Thread: "Something interesting and strange about permutations"

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After solving the MC5D, I have discovered something a bit strange
abo=
ut permutations.

As everyone who read the solution for MC4D know, we=
can permutate the
4-color hypercubies by doing the 3-color series two t=
imes (one of them

the reverse).

But, why we cannot permutate the 3-color pieces with d=
oing two times a
2-color permutation with 2 moves on MC2D?

Becaus=
e the face rotation is different.

When rotating a "face" i=
n MC2D, the move is like this:


1 2 3 --> 3 2 1

In 3D, the same movement should be:

1 =
2 3 --> 3 2 1
4 5 6 --> 6 5 4
7 8 9 --> 9 8 7

But tha=
t's not what we really do with a rubik's cube, it is this (it

would we for example U2, if it is "U" face):

1 2 3 --> =
9 8 7
4 5 6 --> 6 5 4
7 8 9 --> 3 2 1

If you see, this a=
lgorythm (2-color permutation in MC2D) doesn't only
do 4-6 permutati=
on, also 2-8, which don't happen in MC4D with 3-color

series.

By doing the previous movement (the unreal one) we only affe=
ct two
faces which change their stickers (the same as MC2D), but with a<=
br>rubik's cube (and also MC4D and 5D) we are affecting 4 adjacent face=
s

(the other keep still the same stickers). So with the unreal movement
we=
would be able 3-color pieces by doing the sequence: ( F - R ) U ( R
- F=
) U

However, in MC4D we do movements that only affect 4 faces, and =
that

allows us to easily permutate the 4-color hypercubies by doing the
3-col=
or series algorythm. The fact I'm thinking now is if in MC4D and
MC5=
D all adjacent faces should be affected to make the rotation real,
and w=
e are understanding higher dimensional puzzles wrongly.


I hope that you understand what I have said.

Greetings
Lucas<=
br>

lockquote>

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---

From: "lucas_awad" <lucasawad@gmail.com>
Date: Sat, 09 Aug 2008 18:35:25 -0000
Subject: Re: Something interesting and strange about permutations

Yes, that was what I wanted to say.

About the faces not affected (not changing stickers with others) I
will compare MC3D (with the called "unreal movements") with MC4D.

If we have an MC3D with that kind of moves, we will be doing moves in
two ways because we have 4 2-color cubies per face, as we do in 3 ways
in MC4D, with 6 2-color cubies per face.

So moving in two different axes is:

1 2 3 ---> 3 2 1=20
4 5 6 ---> 6 5 4 (from 4th and 6th cubies)
7 8 9 ---> 9 8 7

1 2 3 ---> 7 8 9
4 5 6 ---> 4 5 6 (from 2nd and 8th cubies)
7 8 9 ---> 1 2 3

1 2 3 ---> 3 2 1 ---> 9 8 7
4 5 6 ---> 6 5 4 ---> 6 5 4 (we can also get a U2 move)
7 8 9 ---> 9 8 7 ---> 3 2 1

If we pick the movement from 3-color pieces, we can do also U and U'
moves, like a move mirroring from both 1st and 2nd cubies (but this is
going out the 2-color pieces possible moves, so I don't see it really
possible, but is what we do in a rubik's cube, moving like that).

Extending it to MC4D and doing always three movements from 4-color,
3-color and then 2-color cubies, we always get a move that is like a
4-color move or a U2 (a combination of two 4-color moves).

So what I think, perhaps I am wrong, is that those limited moves are
the really possible, but as I said, perhaps I am wrong.





--- In 4D_Cubing@yahoogroups.com, "Roice Nelson" wrote:
>
> Hi Lucas,
>=20
> It took me a bit, but I think I'm now mostly following what you are
saying
> here. If I am interpreting correctly, I think what you have effectively
> discovered is how the MC2D rotations are not analogous to any of the
other
> puzzles (except David Vanderschel's extended MC3D functionality)
because the
> MC2D face rotations allow mirroring. The MC4D and MC5D puzzles
don't permit
> mirroring twists, which is in more strict analogy with the physical 3D
> Rubik's cube. There is flexibility in how exactly we want to carry
over the
> analogy of twists, but I like the MC4D
> FAQdescription of what it
> means to make a twist, which says "Take the face you
> want to twist and remove it from the larger object. Turn it around
any way
> you like without flipping it over, and then put it back so that it fits
> exactly like it did before.". If we were to adhere to this in MC2D, no
> scrambling twists would be possible, and hence it would be
degenerately easy
> to solve
:)
>=20
> I found your observation about MC4D twists "only affecting 4 faces"
> intriguing! All the MC4D twists (except the identity) do in fact
affect all
> 6 adjacent faces, but it sounds like you are making a distinction
with the
> 3D case where there is no possibility to make a twist and have all
stickers
> on an adjacent face remain the same color. In MC4D, all the adjacent *
> cubies* are getting shuffled around, but some twists (not all!)
allow the
> sticker colors to remain the same on 2 of the 6 adjacent faces.=20
This was a
> cool point for you to make, as I have never explicitly focused on that
> contrast with the 3D puzzle before. Likewise in MC5D, the cubies on 8
> adjacent faces are always affected with every twist, but some twists
allow
> stickers on up to 4 of the 8 adjacent faces to not change color (in
our MC5D
> implementation, this is actually the only possibility since the
twists are
> not fully worked out). I don't think that we are understanding the
higher
> dimensional puzzles wrongly, but that this different behavior arises
due to
> the extra space in the higher dimensions.
>=20
> Also, it is possible on the 3D, 4D, and 5D cubes to build a 3-color
series
> based on two 2-color series, although the 2-color series require 4 moves
> instead of 2. An example in the 3D case (with the 2-color series in
> parenthesis):
>=20
> (R'FRF') B' (FR'F'R) B
>=20
> Anyway, I hope I was on the right track and that these ramblings are
> usefully related to your thoughts...
>=20
> Roice
>=20
> P.S. As a short aside, it is an interesting fact that the motion of any
> rotation can equivalently be described as a set of two reflections,
which is
> why your U2 example is achievable as two of the "unreal movements".
Visual
> Complex Analysis is a fantastic source
to learn
> much more about this.
>=20
>=20
> On 8/6/08, lucas_awad wrote:
> >
> > After solving the MC5D, I have discovered something a bit strange
> > about permutations.
> >
> > As everyone who read the solution for MC4D know, we can permutate the
> > 4-color hypercubies by doing the 3-color series two times (one of them
> > the reverse).
> >
> > But, why we cannot permutate the 3-color pieces with doing two times a
> > 2-color permutation with 2 moves on MC2D?
> >
> > Because the face rotation is different.
> >
> > When rotating a "face" in MC2D, the move is like this:
> >
> > 1 2 3 --> 3 2 1
> >
> > In 3D, the same movement should be:
> >
> > 1 2 3 --> 3 2 1
> > 4 5 6 --> 6 5 4
> > 7 8 9 --> 9 8 7
> >
> > But that's not what we really do with a rubik's cube, it is this (it
> > would we for example U2, if it is "U" face):
> >
> > 1 2 3 --> 9 8 7
> > 4 5 6 --> 6 5 4
> > 7 8 9 --> 3 2 1
> >
> > If you see, this algorythm (2-color permutation in MC2D) doesn't only
> > do 4-6 permutation, also 2-8, which don't happen in MC4D with 3-color
> > series.
> >
> > By doing the previous movement (the unreal one) we only affect two
> > faces which change their stickers (the same as MC2D), but with a
> > rubik's cube (and also MC4D and 5D) we are affecting 4 adjacent faces
> > (the other keep still the same stickers). So with the unreal movement
> > we would be able 3-color pieces by doing the sequence: ( F - R ) U ( R
> > - F ) U
> >
> > However, in MC4D we do movements that only affect 4 faces, and that
> > allows us to easily permutate the 4-color hypercubies by doing the
> > 3-color series algorythm. The fact I'm thinking now is if in MC4D and
> > MC5D all adjacent faces should be affected to make the rotation real,
> > and we are understanding higher dimensional puzzles wrongly.
> >
> > I hope that you understand what I have said.
> >
> > Greetings
> > Lucas
> >
> >=20
> >
>

---

From: "spel_werdz_rite" <spel_werdz_rite@yahoo.com>
Date: Sun, 10 Aug 2008 22:08:00 -0000
Subject: Re: Something interesting and strange about permutations

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Well, there's a simple way to think of this that I believe I brought
up to Roice about a few months ago. The axis of rotation of an a
figure in N-space will always be composed of a segment of N - 2
dimensions. To put this in visual terms, look at a cube (or any 3D
solid) and rotate it. You can imagine a line about which the figure
rotates. When you rotate a piece on MC4D, you will always see 3
stickers that stay in place (maybe not orientation) through which a
line could pass. The same holds true for any 2D solid. Rotate a dollar
bill, or a face on a Rubik's cube. There exists a unique point, as
there did a unique line in 3D through which any part of matter in it
stays stationary. Anybody familiar with vectors can imply this as curl
F =3D 0. Even if you look at a face on MC5D, you will see
a 3x3 array of stickers that do not move. A 4D face's movement has a
2D plane of rotation. But in the case of rotating a line, this would
imply -1D axis, which, of course, does not exist! You may think of
twisting it through 2-Space as done in MC2D, but this is analogous to
turning a shirt inside out. This process is not allowed in out concept
of "twisty puzzles." Thus, MC2D is really a non-existent puzzle, even
a misnomer! Magic Cube 2D!
-Nelson G.

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Well, there's a simple way to think of this that I believe I brought
up =
to Roice about a few months ago. The axis of rotation of an a
figure&nbs=
p; in N-space will always be composed of a segment of N - 2
dimensions. =
To put this in visual terms, look at a cube (or any 3D
solid) and rotate=
it. You can imagine a line about which the figure
rotates. When you rot=
ate a piece on MC4D, you will always see 3
stickers that stay in place (=
maybe not orientation) through which a
line could pass. The same holds t=
rue for any 2D solid. Rotate a dollar
bill, or a face on a Rubik's cube.=
There exists a unique point, as
there did a unique line in 3D through w=
hich any part of matter in it
stays stationary. Anybody familiar with ve=
ctors can imply this as curl
F =3D 0. Even if you look at =
a face on MC5D, you will see
a 3x3 array of stickers that do not move. A=
4D face's movement has a
2D plane of rotation. But in the case of rotat=
ing a line, this would
imply -1D axis, which, of course, does not exist!=
You may think of
twisting it through 2-Space as done in MC2D, but this =
is analogous to
turning a shirt inside out. This process is not allowed =
in out concept
of "twisty puzzles." Thus, MC2D is really a non-existent =
puzzle, even
a misnomer! Magic Cube 2D!
-Nelson G.

--2-5212728862-0881794079=:7--

---

From: Melinda Green <melinda@superliminal.com>
Date: Sun, 10 Aug 2008 19:41:45 -0700
Subject: Re: [MC4D] Re: Something interesting and strange about permutations

MC2D may not be a proper analogy but it is not a misnomer because a
square is definitely a 2D cube. Or if you want to be completely strict,
only the 3D version is a cube but all of these puzzles are based on the
"measure polytope" in various dimensions.

Regarding rotations, I really don't think that it is helpful to try to
think of N-D rotations as involving rotation axes. The fact that 3D
rotations are easily visualized as happening *about* an axis is really
just a quirk of three dimensions. A better way to think of rotations is
that they always occur *within* a 2D plane. In other words, while an
object moves under the influence of any single rotation in any number of
dimensions, any point of that object will move in a circular arc within
a single 2D plane. In 3 dimensions there will be a single rotation axis
that cuts through the centers of rotation of all those parallel planes
but in 4 dimensions there can be more than one axis that does that, so
try to forget about axes and just look for the planes of rotation.

Now as to the legitimacy of classifying MC2D with the other puzzles, it
all depends upon how we want to define these puzzles. We can choose to
allow mirror operations or not, and we can allow twists involving higher
dimensions or not. I don't have a strong opinion on the best choice, and
I'm perfectly happy if there does appear to be a best choice which does
not allow a valid 2D puzzle. For me, the most interesting thing about
MC2D as implemented is that one can easily sketch the entire state graph
for the puzzle (8 states!) and thereby begin to get an idea of what the
topology of other similar puzzles might look like.

-Melinda

spel_werdz_rite wrote:
> Well, there's a simple way to think of this that I believe I brought
> up to Roice about a few months ago. The axis of rotation of an a
> figure in N-space will always be composed of a segment of N - 2
> dimensions. To put this in visual terms, look at a cube (or any 3D
> solid) and rotate it. You can imagine a line about which the figure
> rotates. When you rotate a piece on MC4D, you will always see 3
> stickers that stay in place (maybe not orientation) through which a
> line could pass. The same holds true for any 2D solid. Rotate a dollar
> bill, or a face on a Rubik's cube. There exists a unique point, as
> there did a unique line in 3D through which any part of matter in it
> stays stationary. Anybody familiar with vectors can imply this as curl
> *F* = *0*. Even if you look at a face on MC5D, you will see
> a 3x3 array of stickers that do not move. A 4D face's movement has a
> 2D plane of rotation. But in the case of rotating a line, this would
> imply -1D axis, which, of course, does not exist! You may think of
> twisting it through 2-Space as done in MC2D, but this is analogous to
> turning a shirt inside out. This process is not allowed in out concept
> of "twisty puzzles." Thus, MC2D is really a non-existent puzzle, even
> a misnomer! Magic /Cube/ 2D!
> -Nelson G.

---

From: David Vanderschel <DvdS@Austin.RR.com>
Date: 11 Aug 2008 17:16:57 -0500
Subject: Re: [MC4D] Something interesting and strange about permutations

On Sunday, August 10, "Melinda Green" wrote:
>MC2D may not be a proper analogy but it is not a
>misnomer because a square is definitely a 2D cube.

I would agree that, in the context of multidimensional
puzzles, a square for MC2D is every bit as much a cube
(2-cube, in this case) as is the tesseract (4-cube) in
the MC4D case. So, in saying "misnomer", with
emphasis on "cube", I think Nelson got a bit carried
away. However, and more importantly, he correctly
observed that, unless you allow reflecting twists, the
'puzzle' does not twist at all. (So Nelson's
"misnomer" could probably be taken appropriately with
respect to "magic".)

The reference to "equivalent 2D puzzle" regarding MC2D
on the Superliminal MC4D page is misleading. In the
description for MC2D, it needs to be made clear that,
to permit it to be at all interesting, mirroring
twists are allowed and that this does violate the
analogy with MC4D and the regular 3D puzzle. Melinda,
if you fix the description, this would also be a good
point to mention that MC3D also allows the mirroring
twists extension in the context of the 3D puzzle. In
either case (and in higher dimensions as well), it
could be argued that reflections can be thought of as
being achieved by embedding the n-dimensional puzzle
in (n+1)-dimensional space, so that reflection can be
achieved by performing a 180 degree rotation for which
the extra spatial axis is in the plane of rotation.

