consider the 4x4x4 cube, it turns out the the 2-face cubies
cannot have invisible orientation problems. If they
are both correctly positioned as to colour alone, then they are
either both correctly oriented or else they are both
incorrectly oriented in a visible manner. The way in which the 4x4x4
cube was constructed was to have a central
sphere, with grooves along the octants. The 1-face cubies slid along
those grooves, carrying the 2- and 3-face cubies
with them (just about - the cube tended to explode if roughly
handled!). The fact that the 2- and 3-face cubies had
to slide around a central sphere meant that they had curved backs,
and the nature of those curves made it impossible
for the 2-face cubies to exhibit invisible orientation problems.
There were, however, invisible orientation problems
for the 1-face cubies. The four such cubies for each 4x4 face of the
cube could be permuted. Once a cubie was in a
particular position, however, its orientation was fixed (one corner
of the cubie would always point to the centre of
the 4x4 face). It was reasonably simple, then, to mark the 1-face
cubies in such a manner to force a unique solution.
Although the arguments given above for these limitations to the
possible orientations are mechanical in nature, the
device's mechanism did permit all possible cube rotations, and so I
would regard these limitations as theoretically
justifiable as well.
If we considered the 5x5x5 cube, the nine 1-face cubies on each 5x5
face would split into 3 groups - the four
corners, the four edges, and the centre. The first two groups could
have their elements permuted, and the central
element could be rotated. If we consider the 3 2-face cubies on each
edge, I would expect that the central one could
be flipped, and possibly the other two could be swapped (although I
suspect that similar considerations to those for
the 2-face cubies for the 4x4x4 cube would prevent this).
Mark
... and then there's the "very invisible" orientation and permutation issues:
that is, the orientations and permutations of the 0-sticker cubies
(i.e. internal cubies, which may or may not have physical manifestation,
depending on your faith, since you can never see them),
on 4x4x4 and larger cubes and hypercubes.
Don
--
Don Hatch
hatch@plunk.org
http://www.plunk.org/~hatch/
>... and then there's the "very invisible" orientation and
>permutation issues:
>that is, the orientations and permutations of the 0-sticker cubies
>(i.e. internal cubies, which may or may not have physical
>manifestation,
>depending on your faith, since you can never see them),
>on 4x4x4 and larger cubes and hypercubes.
Yes, but these can be ignored for the 3^3 and 3^4 cubes.
In the 3^3 cube, the orientation of the 0-face cubie in the centre is
fixed by the 6 1-face cubies which surround it - each side of the 0-
face cubie always points to one particularly coloured 1-face cubie.
Once the rest of the cube has been solved, therefore, determining the
orientation of the central 0-face cubie is simply a matter of
deciding which coloured face will be uppermost, and so on. Taking the
cube, and rolling it along the table performs rotations of its cubies
which affect the orientation of the 0-face cubie, but does not affect
the relative orientations of the constituent cubies of the cube.
Similarly, the orientation of the central 0-face cubie of the 3^4
cube is fixed by the fact that each of its faces points to one
particular 1-face cubie. Again, then, the orientation of the 0-face
cubie is simply a matter of which (global) 4D rotations you wish to
apply to the cube to pick which colour to display in which position.
There is, probably, a real problem to be considered with the 4^3
cube, and similar. The 4^3 cube contains 8 internal cubies, which
themselves represent a 2^3 cube in their own right. I have not
considered to what extent the central 2^3 cube can be manipulated
independently of the external faces of the 4^3 cube (physical
implementations of the 4^3 cube do not display these central cubies,
so tracking them is more difficult). However, a computer simulation
of the 4^3 cube could do this.
Mark
Sorry about resurrecting the thread; I didn't check the group in a
while and missed out on this interesting discussion.
I am pretty familiar with the center face orientation supergroup in
three dimensions (occasionally I amuse myself by solving my 5^3 cube
corners and edges first, and leaving the 3x3 centers for last. Some
3-d analogs of the commutator Mark described in message 39 can be
useful here).
I'm a little sad that I didn't think to consider the supergroup in
four dimensions; that the 2-color cubies contribute to it was a
little surprising at first. I expect there's a codimension two issue
here.
Regarding the edge cubies in 4^3 and 5^3 cases Mark discusses below,
those constraints are ultimately algebraic, not physical. They
follow from the permutation rules of the cube. If a physical cube
were designed so that swapping an edge pair (in 4^3) without flipping
them or swapping the central edge (in 5^3) with a non-central edge
were possible, doing so seems to me exactly analogous to popping out
a corner and putting it back twisted.
rb
--- In 4D_Cubing@yahoogroups.com, "mahdeltaphi"
wrote:
> [...]
>
> If we consider the 4x4x4 cube, it turns out the the 2-face cubies
> cannot have invisible orientation problems. If they
> are both correctly positioned as to colour alone, then they are
> either both correctly oriented or else they are both
> incorrectly oriented in a visible manner. The way in which the
4x4x4
> cube was constructed was to have a central
> sphere, with grooves along the octants. The 1-face cubies slid
along
> those grooves, carrying the 2- and 3-face cubies
> with them (just about - the cube tended to explode if roughly
> handled!). The fact that the 2- and 3-face cubies had
> to slide around a central sphere meant that they had curved backs,
> and the nature of those curves made it impossible
> for the 2-face cubies to exhibit invisible orientation problems.
> There were, however, invisible orientation problems
> for the 1-face cubies. The four such cubies for each 4x4 face of
the
> cube could be permuted. Once a cubie was in a
> particular position, however, its orientation was fixed (one corner
> of the cubie would always point to the centre of
> the 4x4 face). It was reasonably simple, then, to mark the 1-face
> cubies in such a manner to force a unique solution.