>Regarding rotations, I really don't think that it is
>helpful to try to think of N-D rotations as involving
>rotation axes. The fact that 3D rotations are easily
>visualized as happening *about* an axis is really
>just a quirk of three dimensions. A better way to
>think of rotations is that they always occur *within*
>a 2D plane.

I don't think that there is an important distinction
to be made here. The (n-2)-dimensional subspace
orthogonal to the plane of rotation is also called the
"fixed space" for the rotation. In 3D, it is just a
line. In 4D, the fixed space for a rotation is a 2D
subspace. Defining a rotation in terms of its fixed
space or its plane of rotation are essentially
equivalent, since the two subspaces are always related
by orthogonality. When Nelson wrote, "The axis of
rotation of a figure in N-space will always be
composed of a segment of N - 2 dimensions.", his "axis
of rotation" would more appropriately be referred to
as the "fixed space for the rotation" and his
"segment" would more appropriately be referred to as
"subspace" or "hyperplane".

>In other words, while an object moves under the
>influence of any single rotation in any number of
>dimensions, any point of that object will move in a
>circular arc within a single 2D plane. In 3
>dimensions there will be a single rotation axis that
>cuts through the centers of rotation of all those
>parallel planes but in 4 dimensions there can be more
>than one axis that does that, so try to forget about
>axes and just look for the planes of rotation.

I think this is poor advice. It is often very useful
to be able think about a rotation in terms of its
fixed space. Indeed, the reasoning for what to use
for a rotation often involves thinking about the
aspects of state that you do NOT want to change.
I.e., it is a constraint on the fixed space that
may motivate the choice of rotation plane.

>Now as to the legitimacy of classifying MC2D with the
>other puzzles, it all depends upon how we want to
>define these puzzles. We can choose to allow mirror
>operations or not, and we can allow twists involving
>higher dimensions or not. I don't have a strong
>opinion on the best choice, and I'm perfectly happy
>if there does appear to be a best choice which does
>not allow a valid 2D puzzle.

There does not have to be just one choice; and, in the
presence of multiple possibilities, there is no need
to evaluate any choice as "best". E.g., the 3D puzzle
is interesting whether or not you allow reflecting
twists. Why try to exclude a choice? Regarding
mirroring, only one choice makes sense in the 2D case;
but, for a higher dimension, in addition to the
non-mirroring twists normally considered, one can also
consider a variation which permits reflections. (You
could argue that making this sort of choice is
analogous to whether or not one regards the
orientation of face stickers to be relevant for the 3D
puzzle. By making a different choice, you create a
somewhat different puzzle.)

>For me, the most interesting thing about MC2D as
>implemented is that one can easily sketch the entire
>state graph for the puzzle (8 states!) and thereby
>begin to get an idea of what the topology of other
>similar puzzles might look like.

Where does this "8 states!" come from? Orientation of
a corner 2-cubie depends only on its position.
However, the 4 corner 2-cubies can be permuted in all
24 different ways. In what sense can 3 different
permutations all be regarded as the same state? I had
pointed out the apparent discrepancy in Melinda's
analysis in somewhat greater detail a couple years
ago:
http://games.groups.yahoo.com/group/4D_Cubing/message/330
Since Melinda is now repeating the dubious claim, I
wonder if she ever saw my old message replying to
hers. In that old message, I also touched on some of
the other issues which have arisen again in the
current discussion as well as some other issues which
have not rearisen (yet).

Regards,
David V.

---

From: Melinda Green <melinda@superliminal.com>
Date: Mon, 11 Aug 2008 22:13:02 -0700
Subject: Re: [MC4D] Something interesting and strange about permutations

David Vanderschel wrote:
> [...]
>> For me, the most interesting thing about MC2D as
>> implemented is that one can easily sketch the entire
>> state graph for the puzzle (8 states!) and thereby
>> begin to get an idea of what the topology of other
>> similar puzzles might look like.
>>
>
> Where does this "8 states!" come from? Orientation of
> a corner 2-cubie depends only on its position.
> However, the 4 corner 2-cubies can be permuted in all
> 24 different ways. In what sense can 3 different
> permutations all be regarded as the same state? I had
> pointed out the apparent discrepancy in Melinda's
> analysis in somewhat greater detail a couple years
> ago:
> http://games.groups.yahoo.com/group/4D_Cubing/message/330
> Since Melinda is now repeating the dubious claim, I
> wonder if she ever saw my old message replying to
> hers. In that old message, I also touched on some of
> the other issues which have arisen again in the
> current discussion as well as some other issues which
> have not rearisen (yet).

David,

I arrived at the conclusion that MC2D contains exactly 8 states by
simply recording every sticker pattern that I was able to produce using
the puzzle, ignoring all duplicates due to color or positional
symmetries. I'm pretty sure there are only 8 of them. I gave each of
them a number and then mapped out all the possible transitions between
the those states to produce the complete graph for the puzzle. Perhaps
there are 24 states if you don't mind duplicates, but since arriving at
one state gives exactly the same options as arriving at symmetrical
twins, it seems best to collapse all duplicates into the same logical
state and leave only a graph containing all the truly unique states. I
attempted to include an ASCII diagram in the post that you replied to
(http://games.groups.yahoo.com/group/4D_Cubing/message/329) but
unfortunately Yahoo stripped out my spacing characters and left a bit of
a mess. I should probably sketch it up again in Visio or other
diagraming tool for clarity.

-Melinda

---

From: "Roice Nelson" <roice3@gmail.com>
Date: Wed, 13 Aug 2008 01:21:04 -0500
Subject: Re: [MC4D] Something interesting and strange about permutations

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In reference to how to think about rotations, I thought I'd also share again
the link to the wikipedia article on the 4-dimensional rotation group
SO(4).
The reason being that all of this discussion of how to think about rotations
applies only to "simple rotations". In the 4D case, there are additionally
"double rotations", which leave *only the origin fixed* during the motion
(in contrast with all 3D rotations), and "isoclinic rotations", which are a
special case of double rotations with different properties (double rotations
generally leave only 2 planes "invariant as a whole" or invariant in the
sense of being rotated in themselves, while isoclinic and simple rotations
leave an infinite number of planes invariant as a whole). And one can even
further distinguish between two types of isoclinic rotations.

All of the more complex rotations are built up from two simple rotations, so
you might say that the extra classification is unnecessary. However, during
my investigations while coding Magic120Cell, I was presented with thinking
about how to make any arbitrary 4D rotation in a single step, and found
these additional motions and their various properties worthwhile to think
about. In 5D, I imagine the different possible combinations of simple
rotations produce yet more unique behaviors.

As for the simple rotations, I do like that there is the possibility of
looking at them in dual ways (to allow any extra insight one might gain from
the various perspectives). For myself, my mental model in 3D is still
biased towards thinking of rotations as acting about an axis, probably
because of how standard schooling presents this. My mental model in 4D is
heavily biased towards thinking about rotations as the motion through a 2D
plane instead of about a fixed subspace. This is likely due to my
understanding being built up from my coding efforts and this being the more
natural way to perform the calculations.

Take Care,
Roice


On Mon, Aug 11, 2008 at 5:16 PM, David Vanderschel wrote:

>
> >Regarding rotations, I really don't think that it is
> >helpful to try to think of N-D rotations as involving
> >rotation axes. The fact that 3D rotations are easily
> >visualized as happening *about* an axis is really
> >just a quirk of three dimensions. A better way to
> >think of rotations is that they always occur *within*
> >a 2D plane.
>
>
> I don't think that there is an important distinction
> to be made here. The (n-2)-dimensional subspace
> orthogonal to the plane of rotation is also called the
> "fixed space" for the rotation. In 3D, it is just a
> line. In 4D, the fixed space for a rotation is a 2D
> subspace. Defining a rotation in terms of its fixed
> space or its plane of rotation are essentially
> equivalent, since the two subspaces are always related
> by orthogonality. When Nelson wrote, "The axis of
> rotation of a figure in N-space will always be
> composed of a segment of N - 2 dimensions.", his "axis
> of rotation" would more appropriately be referred to
> as the "fixed space for the rotation" and his
> "segment" would more appropriately be referred to as
> "subspace" or "hyperplane".
>
> >In other words, while an object moves under the
> >influence of any single rotation in any number of
> >dimensions, any point of that object will move in a
> >circular arc within a single 2D plane. In 3
> >dimensions there will be a single rotation axis that
> >cuts through the centers of rotation of all those
> >parallel planes but in 4 dimensions there can be more
> >than one axis that does that, so try to forget about
> >axes and just look for the planes of rotation.
>
>
> I think this is poor advice. It is often very useful
> to be able think about a rotation in terms of its
> fixed space. Indeed, the reasoning for what to use
> for a rotation often involves thinking about the
> aspects of state that you do NOT want to change.
> I.e., it is a constraint on the fixed space that
> may motivate the choice of rotation plane.
>
> .
>
>
>

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In reference to how to think about rotations, I thought I'd also share again the link to the wikipedia article on the 4-dimensional rotation group SO(4).  The reason being that all of this discussion of how to think about rotations applies only to "simple rotations".  In the 4D case, there are additionally "double rotations", which leave only the origin fixed during the motion (in contrast with all 3D rotations), and "isoclinic rotations", which are a special case of double rotations with different properties (double rotations generally leave only 2 planes "invariant as a whole" or invariant in the sense of being rotated in themselves, while isoclinic and simple rotations leave an infinite number of planes invariant as a whole).  And one can even further distinguish between two types of isoclinic rotations.

 

All of the more complex rotations are built up from two simple rotations, so you might say that the extra classification is unnecessary.  However, during my investigations while coding Magic120Cell, I was presented with thinking about how to make any arbitrary 4D rotation in a single step, and found these additional motions and their various properties worthwhile to think about.  In 5D, I imagine the different possible combinations of simple rotations produce yet more unique behaviors.

 

As for the simple rotations, I do like that there is the possibility of looking at them in dual ways (to allow any extra insight one might gain from the various perspectives).  For myself, my mental model in 3D is still biased towards thinking of rotations as acting about an axis, probably because of how standard schooling presents this.  My mental model in 4D is heavily biased towards thinking about rotations as the motion through a 2D plane instead of about a fixed subspace.  This is likely due to my understanding being built up from my coding efforts and this being the more natural way to perform the calculations.

 

Take Care,

Roice

 

 

On Mon, Aug 11, 2008 at 5:16 PM, David Vanderschel <DvdS@austin.rr.com> wrote:

>Regarding rotations, I really don't think that it is
>helpful to try to think of N-D rotations as involving
>rotation axes. The fact that 3D rotations are easily
>visualized as happening *about* an axis is really

>just a quirk of three dimensions. A better way to
>think of rotations is that they always occur *within*
>a 2D plane.

 

I don't think that there is an important distinction
to be made here. The (n-2)-dimensional subspace

orthogonal to the plane of rotation is also called the
"fixed space" for the rotation. In 3D, it is just a
line. In 4D, the fixed space for a rotation is a 2D
subspace. Defining a rotation in terms of its fixed

space or its plane of rotation are essentially
equivalent, since the two subspaces are always related
by orthogonality. When Nelson wrote, "The axis of
rotation of a figure in N-space will always be
composed of a segment of N - 2 dimensions.", his "axis

of rotation" would more appropriately be referred to
as the "fixed space for the rotation" and his
"segment" would more appropriately be referred to as
"subspace" or "hyperplane".

>In other words, while an object moves under the
>influence of any single rotation in any number of
>dimensions, any point of that object will move in a
>circular arc within a single 2D plane. In 3

>dimensions there will be a single rotation axis that
>cuts through the centers of rotation of all those
>parallel planes but in 4 dimensions there can be more
>than one axis that does that, so try to forget about

>axes and just look for the planes of rotation.

 

I think this is poor advice. It is often very useful
to be able think about a rotation in terms of its
fixed space. Indeed, the reasoning for what to use

for a rotation often involves thinking about the
aspects of state that you do NOT want to change.
I.e., it is a constraint on the fixed space that
may motivate the choice of rotation plane.

.

 

 

------=_Part_74970_11656805.1218608464740--

---

From: "lucas_awad" <lucasawad@gmail.com>
Date: Thu, 14 Aug 2008 18:13:39 -0000
Subject: Re: Something interesting and strange about permutations

So we are saying that MC2D does reflection moves and MC3D simple rotation.

What I'm trying to see is if MC4D should have some different type of
movement, not a reflection and not a simple rotation allowed. The same
with MC5D, with moves that cannot be done in MC4D. Because I think
that if we go up in dimensions we mustn't be able to do the same kind
of movements that we do in a lower dimensional puzzle.

This obviously would make MC4D and MC5D a lot more difficult, but it
is ok if we compare the difficulties between MC2D and MC3D, where the
first can be solved without trying to, I mean, with random moves.

I know that a rubik's cube was done to be a rotation puzzle, but if we
go up or down in dimensions, I think that we shouldn't allow (in
higher) or we can't (in lower) implement the simple rotation around an
axe (one axe).

Also, what I think is that we cannot know the exactly behavior that a
higher dimensional puzzle would have, because we live only in a 3D
space, and we cannot see anything higher than 3D, and cannot imagine a
world in more than 3D, as I think that it cannot be imagined to live
in a 2D world, as a 2D object would be invisible to our eyes. But, us
I said, that is only what I think.

---

From: "Roice Nelson" <roice3@gmail.com>
Date: Wed, 17 Sep 2008 22:24:31 -0500
Subject: Re: [MC4D] Re: Something interesting and strange about permutations

------=_Part_22148_21569170.1221708271501
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Hi Lucas,

Sorry for the very long delay in responding to this. I didn't want to leave
the possible issues you raised unresolved in the thread, but hadn't taken
the time to write out a response until now. I believe we can know the
behavior of the higher dimensional puzzles exactly if we are precise with
our analogies. In a book I read recently, Donal O'Shea wrote about
mathematics "absolute precision buys the freedom to dream meaningfully", and
I agree!

So anyway, I am afraid I have to dissent with the statement "if we go up in
dimensions we mustn't be able to do the same kind of movements that we do in
a lower dimensional puzzle". It seems this is observing a pattern that was
the result of implementation choices that were made rather than observing a
trend through the sequence of dimensions while explicitly controlling the
analogies. To make MC2D interesting, Melinda decided to allow reflection
based twists, but there is nothing fundamental about lower-d puzzles being
able to do movements that the higher-d puzzles can not. On the contrary, as
one moves up the dimension ladder, the capability for additional motions
only increases. There is no motion capable of being done in 2D but not 3D,
or in 3D but not 4D. The set of motions in higher dimensions is a superset,
containing all the lower-d motions plus more that are available because of
the extra space.

I'd argue the reason for the higher difficulty of MC3D vs. MC2D has much
more to do with size of the state spaces of the two puzzles than the motions
allowed in these particular implementations.

To figure out our options for making a twist, we can catalogue all the
possible "similarity" (or shape preserving) motions in any given dimension
of Euclidean space, and these are translation, scaling, rotation, and
reflection. There are no more I am aware of that show up for higher
dimensions, though rotations do get much more interesting as we climb to
higher spaces. Trying to use either translation or scaling as a basis for
twisting would only serve to put the puzzle in quite a different,
unusable form (imagine a 3D cube "twisted" to have one face scaled to twice
the size of all the others). This leaves rotation and reflection as the
only two motions whereby the overall puzzle shape is the same before and
after a twist. One can't physically reflect an object within a given
dimension without either (1) having short term access to a higher dimension
that the object could temporarily move through or (2) if the space had a
certain topology (e.g. a mobius strip or klein bottle), moving the object
through a path that flipped it (but a topology like this of course has not
been observed in our universe to date). Hence the analogical argument for
disallowing reflections on any of these puzzles. But we can of course
loosen the analogy and choose to include them in software implementations if
we want it as a unique extension. And we can do this for puzzles of any
dimension.

Aside: If one chose to completely disallow rotations but allow a minimum
set of reflections for twisting, you could still get all the possible
permutations a puzzle would have with rotations alone (and more actually).
This is because of a property that previously came up, that a rotation can
equivalently be expressed as a set of 2 reflections. Writing this paragraph
made me realize the 3D puzzle reflection extension is more interesting than
in the 2D case because there are similarity reflections through diagonal
axes of a face in addition to coordinate aligned ones. I just checked
David's MC3D implementation and saw that he handles this, distinguishing
reflections by whether an edge or corner is clicked. Nice! (maybe I knew
this in the past and my mind is just failing me)

Well, I'll stop prattling about this. I hope I wasn't too disagreeable on
this topic and just as you said, this is only what I think :) But I really
do think MC4D has it right when comes to how the twisting is performed.

Take Care,
Roice

On 8/14/08, lucas_awad wrote:
>
> So we are saying that MC2D does reflection moves and MC3D simple
> rotation.
>
> What I'm trying to see is if MC4D should have some different type of
> movement, not a reflection and not a simple rotation allowed. The same
> with MC5D, with moves that cannot be done in MC4D. Because I think
> that if we go up in dimensions we mustn't be able to do the same kind
> of movements that we do in a lower dimensional puzzle.
>
> This obviously would make MC4D and MC5D a lot more difficult, but it
> is ok if we compare the difficulties between MC2D and MC3D, where the
> first can be solved without trying to, I mean, with random moves.
>
> I know that a rubik's cube was done to be a rotation puzzle, but if we
> go up or down in dimensions, I think that we shouldn't allow (in
> higher) or we can't (in lower) implement the simple rotation around an
> axe (one axe).
>
> Also, what I think is that we cannot know the exactly behavior that a
> higher dimensional puzzle would have, because we live only in a 3D
> space, and we cannot see anything higher than 3D, and cannot imagine a
> world in more than 3D, as I think that it cannot be imagined to live
> in a 2D world, as a 2D object would be invisible to our eyes. But, us
> I said, that is only what I think.
>
>
>

------=_Part_22148_21569170.1221708271501
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Hi Lucas,

 

Sorry for the very long delay in responding to this.  I didn'=
t want to leave the possible issues you raised unresolved in the thread, bu=
t hadn't taken the time to write out a response until now.  I =
;believe we can know the behavior of the higher dimensional puzzles exactly=
if we are precise with our analogies.  In a book I read recently=
, Donal O'Shea wrote about mathematics "absolute precision bu=
ys the freedom to dream meaningfully", and I agree!

 

So anyway, I am afraid I have to dissent with the statement "if w=
e go up in dimensions we mustn't be able to do the same kind of movemen=
ts that we do in a lower dimensional puzzle".  It seems this is o=
bserving a pattern that was the result of implementation choices that =
were made rather than observing a trend through the sequence of dimensions =
while explicitly controlling the analogies.  To make MC2D interes=
ting, Melinda decided to allow reflection based twists, but there is nothin=
g fundamental about lower-d puzzles being able to do movements that the hig=
her-d puzzles can not.  On the contrary, as one moves up the dimension=
ladder, the capability for additional motions only increases.  There =
is no motion capable of being done in 2D but not 3D, or in 3D but not 4D.&n=
bsp; The set of motions in higher dimensions is a superset, containing all&=
nbsp;the lower-d motions plus more that are available because of the extra =
space.

I'd argue the reason for the higher difficulty=
of MC3D vs. MC2D has much more to do with size of the state spaces of the =
two puzzles than the motions allowed in these particular implementations.r>

 

To figure out our options for making a twist, we can catalogue all the=
possible "similarity" (or shape preserving) motions in any =
given dimension of Euclidean space, and these are translation, scaling, rot=
ation, and reflection.  There are no more I am aware of that show up f=
or higher dimensions, though rotations do get much more interesting as we c=
limb to higher spaces.  Trying to use either translation or scaling as=
a basis for twisting would only serve to put the puzzle in quite a differe=
nt, unusable form (imagine a 3D cube "twisted" to have =
one face scaled to twice the size of all the others).  This leaves rot=
ation and reflection as the only two motions whereby the overall puzzle sha=
pe is the same before and after a twist.  One can't physically ref=
lect an object within a given dimension without either (1) having short ter=
m access to a higher dimension that the object could temporarily move throu=
gh or (2) if the space had a certain topology (e.g. a mobius strip or =
klein bottle), moving the object through a path that flipped it (but a=
topology like this of course has not been observed in our universe to date=
).  Hence the analogical argument for disallowing reflections on any o=
f these puzzles.  But we can of course loosen the analogy and choose t=
o include them in software implementations if we want it as a unique extens=
ion.  And we can do this for puzzles of any dimension.

 

Aside:  If one chose to completely disallow rotations but allow a=
minimum set of reflections for twisting, you could still get all the possi=
ble permutations a puzzle would have with rotations alone (and more actuall=
y).  This is because of a property that previously came up, that a rot=
ation can equivalently be expressed as a set of 2 reflections.  Writin=
g this paragraph made me realize the 3D puzzle reflection extension is more=
interesting than in the 2D case because there are similarity reflections t=
hrough diagonal axes of a face in addition to coordinate aligned ones.=
 I just checked David's MC3D implementation and saw that he handl=
es this, distinguishing reflections by whether an edge or corner is clicked=
.  Nice!  (maybe I knew this in the past and my mind is just fail=
ing me)

 

Well, I'll stop prattling about this.  I hope I wasn't to=
o disagreeable on this topic and just as you said, this is only what I thin=
k :)  But I really do think MC4D has it right when comes to how the tw=
isting is performed.

 

Take Care,

Roice
 

On 8/14/08, =
lucas_awad <>lucasawad@gmail.com> wrote:=20

0px 0.8ex;border-left:#ccc 1px solid">

:0px;margin:0px;width:470px;padding-top:0px">

So we are saying that MC2D does reflection moves and MC3D simple rotatio=
n.

What I'm trying to see is if MC4D should have some different =
type of
movement, not a reflection and not a simple rotation allowed. Th=
e same


with MC5D, with moves that cannot be done in MC4D. Because I think
that =
if we go up in dimensions we mustn't be able to do the same kind
of =
movements that we do in a lower dimensional puzzle.

This obviously w=
ould make MC4D and MC5D a lot more difficult, but it


is ok if we compare the difficulties between MC2D and MC3D, where the
fi=
rst can be solved without trying to, I mean, with random moves.

I kn=
ow that a rubik's cube was done to be a rotation puzzle, but if we


go up or down in dimensions, I think that we shouldn't allow (in
hig=
her) or we can't (in lower) implement the simple rotation around an
=
axe (one axe).

Also, what I think is that we cannot know the exactly=
behavior that a


higher dimensional puzzle would have, because we live only in a 3D
space=
, and we cannot see anything higher than 3D, and cannot imagine a
world =
in more than 3D, as I think that it cannot be imagined to live
in a 2D w=
orld, as a 2D object would be invisible to our eyes. But, us


I said, that is only what I think.

ite" width=3D"1">

------=_Part_22148_21569170.1221708271501--

---

From: Melinda Green <melinda@superliminal.com>
Date: Fri, 26 Sep 2008 00:02:24 -0700
Subject: [MC4D] Re: Something interesting and strange about permutations

This subject of dimensional analogy is very interesting to me. I think
that the way I approach it is to ask a subtly different question than
both of you guys do. You each seem to be asking what *is* the correct
analogy in each dimension whereas I prefer to ask what *should* we
choose the right analogy to be. What we're looking for is the best
definition of an N-dimensional set of puzzles, but "best" in this case
is not the answer to a mathematical question, rather it *is* the
question. "Best" is the question that gives the Rubik's cube as the
answer in 3D plus puzzles that please us the most in the other
dimensions. From my perspective Lucas is making a suggestion which is
entirely reasonable (I.E. not wrong) but which Roice does not find
satisfying. It doesn't work very well for me either but it is not wrong.
Roice on the other hand is suggesting a definition based on
rotations--one that I preferred too, at least maybe until now. My shift
in thinking didn't come from the realization that MC2D didn't seem to
fit perfectly into this definition. It would be nice if it did fit but I
was perfectly happy for it to be an exception, mostly useful for
illustrating state graph properties for these puzzles. By the way, I
added a nice image of the MC2D state graph to the applet page along with
some descriptive text. See http://superliminal.com/cube/mc2d.html.

The thing that really struck me was Roice's observation that rotations
can always be described with pairs of reflections. This started me
thinking that perhaps the "best" analogy might only involve reflection
moves. Looked at this way, perhaps the original Rubik's cube is the
oddity which needed to use planar rotations to satisfy the practical
demands of 3D objects in the physical world. It certainly makes for a
fun and satisfying puzzle but perhaps we shouldn't be more focused on
the way that the plastic puzzle operates than the mathematical group
that it operates upon.

So now we have the basis for defining a new analogy that can reproduce
our puzzles in N dimensions:

"A valid move is any combination of reflections of a hyperface
that leaves its orientation unchanged."

MC2D moves only do interesting things with odd numbers of reflections,
and the Rubik's cube is only physically implementable for even numbers.
Looked at this way, perhaps the "best" version of MC3D would allow both
odd and even numbers of reflections per move but with the option to
restrict the available moves to even numbers of reflections in order to
satisfy people with a nostalgia for physical reality. ;-) Looking at
David's implementation, I now see that this is exactly what he did
although it is the default mode and that each reflecting click must be
preceded by ctrl-q. David: I would love if you would add a new toggle so
that plain clicks always perform reflection moves and ctrl-q clicks
perform rotations.

I purposely call all of these operations "moves" instead of twists
because thinking about twisting drags in all the problems with rotations
that we've been struggling with. I kept saying that it was better to
think about planes of rotation rather than axes of rotation, but that
seems unnatural for a lot of people. If we base the discussions on
reflections, then this suddenly becomes quite natural.

What do you people think? Is this a good basis for defining
N-dimensional twisty puzzles? If so, the only things that remain to
figure out are the best user interface for computer implementations
based on this model, and how best to animate the moves, if at all. I'm
not signing up to implement anything anytime soon but I do enjoy the
thought exercise. I did not have any sense for what would make for a
good user interaction model but I think that David may have pointed us
in the right direction. I do have some ideas for animations that might
work. Let's start with a single reflection move. These can be
reflections about a point, line, or any space of dimension lower than
the puzzle itself. The simplest animation would seem to be a linear
interpolation of the beginning and ending vertex positions. That would
leave a moment of degeneracy in the middle when the part being moved
gets flattened into that point or line, etc. but that's fine. Imagine
that happening in MC2D. A 3x1 slice would collapse into a 0x1 line at
the midpoint of the motion. It is interesting to notice that that is
exactly what you would see in the current projection if the motion was
implemented as a 3D twist coming out of the plane and then back as some
people have mentioned. Maybe an equivalent reflection move on MC3D would
involve an affine 4D rotation in order to flip over a 2x2x1 slice,
leaving it turned inside-out? It's an interesting thought.

And then what about those pairs of reflections that Roice says can
produce rotations? How might we animate those? It seems like we would
have the same two natural choices. We could perform a linear
interpolation of the vertex positions, or maybe we could find pure
rotation matrices that achieve the same results. Even if all rotations
can be expressed as pairs of reflections, it might not follow that all
pairs of reflections can be expressed as rotations, but if it is true
then we will have found a way to redefine all of our puzzles, including
the original Rubik's cube. So now we have come full circle and it is
time to ask what have we gained. First we might have gained a simpler
way to way to define the puzzles we already know and with some new
moves. Second, it might show us how to implement these puzzles in any
number of dimensions. And finally, it might give us back all our
familiar puzzles (Rubik's cube, MC4D, Hyperminx, etc.) as special cases
in which moves consist of pairs of reflections. Oh, and it gives us an
MC2D that is *not* a special case! And I swear that was not my
intention! :-)

-melinda

Roice Nelson wrote:
> Hi Lucas,
>
> Sorry for the very long delay in responding to this. I didn't want to
> leave the possible issues you raised unresolved in the thread, but
> hadn't taken the time to write out a response until now. I believe we
> can know the behavior of the higher dimensional puzzles exactly if we
> are precise with our analogies. In a book I read recently,
> Donal O'Shea wrote about mathematics "absolute precision buys the
> freedom to dream meaningfully", and I agree!
>
> So anyway, I am afraid I have to dissent with the statement "if we go
> up in dimensions we mustn't be able to do the same kind of movements
> that we do in a lower dimensional puzzle". It seems this is observing
> a pattern that was the result of implementation choices that were made
> rather than observing a trend through the sequence of dimensions while
> explicitly controlling the analogies. To make MC2D interesting,
> Melinda decided to allow reflection based twists, but there is nothing
> fundamental about lower-d puzzles being able to do movements that the
> higher-d puzzles can not. On the contrary, as one moves up the
> dimension ladder, the capability for additional motions only
> increases. There is no motion capable of being done in 2D but not 3D,
> or in 3D but not 4D. The set of motions in higher dimensions is a
> superset, containing all the lower-d motions plus more that are
> available because of the extra space.
>
> I'd argue the reason for the higher difficulty of MC3D vs. MC2D has
> much more to do with size of the state spaces of the two puzzles than
> the motions allowed in these particular implementations.
>
> To figure out our options for making a twist, we can catalogue all the
> possible "similarity" (or shape preserving) motions in any given
> dimension of Euclidean space, and these are translation, scaling,
> rotation, and reflection. There are no more I am aware of that show
> up for higher dimensions, though rotations do get much more
> interesting as we climb to higher spaces. Trying to use either
> translation or scaling as a basis for twisting would only serve to put
> the puzzle in quite a different, unusable form (imagine a 3D cube
> "twisted" to have one face scaled to twice the size of all the
> others). This leaves rotation and reflection as the only two motions
> whereby the overall puzzle shape is the same before and after a
> twist. One can't physically reflect an object within a given
> dimension without either (1) having short term access to a higher
> dimension that the object could temporarily move through or (2) if the
> space had a certain topology (e.g. a mobius strip or klein bottle),
> moving the object through a path that flipped it (but a topology like
> this of course has not been observed in our universe to date). Hence
> the analogical argument for disallowing reflections on any of these
> puzzles. But we can of course loosen the analogy and choose to
> include them in software implementations if we want it as a unique
> extension. And we can do this for puzzles of any dimension.
>
> Aside: If one chose to completely disallow rotations but allow a
> minimum set of reflections for twisting, you could still get all the
> possible permutations a puzzle would have with rotations alone (and
> more actually). This is because of a property that previously came
> up, that a rotation can equivalently be expressed as a set of 2
> reflections. Writing this paragraph made me realize the 3D puzzle
> reflection extension is more interesting than in the 2D case because
> there are similarity reflections through diagonal axes of a face in
> addition to coordinate aligned ones. I just checked David's MC3D
> implementation and saw that he handles this, distinguishing
> reflections by whether an edge or corner is clicked. Nice! (maybe I
> knew this in the past and my mind is just failing me)
>
> Well, I'll stop prattling about this. I hope I wasn't too
> disagreeable on this topic and just as you said, this is only what I
> think :) But I really do think MC4D has it right when comes to how
> the twisting is performed.
>
> Take Care,
> Roice

---

From: "Jenelle Levenstein" <jenelle.levenstein@gmail.com>
Date: Fri, 26 Sep 2008 10:43:39 -0500
Subject: Re: [MC4D] Re: Something interesting and strange about permutations

------=_Part_24466_14330550.1222443819144
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Content-Transfer-Encoding: 7bit
Content-Disposition: inline

If you implement moves as reflections then each move will turn some of the
pieces on the cube inside out. Do you think that if you implemented the 3^3
cube with your new definition of moves it would be easier or more difficult
to solve? Although it may be possible to do all the rotations on a 3D cube
using reflections it would take a long time for anyone to get used to and
the possibility of being able to put a piece in position inside out will
make solving more complicated. However it sound like a very interesting
puzzle.