> Although the arguments given above for these limitations to the
> possible orientations are mechanical in nature, the
> device's mechanism did permit all possible cube rotations, and so I
> would regard these limitations as theoretically
> justifiable as well.
>
> [...]
I fully agree that the fundamental constraints I discussed are
algebraic rather than mechanical. That the cubes can be constructed
physically merely (! - apologies to the engineers who worked out how
to do it) represents the fact that the algebra can be implemented in
a concrete form. However, the mechanical constraints in the 4^3 case
are such as to provide a ready heuristic proof, without the need to
set out the full algebraic proof of the problem.
My main interest in raising this thread in the first instance was
that the 3^4 cube represented a case where the cube could not be
represented physically, but only through computer simulation, and
the MC4D visualisation of the cube did not allow for ready
consideration of the problems involved.
DV has sent me an alternative visualisation of the 3^4 cube which
does permit such consideration - I just have not had time to get to
grips with the problem recently!
Mark
--- In 4D_Cubing@yahoogroups.com, "rbreiten"
> Sorry about resurrecting the thread; I didn't check the group in a
> while and missed out on this interesting discussion.
>
> I am pretty familiar with the center face orientation supergroup
in
> three dimensions (occasionally I amuse myself by solving my 5^3
cube
> corners and edges first, and leaving the 3x3 centers for last.
Some
> 3-d analogs of the commutator Mark described in message 39 can be
> useful here).
>
> I'm a little sad that I didn't think to consider the supergroup in
> four dimensions; that the 2-color cubies contribute to it was a
> little surprising at first. I expect there's a codimension two
issue
> here.
>
> Regarding the edge cubies in 4^3 and 5^3 cases Mark discusses
below,
> those constraints are ultimately algebraic, not physical. They
> follow from the permutation rules of the cube. If a physical cube
> were designed so that swapping an edge pair (in 4^3) without
flipping
> them or swapping the central edge (in 5^3) with a non-central edge
> were possible, doing so seems to me exactly analogous to popping
out
> a corner and putting it back twisted.
>
> rb
>
> --- In 4D_Cubing@yahoogroups.com, "mahdeltaphi"
> wrote:
>
> > [...]
> >
> > If we consider the 4x4x4 cube, it turns out the the 2-face
cubies
> > cannot have invisible orientation problems. If they
> > are both correctly positioned as to colour alone, then they are
> > either both correctly oriented or else they are both
> > incorrectly oriented in a visible manner. The way in which the
> 4x4x4
> > cube was constructed was to have a central
> > sphere, with grooves along the octants. The 1-face cubies slid
> along
> > those grooves, carrying the 2- and 3-face cubies
> > with them (just about - the cube tended to explode if roughly
> > handled!). The fact that the 2- and 3-face cubies had
> > to slide around a central sphere meant that they had curved
backs,
> > and the nature of those curves made it impossible
> > for the 2-face cubies to exhibit invisible orientation problems.
> > There were, however, invisible orientation problems
> > for the 1-face cubies. The four such cubies for each 4x4 face of
> the
> > cube could be permuted. Once a cubie was in a
> > particular position, however, its orientation was fixed (one
corner
> > of the cubie would always point to the centre of
> > the 4x4 face). It was reasonably simple, then, to mark the 1-
face
> > cubies in such a manner to force a unique solution.
> > Although the arguments given above for these limitations to the
> > possible orientations are mechanical in nature, the
> > device's mechanism did permit all possible cube rotations, and
so I
> > would regard these limitations as theoretically
> > justifiable as well.
> >
> > [...]
--- In 4D_Cubing@yahoogroups.com, "mahdeltaphi"
wrote:
>[...]
> There is, probably, a real problem to be considered with the 4^3
> cube, and similar. The 4^3 cube contains 8 internal cubies, which
> themselves represent a 2^3 cube in their own right. I have not
> considered to what extent the central 2^3 cube can be manipulated
> independently of the external faces of the 4^3 cube (physical
> implementations of the 4^3 cube do not display these central
> cubies, so tracking them is more difficult). However, a computer
> simulation of the 4^3 cube could do this.
OK I finally got around to thinking about this (and playing with my
5^3 cube since I couldn't find my 4^3). Let's pretend that my
missing 4^3 cube is actually 64 little (fully distinguishable and
orientable) cubes stacked into a 4^3 cube that magically can be moved
only according to the regular rules. We are interested in the group
of permutations G of the internal 2^3 sub-cube that leave the outer
shell fixed.
Rotating an outer face a quarter turn is an 4-cycle on its corners,
two parallel 4-cycles on its edges, and a 4-cycle on its center
faces. Rotating an inner layer a quarter turn is a 4-cycle on its
four edges and a 4-cycle on a face of the internal cube (4-cycles are
odd permutations of course).
If the outer shell is solved, all the edges are in an even
permutation (identity). This restricts the internal cube to an even
number of quarter turns on its faces. Depending on your algorithm
for solving 4^3, this may be a familiar fact (oops, got an edge
pair "flipped").
Sloppily last night I decided that G was generated by half turns of
the internal cube's faces; clearly this is a subgroup of G as it is
easy to construct a sequence of moves that leaves the outer shell
fixed but rotates a face of the internal cube a half turn.
Today, though, I think that by using conjugations I can probably
construct a sequence of moves that fixes the shell but rotates two
faces of the internal cube a quarter turn each, which means that the
only restriction on the corners of the internal cube would be that
they are in an even permutation.
Intuitively this makes sense since it is a very similar restriction
to that of the supergroup problem of the 1-color faces on 3^3.
I will write down the first sequence and try to construct the second
one over the weekend.
rb