On Fri, Sep 26, 2008 at 2:02 AM, Melinda Green wrote:

> This subject of dimensional analogy is very interesting to me. I think
> that the way I approach it is to ask a subtly different question than
> both of you guys do. You each seem to be asking what *is* the correct
> analogy in each dimension whereas I prefer to ask what *should* we
> choose the right analogy to be. What we're looking for is the best
> definition of an N-dimensional set of puzzles, but "best" in this case
> is not the answer to a mathematical question, rather it *is* the
> question. "Best" is the question that gives the Rubik's cube as the
> answer in 3D plus puzzles that please us the most in the other
> dimensions. From my perspective Lucas is making a suggestion which is
> entirely reasonable (I.E. not wrong) but which Roice does not find
> satisfying. It doesn't work very well for me either but it is not wrong.
> Roice on the other hand is suggesting a definition based on
> rotations--one that I preferred too, at least maybe until now. My shift
> in thinking didn't come from the realization that MC2D didn't seem to
> fit perfectly into this definition. It would be nice if it did fit but I
> was perfectly happy for it to be an exception, mostly useful for
> illustrating state graph properties for these puzzles. By the way, I
> added a nice image of the MC2D state graph to the applet page along with
> some descriptive text. See http://superliminal.com/cube/mc2d.html.
>
> The thing that really struck me was Roice's observation that rotations
> can always be described with pairs of reflections. This started me
> thinking that perhaps the "best" analogy might only involve reflection
> moves. Looked at this way, perhaps the original Rubik's cube is the
> oddity which needed to use planar rotations to satisfy the practical
> demands of 3D objects in the physical world. It certainly makes for a
> fun and satisfying puzzle but perhaps we shouldn't be more focused on
> the way that the plastic puzzle operates than the mathematical group
> that it operates upon.
>
> So now we have the basis for defining a new analogy that can reproduce
> our puzzles in N dimensions:
>
> "A valid move is any combination of reflections of a hyperface
> that leaves its orientation unchanged."
>
> MC2D moves only do interesting things with odd numbers of reflections,
> and the Rubik's cube is only physically implementable for even numbers.
> Looked at this way, perhaps the "best" version of MC3D would allow both
> odd and even numbers of reflections per move but with the option to
> restrict the available moves to even numbers of reflections in order to
> satisfy people with a nostalgia for physical reality. ;-) Looking at
> David's implementation, I now see that this is exactly what he did
> although it is the default mode and that each reflecting click must be
> preceded by ctrl-q. David: I would love if you would add a new toggle so
> that plain clicks always perform reflection moves and ctrl-q clicks
> perform rotations.
>
> I purposely call all of these operations "moves" instead of twists
> because thinking about twisting drags in all the problems with rotations
> that we've been struggling with. I kept saying that it was better to
> think about planes of rotation rather than axes of rotation, but that
> seems unnatural for a lot of people. If we base the discussions on
> reflections, then this suddenly becomes quite natural.
>
> What do you people think? Is this a good basis for defining
> N-dimensional twisty puzzles? If so, the only things that remain to
> figure out are the best user interface for computer implementations
> based on this model, and how best to animate the moves, if at all. I'm
> not signing up to implement anything anytime soon but I do enjoy the
> thought exercise. I did not have any sense for what would make for a
> good user interaction model but I think that David may have pointed us
> in the right direction. I do have some ideas for animations that might
> work. Let's start with a single reflection move. These can be
> reflections about a point, line, or any space of dimension lower than
> the puzzle itself. The simplest animation would seem to be a linear
> interpolation of the beginning and ending vertex positions. That would
> leave a moment of degeneracy in the middle when the part being moved
> gets flattened into that point or line, etc. but that's fine. Imagine
> that happening in MC2D. A 3x1 slice would collapse into a 0x1 line at
> the midpoint of the motion. It is interesting to notice that that is
> exactly what you would see in the current projection if the motion was
> implemented as a 3D twist coming out of the plane and then back as some
> people have mentioned. Maybe an equivalent reflection move on MC3D would
> involve an affine 4D rotation in order to flip over a 2x2x1 slice,
> leaving it turned inside-out? It's an interesting thought.
>
> And then what about those pairs of reflections that Roice says can
> produce rotations? How might we animate those? It seems like we would
> have the same two natural choices. We could perform a linear
> interpolation of the vertex positions, or maybe we could find pure
> rotation matrices that achieve the same results. Even if all rotations
> can be expressed as pairs of reflections, it might not follow that all
> pairs of reflections can be expressed as rotations, but if it is true
> then we will have found a way to redefine all of our puzzles, including
> the original Rubik's cube. So now we have come full circle and it is
> time to ask what have we gained. First we might have gained a simpler
> way to way to define the puzzles we already know and with some new
> moves. Second, it might show us how to implement these puzzles in any
> number of dimensions. And finally, it might give us back all our
> familiar puzzles (Rubik's cube, MC4D, Hyperminx, etc.) as special cases
> in which moves consist of pairs of reflections. Oh, and it gives us an
> MC2D that is *not* a special case! And I swear that was not my
> intention! :-)
>
> -melinda
>
>
> Roice Nelson wrote:
> > Hi Lucas,
> >
> > Sorry for the very long delay in responding to this. I didn't want to
> > leave the possible issues you raised unresolved in the thread, but
> > hadn't taken the time to write out a response until now. I believe we
> > can know the behavior of the higher dimensional puzzles exactly if we
> > are precise with our analogies. In a book I read recently,
> > Donal O'Shea wrote about mathematics "absolute precision buys the
> > freedom to dream meaningfully", and I agree!
> >
> > So anyway, I am afraid I have to dissent with the statement "if we go
> > up in dimensions we mustn't be able to do the same kind of movements
> > that we do in a lower dimensional puzzle". It seems this is observing
> > a pattern that was the result of implementation choices that were made
> > rather than observing a trend through the sequence of dimensions while
> > explicitly controlling the analogies. To make MC2D interesting,
> > Melinda decided to allow reflection based twists, but there is nothing
> > fundamental about lower-d puzzles being able to do movements that the
> > higher-d puzzles can not. On the contrary, as one moves up the
> > dimension ladder, the capability for additional motions only
> > increases. There is no motion capable of being done in 2D but not 3D,
> > or in 3D but not 4D. The set of motions in higher dimensions is a
> > superset, containing all the lower-d motions plus more that are
> > available because of the extra space.
> >
> > I'd argue the reason for the higher difficulty of MC3D vs. MC2D has
> > much more to do with size of the state spaces of the two puzzles than
> > the motions allowed in these particular implementations.
> >
> > To figure out our options for making a twist, we can catalogue all the
> > possible "similarity" (or shape preserving) motions in any given
> > dimension of Euclidean space, and these are translation, scaling,
> > rotation, and reflection. There are no more I am aware of that show
> > up for higher dimensions, though rotations do get much more
> > interesting as we climb to higher spaces. Trying to use either
> > translation or scaling as a basis for twisting would only serve to put
> > the puzzle in quite a different, unusable form (imagine a 3D cube
> > "twisted" to have one face scaled to twice the size of all the
> > others). This leaves rotation and reflection as the only two motions
> > whereby the overall puzzle shape is the same before and after a
> > twist. One can't physically reflect an object within a given
> > dimension without either (1) having short term access to a higher
> > dimension that the object could temporarily move through or (2) if the
> > space had a certain topology (e.g. a mobius strip or klein bottle),
> > moving the object through a path that flipped it (but a topology like
> > this of course has not been observed in our universe to date). Hence
> > the analogical argument for disallowing reflections on any of these
> > puzzles. But we can of course loosen the analogy and choose to
> > include them in software implementations if we want it as a unique
> > extension. And we can do this for puzzles of any dimension.
> >
> > Aside: If one chose to completely disallow rotations but allow a
> > minimum set of reflections for twisting, you could still get all the
> > possible permutations a puzzle would have with rotations alone (and
> > more actually). This is because of a property that previously came
> > up, that a rotation can equivalently be expressed as a set of 2
> > reflections. Writing this paragraph made me realize the 3D puzzle
> > reflection extension is more interesting than in the 2D case because
> > there are similarity reflections through diagonal axes of a face in
> > addition to coordinate aligned ones. I just checked David's MC3D
> > implementation and saw that he handles this, distinguishing
> > reflections by whether an edge or corner is clicked. Nice! (maybe I
> > knew this in the past and my mind is just failing me)
> >
> > Well, I'll stop prattling about this. I hope I wasn't too
> > disagreeable on this topic and just as you said, this is only what I
> > think :) But I really do think MC4D has it right when comes to how
> > the twisting is performed.
> >
> > Take Care,
> > Roice
>
>
>

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If you implement moves as reflections then each move will turn some of the pieces on the cube inside out. Do you think that if you implemented the 3^3 cube with your new definition of moves it would be easier or more difficult to solve? Although it may be possible to do all the rotations on a 3D cube using reflections it would take a long time for anyone to get used to and the possibility of being able to put a piece in position inside out will make solving more complicated. However it sound like a very interesting puzzle.

On Fri, Sep 26, 2008 at 2:02 AM, Melinda Green <melinda@superliminal.com> wrote:

This subject of dimensional analogy is very interesting to me. I think

that the way I approach it is to ask a subtly different question than

both of you guys do. You each seem to be asking what *is* the correct

analogy in each dimension whereas I prefer to ask what *should* we

choose the right analogy to be. What we're looking for is the best

definition of an N-dimensional set of puzzles, but "best" in this case

is not the answer to a mathematical question, rather it *is* the

question. "Best" is the question that gives the Rubik's cube as the

answer in 3D plus puzzles that please us the most in the other

dimensions. From my perspective Lucas is making a suggestion which is

entirely reasonable (I.E. not wrong) but which Roice does not find

satisfying. It doesn't work very well for me either but it is not wrong.

Roice on the other hand is suggesting a definition based on

rotations--one that I preferred too, at least maybe until now. My shift

in thinking didn't come from the realization that MC2D didn't seem to

fit perfectly into this definition. It would be nice if it did fit but I

was perfectly happy for it to be an exception, mostly useful for

illustrating state graph properties for these puzzles. By the way, I

added a nice image of the MC2D state graph to the applet page along with

some descriptive text. See http://superliminal.com/cube/mc2d.html.



The thing that really struck me was Roice's observation that rotations

can always be described with pairs of reflections. This started me

thinking that perhaps the "best" analogy might only involve reflection

moves. Looked at this way, perhaps the original Rubik's cube is the

oddity which needed to use planar rotations to satisfy the practical

demands of 3D objects in the physical world. It certainly makes for a

fun and satisfying puzzle but perhaps we shouldn't be more focused on

the way that the plastic puzzle operates than the mathematical group

that it operates upon.



So now we have the basis for defining a new analogy that can reproduce

our puzzles in N dimensions:



"A valid move is any combination of reflections of a hyperface

that leaves its orientation unchanged."



MC2D moves only do interesting things with odd numbers of reflections,

and the Rubik's cube is only physically implementable for even numbers.

Looked at this way, perhaps the "best" version of MC3D would allow both

odd and even numbers of reflections per move but with the option to

restrict the available moves to even numbers of reflections in order to

satisfy people with a nostalgia for physical reality. ;-) Looking at

David's implementation, I now see that this is exactly what he did

although it is the default mode and that each reflecting click must be

preceded by ctrl-q. David: I would love if you would add a new toggle so

that plain clicks always perform reflection moves and ctrl-q clicks

perform rotations.



I purposely call all of these operations "moves" instead of twists

because thinking about twisting drags in all the problems with rotations

that we've been struggling with. I kept saying that it was better to

think about planes of rotation rather than axes of rotation, but that

seems unnatural for a lot of people. If we base the discussions on

reflections, then this suddenly becomes quite natural.



What do you people think? Is this a good basis for defining

N-dimensional twisty puzzles? If so, the only things that remain to

figure out are the best user interface for computer implementations

based on this model, and how best to animate the moves, if at all. I'm

not signing up to implement anything anytime soon but I do enjoy the

thought exercise. I did not have any sense for what would make for a

good user interaction model but I think that David may have pointed us

in the right direction. I do have some ideas for animations that might

work. Let's start with a single reflection move. These can be

reflections about a point, line, or any space of dimension lower than

the puzzle itself. The simplest animation would seem to be a linear

interpolation of the beginning and ending vertex positions. That would

leave a moment of degeneracy in the middle when the part being moved

gets flattened into that point or line, etc. but that's fine. Imagine

that happening in MC2D. A 3x1 slice would collapse into a 0x1 line at

the midpoint of the motion. It is interesting to notice that that is

exactly what you would see in the current projection if the motion was

implemented as a 3D twist coming out of the plane and then back as some

people have mentioned. Maybe an equivalent reflection move on MC3D would

involve an affine 4D rotation in order to flip over a 2x2x1 slice,

leaving it turned inside-out? It's an interesting thought.



And then what about those pairs of reflections that Roice says can

produce rotations? How might we animate those? It seems like we would

have the same two natural choices. We could perform a linear

interpolation of the vertex positions, or maybe we could find pure

rotation matrices that achieve the same results. Even if all rotations

can be expressed as pairs of reflections, it might not follow that all

pairs of reflections can be expressed as rotations, but if it is true

then we will have found a way to redefine all of our puzzles, including

the original Rubik's cube. So now we have come full circle and it is

time to ask what have we gained. First we might have gained a simpler

way to way to define the puzzles we already know and with some new

moves. Second, it might show us how to implement these puzzles in any

number of dimensions. And finally, it might give us back all our

familiar puzzles (Rubik's cube, MC4D, Hyperminx, etc.) as special cases

in which moves consist of pairs of reflections. Oh, and it gives us an

MC2D that is *not* a special case! And I swear that was not my

intention! :-)



-melinda

Roice Nelson wrote:

> Hi Lucas,

>

> Sorry for the very long delay in responding to this. I didn't want to

> leave the possible issues you raised unresolved in the thread, but

> hadn't taken the time to write out a response until now. I believe we

> can know the behavior of the higher dimensional puzzles exactly if we

> are precise with our analogies. In a book I read recently,

> Donal O'Shea wrote about mathematics "absolute precision buys the

> freedom to dream meaningfully", and I agree!

>

> So anyway, I am afraid I have to dissent with the statement "if we go

> up in dimensions we mustn't be able to do the same kind of movements

> that we do in a lower dimensional puzzle". It seems this is observing

> a pattern that was the result of implementation choices that were made

> rather than observing a trend through the sequence of dimensions while

> explicitly controlling the analogies. To make MC2D interesting,

> Melinda decided to allow reflection based twists, but there is nothing

> fundamental about lower-d puzzles being able to do movements that the

> higher-d puzzles can not. On the contrary, as one moves up the

> dimension ladder, the capability for additional motions only

> increases. There is no motion capable of being done in 2D but not 3D,

> or in 3D but not 4D. The set of motions in higher dimensions is a

> superset, containing all the lower-d motions plus more that are

> available because of the extra space.

>

> I'd argue the reason for the higher difficulty of MC3D vs. MC2D has

> much more to do with size of the state spaces of the two puzzles than

> the motions allowed in these particular implementations.

>

> To figure out our options for making a twist, we can catalogue all the

> possible "similarity" (or shape preserving) motions in any given

> dimension of Euclidean space, and these are translation, scaling,

> rotation, and reflection. There are no more I am aware of that show

> up for higher dimensions, though rotations do get much more

> interesting as we climb to higher spaces. Trying to use either

> translation or scaling as a basis for twisting would only serve to put

> the puzzle in quite a different, unusable form (imagine a 3D cube

> "twisted" to have one face scaled to twice the size of all the

> others). This leaves rotation and reflection as the only two motions

> whereby the overall puzzle shape is the same before and after a

> twist. One can't physically reflect an object within a given

> dimension without either (1) having short term access to a higher

> dimension that the object could temporarily move through or (2) if the

> space had a certain topology (e.g. a mobius strip or klein bottle),

> moving the object through a path that flipped it (but a topology like

> this of course has not been observed in our universe to date). Hence

> the analogical argument for disallowing reflections on any of these

> puzzles. But we can of course loosen the analogy and choose to

> include them in software implementations if we want it as a unique

> extension. And we can do this for puzzles of any dimension.

>

> Aside: If one chose to completely disallow rotations but allow a

> minimum set of reflections for twisting, you could still get all the

> possible permutations a puzzle would have with rotations alone (and

> more actually). This is because of a property that previously came

> up, that a rotation can equivalently be expressed as a set of 2

> reflections. Writing this paragraph made me realize the 3D puzzle

> reflection extension is more interesting than in the 2D case because

> there are similarity reflections through diagonal axes of a face in

> addition to coordinate aligned ones. I just checked David's MC3D

> implementation and saw that he handles this, distinguishing

> reflections by whether an edge or corner is clicked. Nice! (maybe I

> knew this in the past and my mind is just failing me)

>

> Well, I'll stop prattling about this. I hope I wasn't too

> disagreeable on this topic and just as you said, this is only what I

> think :) But I really do think MC4D has it right when comes to how

> the twisting is performed.

>

> Take Care,

> Roice

------=_Part_24466_14330550.1222443819144--

---

From: David Vanderschel <DvdS@Austin.RR.com>
Date: 26 Sep 2008 19:23:13 -0500
Subject: Re: [MC4D] Something interesting and strange about permutations

On Friday, September 26, "Melinda Green" wrote:
>... From my perspective Lucas is making a suggestion
>which is entirely reasonable (I.E. not wrong) but
>which Roice does not find satisfying. It doesn't work
>very well for me either but it is not wrong. Roice
>on the other hand is suggesting a definition based on
>rotations-- ...

My perspective on that dialogue is a bit different.
I could not really discern any specific suggestion in
what Lucas wrote. What Roice wrote struck me as a
polite attempt to expose the logical flaws in what
Lucas had written. I thought Roice's observations
were very perceptive. Roice mentioned the reflection
alternative as a point of interest, but I did not
perceive that he was suggesting it in an advocacy
mode.

>one that I preferred too, at least maybe until
>now. My shift in thinking didn't come from the
>realization that MC2D didn't seem to fit perfectly
>into this definition. It would be nice if it did fit
>but I was perfectly happy for it to be an exception,
>mostly useful for illustrating state graph properties
>for these puzzles. By the way, I added a nice image
>of the MC2D state graph to the applet page along with
>some descriptive text. See
>http://superliminal.com/cube/mc2d.html.

I have more to say in this regard, but I will wait and
make those comments in the earlier thread. (I got
behind on everything by trying to watch too much
Olympics coverage. Not caught up yet. :( )

>The thing that really struck me was Roice's
>observation that rotations can always be described
>with pairs of reflections. This started me thinking
>that perhaps the "best" analogy might only involve
>reflection moves. Looked at this way, perhaps the
>original Rubik's cube is the oddity which needed to
>use planar rotations to satisfy the practical demands
>of 3D objects in the physical world. It certainly
>makes for a fun and satisfying puzzle but perhaps we
>shouldn't be more focused on the way that the plastic
>puzzle operates than the mathematical group that it
>operates upon.

If you admit reflections in addition to non-reflecting
reorientations, then you are introducing a lot more
possible permutations of the stickers. But the group
of the original non-reflecting puzzle remains as a
subgroup of the larger group that includes
reflections.

David Smith, you might want to consider extending your
permutation counting exercise to include also puzzles
with mirroring allowed. I am actually curious about
how many-fold is the increase even for the 3x3
3-puzzle. (For myself, the mirroring issue seems more
interesting than the super-supercube considerations.)

>So now we have the basis for defining a new analogy that can reproduce
>our puzzles in N dimensions:

> "A valid move is any combination of reflections of a hyperface
>that leaves its orientation unchanged."

Since all non-reflecting reorientations can be
achieved by reflecting pairs, this simply amounts to
adding reflection to what we already had. I.e., we
are just talking about the binary choice of adding
reflections - or not.

>MC2D moves only do interesting things with odd
>numbers of reflections,

? I do not understand the above statement. Why is a
4-cycle (Rotate corner positions.) not interesting?
[(1,2)(3,4)] * [(2,4)] = [(4,3,2,1)]

>and the Rubik's cube is only physically implementable
>for even numbers. Looked at this way, perhaps the
>"best" version of MC3D would allow both odd and even
>numbers of reflections per move but with the option
>to restrict the available moves to even numbers of
>reflections in order to satisfy people with a
>nostalgia for physical reality. ;-) Looking at
>David's implementation, I now see that this is
>exactly what he did although it is the default mode
>and that each reflecting click must be preceded by
>ctrl-q. David: I would love if you would add a new
>toggle so that plain clicks always perform reflection
>moves and ctrl-q clicks perform rotations.

I am pleased that you can imagine using the feature so
much that you regard the ctrl-q clicks as burdensome.
I don't anticipate doing any maintenance on the
program in the next few months; but I will consider it
when I do. Personally, I regarded the feature as sort
of a 'neat' add-on, not to be used all that much.
Thus I introduced it in a manner which did not impact
the normal non-reflecting mode of operation. The
toggle you suggest would still achieve this. When I
have played with it myself, I used the reflecting
twists rather sparingly, so the prefix was no burden.

IMO, the interest of discovering how to generate any
rotation with two reflections does not last long. But
if what the user wants is a rotation, it does not make
a lot of sense to me to force the user to achieve it
as two reflections.

In working with a 4-puzzle simulator of my own, I
found that I wanted to specify twists incrementally.
(E.g., swap these two axes; then flip that one.) I
came to the realization that things would actually be
easier to think about if I temporarily allowed
reflected states in a slice as long as the user got
the slice back to unmirrored before twisting any other
slice.

>I purposely call all of these operations "moves"
>instead of twists because thinking about twisting
>drags in all the problems with rotations that we've
>been struggling with. I kept saying that it was
>better to think about planes of rotation rather than
>axes of rotation, but that seems unnatural for a lot
>of people. If we base the discussions on reflections,
>then this suddenly becomes quite natural.

I am not sure what the "this" is in the preceding
statement. I agree that we should not be thinking
about "axis of rotation" in dimensions higher than 3,
as I think it is an erroneous concept. I would prefer
to continue to use "axis" to refer to a 1-dimensional
subspace. In the context of rotations in dimensions
higher than 3, the thing analogous to an axis which
one should be talking about is the "fixed space" of
the rotation. The dimension of that fixed space is 2
less than the dimension in which we are operating.
The plane of rotation and the fixed space are uniquely
related to one another by orthogonality; so, from a
mathematical point of view, there is really no
difference in which you want to talk about. There are
definitely circumstances when the fixed space view
offers more insight. There are also circumstances
when it is useful to be able to switch back and forth
between the two in contemplating the significance of a
reorientation.

>What do you people think? Is this a good basis for
>defining N-dimensional twisty puzzles?

IMO, no. As I indicated above, if what a user
imagines he wants is a non-reflecting reorientation,
there is no need to force that user into using
reflections to achieve it. That just makes the puzzle
more tedious without making it more interesting. (If
I had macros and no non-reflecting twists, the first
thing I would do would be to create macros for the
non-reflecting twists.) However, _adding_ the
possibility of reflections (as opposed to restricting
to reflections) remains interesting.

>If so, the only things that remain to figure out are
>the best user interface for computer implementations
>based on this model, and how best to animate the
>moves, if at all.

I don't think there is any difficulty with respect to
animation. My implementation in 3-space was actually
trivial to achieve, as it used machinery which already
existed. You can think of it this way: Say it is an
n-dimensional puzzle. Embed the puzzle in a
(n+1)-dimension space by adding a new coordinate axis.
Now a reflection is a 180 degree rotation in a plane
that includes the new axis. The intersection of that
plane with the orginal subspace of the puzzle is the
(n-1)-dimensional hyperplane in n-space through which
the reflection occurs. What I implemented for
animating the reflection in 3-space is actually what
you would see in 3D (after the 4D scene is projected
down to 3D) while the rotation in 4-space is
proceeding. I don't think that there is any
difficulty extending the approach to higher
dimensions. (In MC3D, you can use the feature to
control the animation with the mouse to examine more
closely what happens at the critical flattening point
of the animation process.)

For nD, the user interface that I would want would
include the possibility to flip (negate) any axis and
the possibility to swap any two axes - combinations of
which can be composed to specify a more complex
reorientation and/or reflection. As it happens, these
are also sufficient to achieve all possible
reorientations; but others can be imagined.

>I'm not signing up to implement anything anytime soon
>but I do enjoy the thought exercise. I did not have
>any sense for what would make for a good user
>interaction model but I think that David may have
>pointed us in the right direction. I do have some
>ideas for animations that might work. Let's start
>with a single reflection move. These can be
>reflections about a point, line, or any space of
>dimension lower than the puzzle itself.

Not exactly. Any reflection will occur with respect
to an (n-1)-dimensional reflection hyperplane. In
that (n-1)-dimensional plane there can be embedded a
hyperplane of lower dimension which one may wish to be
included in the reflection plane. But a hyperplane of
dimension less than n-1 does not uniquely determine a
reflection. It does determine a family of possible
reflection planes which include it.

>The simplest animation would seem to be a linear
>interpolation of the beginning and ending vertex
>positions. That would leave a moment of degeneracy in
>the middle when the part being moved gets flattened
>into that point or line, etc. but that's
>fine.

It's more than fine: It's reality! (Mathematically
speaking, of course.)

>Imagine that happening in MC2D. A 3x1 slice would
>collapse into a 0x1 line at the midpoint of the
>motion. It is interesting to notice that that is
>exactly what you would see in the current projection
>if the motion was implemented as a 3D twist coming
>out of the plane and then back as some people have
>mentioned. Maybe an equivalent reflection move on
>MC3D would involve an affine 4D rotation in order to
>flip over a 2x2x1 slice, leaving it turned
>inside-out? It's an interesting thought.

Indeed. It is what I did and what I explained in
general n-dimensional terms above. I do not
understand the comment about "inside-out". A
reflection does not turn things inside out. E.g.,
look at your image in a mirror. If you try to flip a
slice of the 3-puzzle in 3D, you do wind up
(uselessly) with it being outside in; but it is not
reflected. (I wrote "outside in" because the most
dominant apparent effect is that the 3x3 array of
stickers that had been on the outside are now
invisibly facing inwards.)

>And then what about those pairs of reflections that
>Roice says can produce rotations? How might we
>animate those? It seems like we would have the same
>two natural choices. We could perform a linear
>interpolation of the vertex positions, or maybe we
>could find pure rotation matrices that achieve the
>same results. Even if all rotations can be expressed
>as pairs of reflections, it might not follow that all
>pairs of reflections can be expressed as rotations,

But they can be. One thing that needs to be mentioned
is that the reflection planes cannot be chosen
arbitrarily. They must be such that the puzzle
transforms onto itself in the same space. Given that
restriction, it follows that the result of two
reflections must be a non-reflecting reorientation.
(What else could it be?)

>but if it is true then we will have found a way to
>redefine all of our puzzles, including the original
>Rubik's cube. So now we have come full circle and it
>is time to ask what have we gained. First we might
>have gained a simpler way to way to define the
>puzzles we already know and with some new
>moves.

I don't really see this. It seems to me that the
relevant observation is that all these puzzles can be
extended by adding the possibility of reflecting
'twists'. (Indeed, the 2-puzzle does nothing at all
unless you admit reflections.)

Assuming that some sort of implementation has been
provided, it seems to me that it is up to the puzzler
whether he wants to deal with the potential
complications of introducing mirroring twists. Some
will find the extended puzzle to be interesting.

I have yet to decide whether the 3-puzzle with
reflections is easier or more difficult than without.
A number of the usual parity considerations go out the
window with reflections allowed. You can achieve
things like every cubie placed correctly except for a
single unoriented corner.

>Second, it might show us how to implement these
>puzzles in any number of dimensions.

I am not sure what is the issue here. In principle,
reorientations are not difficult to specify in any
dimension. (Any reorientation can be seen as a
permutation of the axes and negation of a subset of
them. There is no mirroring if and only if the
even/odd parity of the permutation is equal to that of
the number of axes flipped.)

>And finally, it might give us back all our familiar
>puzzles (Rubik's cube, MC4D, Hyperminx, etc.) as
>special cases in which moves consist of pairs of
>reflections.

I.e., the groups of the non-reflecting puzzles remain
as subgroups of the puzzles extended to include
mirroring.

>Oh, and it gives us an MC2D that is *not* a special
>case! And I swear that was not my intention! :-)

Since reflecting twists can be added to extend a
puzzle of any dimension, it can be argued that MC2D
was never a special case in the first place. It is
just that the order-3 n-puzzle for n=2 is especially
dull if the mirroring extension is not invoked.

The 2-puzzle is interesting in that it would appear to
be possible to physically implement a device that
actually permits the reflecting twists by doing
rotations in 3-space. (In this case, I normally refer
to them as "flips" rather than "twists".) I said
"possible". However, I have not devined a mechanism
that would achieve it. (Each corner must be attached
to at least one of the edge pieces adjacent to it.
The trick is to make it release from one of those
edges when you start flipping the other edge. It
would take a little slop I think.)



On Friday, September 26, "Jenelle Levenstein" wrote:
>If you implement moves as reflections then each move
>will turn some of the pieces on the cube inside
>out.

Not true. E.g., for the 3-puzzle, the corner cubies
can exist in two states - mirrored or not -
independent of their current locations. But they are
still right side out. The edge cubies are not chiral
so they cannot be in reflected states.

>Do you think that if you implemented the 3^3 cube
>with your new definition of moves it would be easier
>or more difficult to solve?

The implementation already exists. You can try it
yourself. At first, having the possibility of
mirrored corners will be perceived as a substantial
complication. However, there are some transformations
that are easier to achieve with reflections possible,
so I am not sure it would be perceived as more
difficult in the long run. (Though preservation of
some the previous parities disappears, a new parity on
the number of mirrored corners arises. That number
must always be even.)

When I have worked the puzzle after a scramble with
mirroring allowed, my first project was to get all the
corners to an unmirrored state. Then I proceeded
normally without doing reflections. There are
probably better ways.

>Although it may be possible to do all the rotations
>on a 3D cube using reflections it would take a long
>time for anyone to get used to

Not really difficult to learn. Just tedious to
execute if you have to do it every time what you want
is a simple turn.

>and the possibility of being able to put a piece in
>position inside out will make solving more
>complicated.

As I pointed out above, there is the complication of
mirrored corners - but not "inside out" anything.

>However it sound like a very interesting puzzle.

And you can experience it now!:
http://david-v.home.texas.net/MC3D/


Regards,
David V.

---

From: "lucas_awad" <lucasawad@gmail.com>
Date: Sat, 27 Sep 2008 01:46:24 -0000
Subject: Re: [MC4D] Something interesting and strange about permutations

It is interesting to see how MC3D works with reflections. As it can be
seen, a 90=BA rotation can be done by doing a reflection from an edge
and another one from a corner. 180=BA rotation can be done with two
reflections from different stickers of the same kind.

I want to speak about what I said about the special rotations that
should only be allowed in MC4D.
What I was thinking about was that if we say that a rotation is a set
of two reflections, and we are only allowed to do rotations in a 3D
cube, and reflections in the MC2D, then it would be logical to think
that in MC4D we should be allowed to do only a move that will be a set
of two rotations (where when saying rotation I mean a 90=BA rotation).

But ok, mathematically speaking we don't have to be necessarily not
able to do simple rotation from MC4D to higher dimensions, as
reflections from MC3D.

So I think that the problem to determine the possible movements of a
puzzle is if we think matematically or physically (from the reality of
the rubik cube).

About the difficulty of the puzzles, I think that reflections make the
resolution much easier, although it may be not necessarily more
efficient. A demostration that it is easier is that with reflection we
can permute 3-color pieces doing two times a two-color series of the
MC2D, I mean, the same way as we do with a 4-color series of the MC4D,
that is not much more than doing two times a 3-color series. Perhaps
it's hard to understand that reflections makes it easier, because when
I say easier I mean that a person who has never solved the cube would
have less difficulty, without considering the efficiency in the same way.

---

From: "David Smith" <djs314djs314@yahoo.com>
Date: Sat, 27 Sep 2008 03:29:28 -0000
Subject: Re: [MC4D] Something interesting and strange about permutations

Hello David, nice to meet you. I agree with your observations
very much about this topic.

--- In 4D_Cubing@yahoogroups.com, David Vanderschel wrote:
=20
> David Smith, you might want to consider extending your
> permutation counting exercise to include also puzzles
> with mirroring allowed. I am actually curious about
> how many-fold is the increase even for the 3x3
> 3-puzzle. (For myself, the mirroring issue seems more
> interesting than the super-supercube considerations.)

Yes, I will definitely work on this new problem. By the
way, from my point of view the super-supercube, while being
a generalization of the regular cube, is the most elegant
and simplest form of the Rubik's Cube. In d dimensions,
it's a hypercube subdivided into n^d hypercubies, each of
which is uniquely identifiable in any position or orientation.
Any 1 x n^(d-1) group of hypercubies can rotate freely
around the point at the center of the group of hypercubies
in a manner which brings each hypercubie's position to
a position previously occupied by a hypercubie. If we
include reflections, they can occur about any
(n-1)-dimensional space which contains the point at
the center of the group of hypercubies and is orthogonal
to the faces of the hypercubies it intersects.

> >MC2D moves only do interesting things with odd
> >numbers of reflections,
>=20
> ? I do not understand the above statement. Why is a
> 4-cycle (Rotate corner positions.) not interesting?
> [(1,2)(3,4)] * [(2,4)] =3D [(4,3,2,1)]

You are definitely correct here David, but your example
is unfortunately not (it actually contains 3 reflections).
An even number of reflections can only produce an even
permutation of the cubies, and an odd number of reflections,
an odd permutation. (For example, reflecting two adjacent
sides of MC2D would create a 3-cycle of the corners, which
is not possible with an odd number of reflections. Of
course, you wrote a very lengthy reply which was very
accurate, so this small error is understandable! :)

--- In 4D_Cubing@yahoogroups.com, "lucas_awad" wrote:

> I want to speak about what I said about the special rotations that
> should only be allowed in MC4D.
> What I was thinking about was that if we say that a rotation is a=20
set
> of two reflections, and we are only allowed to do rotations in a 3D
> cube, and reflections in the MC2D, then it would be logical to=20
think
> that in MC4D we should be allowed to do only a move that will be a=20
set
> of two rotations (where when saying rotation I mean a 90=BA=20
rotation).

That's an interesting way to view the analogy of allowable
moves in higher dimensions, Lucas. In my way of thinking,
I would not want to restrict the rotations that could be
done on MC4D, as all rotations are allowable (nothing about
4D space prevents this). MC2D can only perform reflections
due to the fact that at least 3 dimensions are required
for the type of rotation implied by the Rubik's Cube (A
rotation that preserves the shape and position of the layer
of cubies while not breaking away from the rest of the cube).
Reflections however, require the cube to be in a space at
least one dimension higher than the cube itself for the
move to be physically possible. Therefore, when only
considering what is physically possible, when using the
cube in a space of dimension equal to the dimension of the
cube no moves can be performed on MC2D, while any rotation
that meets the restrictions above can be performed on a face
of a higher-dimensional cube. If using the cube in a space
of higher dimension than it, reflections can also be
performed (and hence the only moves allowable on MC2D).
Of course, one can always restrict what moves
can be performed on any permutation puzzle, so your idea
on what moves can be performed on MC4D is not incorrect
by any means; it is just a further restriction of all
possible moves, which results in a slightly different
puzzle (and which represents a subgroup of the group I was
considering, with a smaller order).

I am very close to getting the super-supercube formula.
I have been busy lately while also catching up on some
reading in my spare time, which is why I have not
finished yet (with the experience of figuring out
the other 3D and 4D formulas, it is not that time-
consuming). I will probably post the formula tommorow,
and I will then consider cubes with reflections allowed.

All the Best,
David

---

From: David Vanderschel <DvdS@Austin.RR.com>
Date: 27 Sep 2008 09:19:37 +0000
Subject: Re: [MC4D] Something interesting and strange about permutations

On Friday, September 26, "David Smith" wrote:

>--- In 4D_Cubing@yahoogroups.com, David Vanderschel wrote:

>> David Smith, you might want to consider extending your
>> permutation counting exercise to include also puzzles
>> with mirroring allowed. ...

>Yes, I will definitely work on this new problem. By the
>way, from my point of view the super-supercube, while being
>a generalization of the regular cube, is the most elegant
>and simplest form of the Rubik's Cube. In d dimensions,
>it's a hypercube subdivided into n^d hypercubies, each of
>which is uniquely identifiable in any position or orientation.
>Any 1 x n^(d-1) group of hypercubies can rotate freely
>around the point at the center of the group of hypercubies
>in a manner which brings each hypercubie's position to
>a position previously occupied by a hypercubie.

Actually, this is how I have always viewed the order-3
puzzles (any dimension). And, yes, the view applies
as well to puzzles of order higher than 3. What you
are describing as "1 x n^(d-1) group of hypercubies"
is what I call a "slice". I distinguish between
external slices and internal slices.

My lack of interest stemmed from what I perceived as a
lack of practical physical realizablity, since the
group permutes what amount to stickers that can never
be visible. But I must agree that it can be simulated
in an understandable way.

>If we include reflections, they can occur about any
>(n-1)-dimensional space which contains the point at
>the center of the group of hypercubies and is
>orthogonal to the faces of the hypercubies it
>intersects.

You can also reflect about hyperplanes that are on
diagonals. The reflection planes must be such that
the puzzle transforms onto itself, but that does not
mean they must be axis-aligned. E.g., for the
3-puzzle, a plane of reflection can intersect the
locations of 4 corners in such a way that 2 pairs are
diagonal from each other across opposing faces and 2
other pairs are adjacent along opposing edges. (The
last 2 pairs are what I call "triagonal" from one
another, diagonal within the 4-point rectangle.)

>> >MC2D moves only do interesting things with odd
>> >numbers of reflections,

>> ? I do not understand the above statement. Why is a
>> 4-cycle (Rotate corner positions.) not interesting?
>> [(1,2)(3,4)] * [(2,4)] = [(4,3,2,1)]

>You are definitely correct here David, but your example
>is unfortunately not (it actually contains 3
>reflections).

Perhaps we are not together on the permutation
notation. I am using cycle representation. If you
want to think about it geometrically, assign indices
to the corner positions in clockwise order starting at
the upper left hand corner. Then [(1,2)(3,4)] is a
reflection about the y-axis, [(2,4)] is a reflection
about the NW->SE diagonal, and [(4,3,2,1)] is a
4-cycle rotating all four corners counter clockwise.

>An even number of reflections can only produce an even
>permutation of the cubies,

Not true. Reflections can be odd or even and there is
one of each in the two I composed above. (Fairly
obvious, with 2 2-cycles in one and only 1 2-cycle in
the other.) So the product, a 4-cycle, is odd.
Furthermore, the product is not mirroring. (It is
true that the composition of two reflecting
permutations is always a non-reflecting
reorientation (or identity if the same reflection is
used twice).)

>and an odd number of reflections, an odd
>permutation. (For example, reflecting two adjacent
>sides of MC2D would create a 3-cycle of the corners,
>which is not possible with an odd number of
>reflections. Of course, you wrote a very lengthy
>reply which was very accurate, so this small error is
>understandable! :)

I suppose yours is somewhat larger since you implied
that I had erred when I had not; but I will forgive
you anyway! And please do not hesitate to attempt to
correct me if, in future, it again looks as if I have
erred. Next time it may be so, and I would not wish
to go uncorrected. I do err frequently because I
often have complex thoughts. I learn by being
corrected, and I occasionally assert something I am
not quite sure about with the hope that someone will
point me in the right direction if I turn out to be
confused.


After I posted my previous message it occurred to me
that I missed yet another possibility and an even
larger group. We might call it a SUPER--super-
supercube. I had written,

>If you try to flip a slice of the 3-puzzle in 3D, you
>do wind up (uselessly) with it being outside in; but
>it is not reflected.

I now regret that "(uselessly)" parenthesis. Let us
take the attitude that an order-m n-puzzle is an m^n
stack of nD cubies. (My n is your d and my m is your
n. I prefer to stick with n for the dimension of the
puzzle.) Also let us imagine that _every_ one of the
2n facelets on each cubie has a sticker. Use a normal
scheme for assigning colored stickers to the visible
facelets. I am willing to make all the invisible
stickers be black; but it would not be a necessity.
In the order-3 cases, it is certainly OK as each cubie
is uniquely identified by the combination of colored
stickers it bears. Now I want to say that, in
addition to our familiar twists, you can remove a
slice, flip it over with respect to the axis in which
it is 'flat', and replace it. (I can imagine no real
mechanism, but you can easily simulate the 3D cases
with a pile of cubical blocks. Just a little tedious
to reorient the whole cube or to remove a slice and
replace it flipped - but certainly possible.) Now
this is not mirror reflection; but it is a new kind of
permutation. The colored stickers can be turned to
face inwards and have valid possible internal
positions. Flipping of slices need not be limited to
external slices.

Imagine that all 2n*m^n facelet positions are indexed
relative to fixed spatial coordinates and we identify
the stickers on them in the initial position by the
position they occupy. The permitted alterations to
the pile can be seen as permutations of the 2n*m^n
stickers. There must be some sense in which the
familiar group of the regular order-m n-puzzle is
still in there as a subgroup; but I am not sure yet
how to characterize that, since there are extra
stickers. (May require a quotient.) We now have the
ability to turn over any slice relative to the
dimension in which it has thickness 1. When the
puzzle is scrambled, many initially-visible colored
stickers may no longer be visible; but they can still
be accessed by appropriate manipulation.

Anyone for slice-swapping? ;-)


Regards,
David V.

---

From: "David Smith" <djs314djs314@yahoo.com>
Date: Sat, 27 Sep 2008 13:35:31 -0000
Subject: Re: [MC4D] Something interesting and strange about permutations

Hi David,

--- In 4D_Cubing@yahoogroups.com, David Vanderschel wrote:
>
> On Friday, September 26, "David Smith" wrote:
>=20
> >--- In 4D_Cubing@yahoogroups.com, David Vanderschel wrote:
>=20
> >> David Smith, you might want to consider extending your
> >> permutation counting exercise to include also puzzles
> >> with mirroring allowed. ...
>=20
> >Yes, I will definitely work on this new problem. By the
> >way, from my point of view the super-supercube, while being
> >a generalization of the regular cube, is the most elegant
> >and simplest form of the Rubik's Cube. In d dimensions,
> >it's a hypercube subdivided into n^d hypercubies, each of
> >which is uniquely identifiable in any position or orientation.
> >Any 1 x n^(d-1) group of hypercubies can rotate freely
> >around the point at the center of the group of hypercubies
> >in a manner which brings each hypercubie's position to
> >a position previously occupied by a hypercubie.=20=20
>=20
> Actually, this is how I have always viewed the order-3
> puzzles (any dimension). And, yes, the view applies
> as well to puzzles of order higher than 3. What you
> are describing as "1 x n^(d-1) group of hypercubies"
> is what I call a "slice". I distinguish between
> external slices and internal slices.

Yes, I also refer to these as slices!

> My lack of interest stemmed from what I perceived as a
> lack of practical physical realizablity, since the
> group permutes what amount to stickers that can never
> be visible. But I must agree that it can be simulated
> in an understandable way.

I like to think of the super-supercube as a mathematical
entity, rather than a physical reality. I don't really
think of the hypercubies as having stickers at all. They
are just uniquely identifiable in any position or
orientation.

> >If we include reflections, they can occur about any
> >(n-1)-dimensional space which contains the point at
> >the center of the group of hypercubies and is
> >orthogonal to the faces of the hypercubies it
> >intersects.
>=20
> You can also reflect about hyperplanes that are on
> diagonals. The reflection planes must be such that
> the puzzle transforms onto itself, but that does not
> mean they must be axis-aligned. E.g., for the
> 3-puzzle, a plane of reflection can intersect the
> locations of 4 corners in such a way that 2 pairs are
> diagonal from each other across opposing faces and 2
> other pairs are adjacent along opposing edges. (The
> last 2 pairs are what I call "triagonal" from one
> another, diagonal within the 4-point rectangle.)

Yes, I forgot about these reflections. I suppose I was
thinking about the fact that any reflection can be produced
by the type of reflections I mentioned.

> >> >MC2D moves only do interesting things with odd
> >> >numbers of reflections,
>=20
> >> ? I do not understand the above statement. Why is a
> >> 4-cycle (Rotate corner positions.) not interesting?
> >> [(1,2)(3,4)] * [(2,4)] =3D [(4,3,2,1)]
>=20
> >You are definitely correct here David, but your example
> >is unfortunately not (it actually contains 3
> >reflections).
>=20
> Perhaps we are not together on the permutation
> notation. I am using cycle representation. If you
> want to think about it geometrically, assign indices
> to the corner positions in clockwise order starting at
> the upper left hand corner. Then [(1,2)(3,4)] is a
> reflection about the y-axis, [(2,4)] is a reflection
> about the NW->SE diagonal, and [(4,3,2,1)] is a
> 4-cycle rotating all four corners counter clockwise.
>=20
> >An even number of reflections can only produce an even
> >permutation of the cubies,=20
>=20
> Not true. Reflections can be odd or even and there is
> one of each in the two I composed above. (Fairly
> obvious, with 2 2-cycles in one and only 1 2-cycle in
> the other.) So the product, a 4-cycle, is odd.
> Furthermore, the product is not mirroring. (It is
> true that the composition of two reflecting
> permutations is always a non-reflecting
> reorientation (or identity if the same reflection is
> used twice).)
>=20
> >and an odd number of reflections, an odd
> >permutation. (For example, reflecting two adjacent
> >sides of MC2D would create a 3-cycle of the corners,
> >which is not possible with an odd number of
> >reflections. Of course, you wrote a very lengthy
> >reply which was very accurate, so this small error is
> >understandable! :)
>=20
> I suppose yours is somewhat larger since you implied
> that I had erred when I had not; but I will forgive
> you anyway! And please do not hesitate to attempt to
> correct me if, in future, it again looks as if I have
> erred. Next time it may be so, and I would not wish
> to go uncorrected. I do err frequently because I
> often have complex thoughts. I learn by being
> corrected, and I occasionally assert something I am
> not quite sure about with the hope that someone will
> point me in the right direction if I turn out to be
> confused.

I am quite familiar with cycle notation! I believe
you are making an error as to what a move is in MC2D.
You seem to be talking about reflecting the entire
square! As in MC3D, a move is any rotation or
reflection of a face that preserves the general shape
(i.e. takes positions of cubies to positions of
previous cubies). I believe you are thinking about
the entire puzzle as a face of MC3D, because the moves
you are describing are reflections of a face in that
puzzle.

If you launch MC2D, it will be immediately clear that
there is only one move, and that is to swap two
adjacent corners. Thus, (1,2)(3,4) is actually two
moves, one reflection of the North face and one of
the South face. If you were to actually reflect
the entire puzzle, the centers would also reflect,
and thus would just be a reorientation of the puzzle,
and not a move at all.

> After I posted my previous message it occurred to me
> that I missed yet another possibility and an even
> larger group. We might call it a SUPER--super-=20
> supercube. I had written,
>=20
> >If you try to flip a slice of the 3-puzzle in 3D, you
> >do wind up (uselessly) with it being outside in; but
> >it is not reflected.
>=20
> I now regret that "(uselessly)" parenthesis. Let us
> take the attitude that an order-m n-puzzle is an m^n
> stack of nD cubies. (My n is your d and my m is your
> n. I prefer to stick with n for the dimension of the
> puzzle.) Also let us imagine that _every_ one of the
> 2n facelets on each cubie has a sticker. Use a normal
> scheme for assigning colored stickers to the visible
> facelets. I am willing to make all the invisible
> stickers be black; but it would not be a necessity.
> In the order-3 cases, it is certainly OK as each cubie
> is uniquely identified by the combination of colored
> stickers it bears. Now I want to say that, in
> addition to our familiar twists, you can remove a
> slice, flip it over with respect to the axis in which
> it is 'flat', and replace it. (I can imagine no real
> mechanism, but you can easily simulate the 3D cases
> with a pile of cubical blocks. Just a little tedious
> to reorient the whole cube or to remove a slice and
> replace it flipped - but certainly possible.) Now
> this is not mirror reflection; but it is a new kind of
> permutation. The colored stickers can be turned to
> face inwards and have valid possible internal
> positions. Flipping of slices need not be limited to
> external slices.
>=20
> Imagine that all 2n*m^n facelet positions are indexed
> relative to fixed spatial coordinates and we identify
> the stickers on them in the initial position by the
> position they occupy. The permitted alterations to
> the pile can be seen as permutations of the 2n*m^n
> stickers. There must be some sense in which the
> familiar group of the regular order-m n-puzzle is
> still in there as a subgroup; but I am not sure yet
> how to characterize that, since there are extra
> stickers. (May require a quotient.) We now have the
> ability to turn over any slice relative to the
> dimension in which it has thickness 1. When the
> puzzle is scrambled, many initially-visible colored
> stickers may no longer be visible; but they can still
> be accessed by appropriate manipulation.
>=20
> Anyone for slice-swapping? ;-)

A very interesting idea, and congrats for thinking
of it! Who knows, I may end up doing my permutation
formulas for these puzzles as well! :)

All the Best,
David

---

From: "Roice Nelson" <roice3@gmail.com>
Date: Sat, 27 Sep 2008 14:58:29 -0500
Subject: Re: [MC4D] Something interesting and strange about permutations

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Wow, lots of good discussion and ideas here! I blink and I've fallen
behind! You guys have worked through everything so well, and so I only have
a few comments with which I will participate.

>>Let's start
>>with a single reflection move. These can be
>>reflections about a point, line, or any space of
>>dimension lower than the puzzle itself.

>Not exactly. Any reflection will occur with respect
>to an (n-1)-dimensional reflection hyperplane. In
>that (n-1)-dimensional plane there can be embedded a
>hyperplane of lower dimension which one may wish to be
>included in the reflection plane. But a hyperplane of
>dimension less than n-1 does not uniquely determine a
>reflection. It does determine a family of possible
>reflection planes which include it.

Melinda's thought of allowing reflections across any subspace of
intermediate dimension is interesting and worthy of further study! I had
only previously considered reflections across hyperplanes of dimension n-1
only as you describe, but the behavior of the other cases also appears to be
uniquely determined. Consider point
reflections,
which are easy to think about in any dimension (and only
orientation-reversing in odd dimensions).

However, I do have the feeling (no proof unfortunately) that the n-1
hyperplane reflections are the most fundamental operations. For example,
point reflections in 2D and 3D results in symmetries that can instead be
built up from reflections across lines and planes, respectively. But the
reverse isn't true, e.g. in 2D you can't make a reflection across a line by
composing point reflections (since it is just a 180 degree rotation).
Furthermore, you can't express 2D rotations as a pair of point reflections,
so the "rotations as pairs of reflections" property clearly doesn't extend
to all portions of this larger class of reflections.

Anyway, my intuition says if we limit ourselves to reflections across
hyperplanes as you've described, we can reach all mirror puzzle
permutations, just like we can reach all permutations in the rotation-only
puzzles when limiting rotations to being coordinate-axis aligned. But it
could be visually interesting to somehow support these other reflection
types, just as the edge and corner twists in MC4D bring a lot to that
puzzle.

Btw, one suggestion for naming the puzzle types we've laid out would follow
the language of topology: orientable or nonorientable, the latter being the
case where an odd number of reflection moves was allowed.

>>Even if all rotations can be expressed
>>as pairs of reflections, it might not follow that all
>>pairs of reflections can be expressed as rotations,

>But they can be. One thing that needs to be mentioned
>is that the reflection planes cannot be chosen
>arbitrarily. They must be such that the puzzle
>transforms onto itself in the same space. Given that
>restriction, it follows that the result of two
>reflections must be a non-reflecting reorientation.
>(What else could it be?)

Yep, this is correct. As another aside, without the "transform onto itself"
restriction, compositions of two reflections can result in rotations or
translations (or the identity), depending on whether the two reflection
planes intersect or not (distinct, parallel reflection planes result in
translations). We would limit our reflection planes to going through the
origin of a hyperface because we don't want translations of faces to occur
during moves (also, think about what a single reflection would do to a
face if this was not the case), and with this limitation all pairs of
reflections will equivalently be a rotation for us.

One last thought. It does seem using reflections as a talking point for
discussing face "moves" has been very useful, since they are a more
fundamental motion than rotations. Coxeter
groups,
which describe all the symmetries of regular polytopes among other
things, are "abstract groups that admit a formal description in terms of
mirror symmetries". I think it is natural we have been led in this
direction...

All the best,
Roice

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Wow, lots of good discussion and ideas here!  I blink and I've fallen behind!  You guys have worked through everything so well, and so I only have a few comments with which I will participate.

>>Let's start
>>with a single reflection move. These can be
>>reflections about a point, line, or any space of
>>dimension lower than the puzzle itself. 

>Not exactly. Any reflection will occur with respect
>to an (n-1)-dimensional reflection hyperplane. In
>that (n-1)-dimensional plane there can be embedded a
>hyperplane of lower dimension which one may wish to be

>included in the reflection plane. But a hyperplane of
>dimension less than n-1 does not uniquely determine a
>reflection. It does determine a family of possible
>reflection planes which include it.

Melinda's thought of allowing reflections across any subspace of intermediate dimension is interesting and worthy of further study!  I had only previously considered reflections across hyperplanes of dimension n-1 only as you describe, but the behavior of the other cases also appears to be uniquely determined.  Consider point reflections, which are easy to think about in any dimension (and only orientation-reversing in odd dimensions).  

However, I do have the feeling (no proof unfortunately) that the n-1 hyperplane reflections are the most fundamental operations.  For example, point reflections in 2D and 3D results in symmetries that can instead be built up from reflections across lines and planes, respectively.  But the reverse isn't true, e.g. in 2D you can't make a reflection across a line by composing point reflections (since it is just a 180 degree rotation).  Furthermore, you can't express 2D rotations as a pair of point reflections, so the "rotations as pairs of reflections" property clearly doesn't extend to all portions of this larger class of reflections.

Anyway, my intuition says if we limit ourselves to reflections across hyperplanes as you've described, we can reach all mirror puzzle permutations, just like we can reach all permutations in the rotation-only puzzles when limiting rotations to being coordinate-axis aligned.  But it could be visually interesting to somehow support these other reflection types, just as the edge and corner twists in MC4D bring a lot to that puzzle.

Btw, one suggestion for naming the puzzle types we've laid out would follow the language of topology: orientable or nonorientable, the latter being the case where an odd number of reflection moves was allowed.

>>Even if all rotations can be expressed
>>as pairs of reflections, it might not follow that all

>>pairs of reflections can be expressed as rotations,

>But they can be. One thing that needs to be mentioned
>is that the reflection planes cannot be chosen
>arbitrarily. They must be such that the puzzle

>transforms onto itself in the same space. Given that
>restriction, it follows that the result of two
>reflections must be a non-reflecting reorientation.
>(What else could it be?) 

Yep, this is correct.  As another aside, without the "transform onto itself" restriction, compositions of two reflections can result in rotations or translations (or the identity), depending on whether the two reflection planes intersect or not (distinct, parallel reflection planes result in translations).  We would limit our reflection planes to going through the origin of a hyperface because we don't want translations of faces to occur during moves (also, think about what a single reflection would do to a face if this was not the case), and with this limitation all pairs of reflections will equivalently be a rotation for us. 

One last thought.  It does seem using reflections as a talking point for discussing face "moves" has been very useful, since they are a more fundamental motion than rotations.  Coxeter groups, which describe all the symmetries of regular polytopes among other things, are "abstract groups that admit a formal description in terms of mirror symmetries".  I think it is natural we have been led in this direction...

All the best,

Roice

------=_Part_20037_5012675.1222545509220--

---

From: David Vanderschel <DvdS@Austin.RR.com>
Date: 28 Sep 2008 05:15:49 -0500
Subject: Re: [MC4D] Something interesting and strange about permutations

On Saturday, September 27, "David Smith" wrote:
>I like to think of the super-supercube as a mathematical
>entity, rather than a physical reality. I don't really
>think of the hypercubies as having stickers at all. They
>are just uniquely identifiable in any position or
>orientation.

Then what is the group? For me, the associated group
is the resulting set of sticker permutations. If you
just think about permuting the cubies, then it is
difficult to take into account orientation.

Melinda:
>> >> >MC2D moves only do interesting things with odd
>> >> >numbers of reflections,

Me:
>> >> ? I do not understand the above statement. Why is a
>> >> 4-cycle (Rotate corner positions.) not interesting?
>> >> [(1,2)(3,4)] * [(2,4)] = [(4,3,2,1)]

Smith:
>> >You are definitely correct here David, but your example
>> >is unfortunately not (it actually contains 3
>> >reflections).

>I am quite familiar with cycle notation! I believe
>you are making an error as to what a move is in MC2D.

Quite likely. Remember, this started with my "I do
not understand the above statement."

>You seem to be talking about reflecting the entire
>square!

No. I am talking about permutations and symmetries of
the permutation patterns themselves - i.e., about the
group. MC2D can be thought of as implementing S4 (all
permutations of 4 things). The objects being permuted
are just the 4 corner cubies. The rest of the puzzle
does not really matter except to the extent that it
constrains what constitutes a twist. (Actually, any 3
of the puzzle's 4 adjacent-corner flips will generate
the entire group.) D4, corresponding to rigid motions
of a square, is a subgroup of S4. D4 has permutations
considered to be reflections. Those are what I am
talking about.

So when I say that [(1,2)(3,4)] is a reflection in the
y-axis, I am talking about its effect on the positions
of the corner cubies, independent of the fact that the
edge cubies do not move. (As far as the group is
concerned, the edge cubies do not even exist.)

In the statement that confuses me, I took the
reference to "reflections" as referring to certain of
the group elements. I now realize that she is
probably using the word in the very low level sense
for which I have been saying "twist" or "flip",
because that basic move is a reflection of a single
slice.

When I was still thinking that "reflection" referred
to a group element as opposed to an action on a slice,
I inferred that she must be talking about permutations
that result from more than one flip. So I provided an
example with an even number of group elements (2)
which are reflections and which, on composition,
produce what I consider to be an "interesting"
product. Well, it is interesting, but the exercise
does miss Melinda's point.

>As in MC3D, a move is any rotation or reflection of a
>face that preserves the general shape (i.e. takes
>positions of cubies to positions of previous cubies).
>I believe you are thinking about the entire puzzle as
>a face of MC3D, because the moves you are describing
>are reflections of a face in that puzzle.

I've lost you here. There are a lot more than 4
things being permuted in MC3D. I have the feeling
that you may be thinking more about a transformation
of n-space, where I am talking about permutations on 4
things.

>If you launch MC2D, it will be immediately clear that
>there is only one move, and that is to swap two
>adjacent corners. Thus, (1,2)(3,4) is actually two
>moves, one reflection of the North face and one of
>the South face.

OK. I see another source of my confusion. I had
taken "move" to refer to the result of any sequence of
flips, because it was clear to me that repeating the
same flip merely cancels the effect of the previous
occurrence of it. But I think I misunderstood and
that she intended "move" to refer to what you could do
with a single slice by composing multiple reflections
with respect to it. This is meaningful in higher
dimensions; but, in MC2D, it is trivial, as there is
only one "reflection" of a slice - the basic flip. So
you cannot combine _different_ reflections when
reorienting a given slice. Now I think I understand
that Melinda was just pointing out the degeneracy of
MC2D in the context of composing multiple reflections
on a single slice. The remark may well have been
tongue-in-cheek, since the only reasonable odd number
in this context is 1. The point is so simple that I
was looking for something more profound for it to
mean.



>> After I posted my previous message it occurred to me
>> that I missed yet another possibility and an even
>> larger group. We might call it a SUPER--super-
>> supercube. ...

>A very interesting idea, and congrats for thinking
>of it! Who knows, I may end up doing my permutation
>formulas for these puzzles as well! :)

When I wrote,
>> Anyone for slice-swapping? ;-)
I was sort of poking fun at myself and anyone who
would take these variations too seriously. I was
pointing out that one could continue to invent new
rules for rearranging the cubie pile ad infinitum. At
some point one must draw the line and concentrate on
the variations that appear to be more pleasingly
elegant.

Carrying it to the ultimate extreme (for the order-3
3-puzzle): Suppose we had 27 cubical blocks with
unique identifiers on all their faces. Take a
particular 3x3x3 pile as being the initial state.
Then any other way of piling them determines a
permutation of the identifiers relative to the
original pile. Now suppose we start making random
stacks (busy monkeys implementing the most liberal
rules for rearranging the stack) and noting the
resulting permutations. On average, how many times
must we scramble it before the permutations we have
collected will generate the entire group?

Regarding elegance: I think the big success of the
original Rubik's Cube arose from the fact that it is
an elegant object. Even folks who have no hope of
solving it can marvel at the cleverness of the
mechanism.




On Saturday, September 27, "Roice Nelson" wrote:
>Melinda's thought of allowing reflections across any subspace of
>intermediate dimension is interesting and worthy of further study! I had
>only previously considered reflections across hyperplanes of dimension n-1
>only as you describe, but the behavior of the other cases also appears to be
>uniquely determined. Consider point
>reflections,
>which are easy to think about in any dimension (and only
>orientation-reversing in odd dimensions).

I must confess that I had never contemplated these
other kinds of "reflection" at all. Reflection in an
(n-1)-dimensional hyperplane is the only kind of
reflection that I ever thought about when hearing the
word "reflection". Thus I misinterpreted Melinda's
reference and assumed that reflecting about a given
hyperplane of dimension less than n-1 amounted to
picking a hyperplane of dimension n-1 which contained
the given one. I had never even heard of "point
reflection". But I do see that meaningful
transformations of n-space can be specified this way.
New insight!

Note that a point reflection in 2-space is not a
reflection at all in the sense in which we have been
using "reflection" here; so, in our mirroring context,
you would not really expect it to come up. This is
probably true for other reflection-variation/dimension
combinations as well.

>However, I do have the feeling (no proof
>unfortunately) that the n-1 hyperplane reflections
>are the most fundamental operations. ...

>Anyway, my intuition says if we limit ourselves to
>reflections across hyperplanes as you've described,
>we can reach all mirror puzzle permutations, ...

My own intuition is similar. I think I might even be
able to formalize it; but I am not sure it would worth
it, as it seems clear enough. There are only so many
ways to reorient an n-cube (or an n-slice), mirrored
or not. If you already have all the non-reflecting
reorientations, then you only need one of the
reflections to generate all the rest. (You can get
that by an enumeration and uniqueness argument.)

>Btw, one suggestion for naming the puzzle types we've
>laid out would follow the language of topology:
>orientable or nonorientable, the latter being the
>case where an odd number of reflection moves was
>allowed.

I may not understand what you are driving at here.
If the issue is "reflection allowed or not" why would
"nonorientable" be preferable to "reflection
allowed". At present I don't get a suitable semantic
reaction from "nonorientable".


Regards,
David V.

---

From: "David Smith" <djs314djs314@yahoo.com>
Date: Sun, 28 Sep 2008 12:39:04 -0000
Subject: Re: [MC4D] Something interesting and strange about permutations

Hi David,

I would like to apologize for repetately implying that
you were incorrect about this topic in any way. By=20
doing so, I called into question your intelligence
and experience in these topics (which is clearly much
higher than my own), and demonstrated my own
lack of knowledge. I should have realized after you
corrected me that this confusion was based on a
misunderstanding, not that "I was correct and you
were not". You may think I am being too hard on
myself, but I must say what I feel.

--- In 4D_Cubing@yahoogroups.com, David Vanderschel wrote:
>
> On Saturday, September 27, "David Smith" wrote:
> >I like to think of the super-supercube as a mathematical
> >entity, rather than a physical reality. I don't really
> >think of the hypercubies as having stickers at all. They
> >are just uniquely identifiable in any position or
> >orientation.
>=20
> Then what is the group? For me, the associated group
> is the resulting set of sticker permutations. If you
> just think about permuting the cubies, then it is
> difficult to take into account orientation.

I was entirely wrong here! I completely forgot what the
basic notion of the group of a permutation puzzle was
(i.e. the set of sticker permutations).

> Melinda:
> >> >> >MC2D moves only do interesting things with odd
> >> >> >numbers of reflections,
>=20
> Me:
> >> >> ? I do not understand the above statement. Why is a
> >> >> 4-cycle (Rotate corner positions.) not interesting?
> >> >> [(1,2)(3,4)] * [(2,4)] =3D [(4,3,2,1)]
>=20=20
> Smith:
> >> >You are definitely correct here David, but your example
> >> >is unfortunately not (it actually contains 3
> >> >reflections).
>=20
> >I am quite familiar with cycle notation! I believe
> >you are making an error as to what a move is in MC2D.
>=20
> Quite likely. Remember, this started with my "I do
> not understand the above statement."=20=20

When I made this statement, I was basically saying
that you were incorrect, and I apologize for that.

> >You seem to be talking about reflecting the entire
> >square!=20=20
>=20
> No. I am talking about permutations and symmetries of
> the permutation patterns themselves - i.e., about the
> group. MC2D can be thought of as implementing S4 (all
> permutations of 4 things). The objects being permuted
> are just the 4 corner cubies. The rest of the puzzle
> does not really matter except to the extent that it
> constrains what constitutes a twist. (Actually, any 3
> of the puzzle's 4 adjacent-corner flips will generate
> the entire group.) D4, corresponding to rigid motions
> of a square, is a subgroup of S4. D4 has permutations
> considered to be reflections. Those are what I am
> talking about.
>=20
> So when I say that [(1,2)(3,4)] is a reflection in the
> y-axis, I am talking about its effect on the positions
> of the corner cubies, independent of the fact that the
> edge cubies do not move. (As far as the group is
> concerned, the edge cubies do not even exist.)
>=20
> In the statement that confuses me, I took the
> reference to "reflections" as referring to certain of
> the group elements. I now realize that she is
> probably using the word in the very low level sense
> for which I have been saying "twist" or "flip",
> because that basic move is a reflection of a single
> slice.
>=20
> When I was still thinking that "reflection" referred
> to a group element as opposed to an action on a slice,
> I inferred that she must be talking about permutations
> that result from more than one flip. So I provided an
> example with an even number of group elements (2)
> which are reflections and which, on composition,
> produce what I consider to be an "interesting"
> product. Well, it is interesting, but the exercise
> does miss Melinda's point.

The basic source of our misunderstanding was what
Melinda meant when she said "reflection". I thought
it meant a single move, or a swap of two corners.
Thank you for reminding me about S4 and D4, and
how they relate to MC2D.

> >As in MC3D, a move is any rotation or reflection of a
> >face that preserves the general shape (i.e. takes
> >positions of cubies to positions of previous cubies).
> >I believe you are thinking about the entire puzzle as
> >a face of MC3D, because the moves you are describing
> >are reflections of a face in that puzzle.
>=20
> I've lost you here. There are a lot more than 4
> things being permuted in MC3D. I have the feeling
> that you may be thinking more about a transformation
> of n-space, where I am talking about permutations on 4
> things.=20

Clearly what I said here was poorly phrased. It was
once again based on my assumption that you were
incorrect. I was thinking that you were confusing
the moves of MC2D for the moves of a face of MC3D,
which is of course absurd, as you would definitely
not make such an obviously wrong mistake!

> >If you launch MC2D, it will be immediately clear that
> >there is only one move, and that is to swap two
> >adjacent corners. Thus, (1,2)(3,4) is actually two
> >moves, one reflection of the North face and one of
> >the South face.=20=20
>=20
> OK. I see another source of my confusion. I had
> taken "move" to refer to the result of any sequence of
> flips, because it was clear to me that repeating the
> same flip merely cancels the effect of the previous
> occurrence of it. But I think I misunderstood and
> that she intended "move" to refer to what you could do
> with a single slice by composing multiple reflections
> with respect to it. This is meaningful in higher
> dimensions; but, in MC2D, it is trivial, as there is
> only one "reflection" of a slice - the basic flip. So
> you cannot combine _different_ reflections when
> reorienting a given slice. Now I think I understand
> that Melinda was just pointing out the degeneracy of
> MC2D in the context of composing multiple reflections
> on a single slice. The remark may well have been
> tongue-in-cheek, since the only reasonable odd number
> in this context is 1. The point is so simple that I
> was looking for something more profound for it to
> mean.

I was taking "move" to be a single face reflection,
and that when Melinda said an odd number of moves was
not interesting, that she was referring to a sequence
of moves.

> >> After I posted my previous message it occurred to me
> >> that I missed yet another possibility and an even
> >> larger group. We might call it a SUPER--super-
> >> supercube. ...
>=20
> >A very interesting idea, and congrats for thinking
> >of it! Who knows, I may end up doing my permutation
> >formulas for these puzzles as well! :)
>=20
> When I wrote,
> >> Anyone for slice-swapping? ;-)
> I was sort of poking fun at myself and anyone who
> would take these variations too seriously. I was
> pointing out that one could continue to invent new
> rules for rearranging the cubie pile ad infinitum. At
> some point one must draw the line and concentrate on
> the variations that appear to be more pleasingly
> elegant.=20=20
>=20
> Carrying it to the ultimate extreme (for the order-3
> 3-puzzle): Suppose we had 27 cubical blocks with
> unique identifiers on all their faces. Take a
> particular 3x3x3 pile as being the initial state.
> Then any other way of piling them determines a
> permutation of the identifiers relative to the
> original pile. Now suppose we start making random
> stacks (busy monkeys implementing the most liberal
> rules for rearranging the stack) and noting the
> resulting permutations. On average, how many times
> must we scramble it before the permutations we have
> collected will generate the entire group?
>=20
> Regarding elegance: I think the big success of the
> original Rubik's Cube arose from the fact that it is
> an elegant object. Even folks who have no hope of
> solving it can marvel at the cleverness of the
> mechanism.

Yes, the Rubik's Cube is a very simple and elegant
physical representation of a large group. Your
super-super-supercube concept was, as you explained,
just a further definition of how to rearrange
the cubies. Calculating the formulas for endless
redefinitions of the rules would be pointless! As
you said, we should concentrate on what we find to
be a worthwhile and satisfying concept.

> On Saturday, September 27, "Roice Nelson" wrote:
> >Melinda's thought of allowing reflections across any subspace of
> >intermediate dimension is interesting and worthy of further=20
study! I had
> >only previously considered reflections across hyperplanes of=20
dimension n-1
> >only as you describe, but the behavior of the other cases also=20
appears to be
> >uniquely determined. Consider point
> >reflections,
> >which are easy to think about in any dimension (and only
> >orientation-reversing in odd dimensions).
>=20
> I must confess that I had never contemplated these
> other kinds of "reflection" at all. Reflection in an
> (n-1)-dimensional hyperplane is the only kind of
> reflection that I ever thought about when hearing the
> word "reflection". Thus I misinterpreted Melinda's
> reference and assumed that reflecting about a given
> hyperplane of dimension less than n-1 amounted to
> picking a hyperplane of dimension n-1 which contained
> the given one. I had never even heard of "point
> reflection". But I do see that meaningful
> transformations of n-space can be specified this way.
> New insight!

I too had never heard of such a reflection but it
does make sense, and is a completely analogous
way to define reflections about a space at least
2 dimensions lower than the object one is
considering.

Regarding my statements here, I think you may
feel that I should not be so critical of myself,
but I felt I must apologize for thinking and saying
you were incorrect (especially the second time),
as I should have realized that this was just
a misunderstanding since you explained yourself
so clearly in your last post.

All the Best,
David

---

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