Thread: "definition of a twist"

This is my first email to the group so hello group,

I read that I don't need to apologize for length...fewf. I've been havi=
ng conversations with both Melinda and Roice on what appears to be (from my=
perspective) all 3 programs, MC4D, MC5D, and MC7D, using too strict of a d=
efinition of a "twist." More specifically, the twists in these programs tw=
ist an n-1 dimensional face, the definition of a twist that I can see is th=
at the face being twisted has to be anywhere from 2 dimensional up to n-1 d=
imensional. This would allow some 2-d atomic twists in MC4D and MC5D which=
I've edited and posted in Photos > More possible twists. You'll note that=
my twists in 4D make your famous Klaus parity errors quite simple--possibl=
y too simple for your liking (see 2-4 below). Going with Roice's suggestio=
n, I will continue our conversation with the whole group.=20=20

(1) I installed Nate Berglund's program (http://people.math.gatech.edu/~be=
rglund/Rubik/index.html), and yes indeed those are exactly the missing move=
s for 4D.

(2) Great question: do these twists make for more possible states? We do=
need the group's help here. For the 3^4 cube, Matthew Sheerin, or Klaus m=
ay be able to help answer. I'm referring to messages #695, 772, 778, and P=
hotos > "parity problems" by Klaus: http://groups.yahoo.com/group/4D_Cubin=
g/photos/album/565962423/pic/list. Matt says he posted a solution to these=
, but I can't find it. I notice Klaus's Oct. 13 parity error has the color=
s across from each other. Therefore we still need to know if my rot_A1 twi=
st is solvable using current MC4D twists as well as Klaus's Nov. 14 parity =
error (same as my rot_A2 twist...same as my rot_B1 twist in a sense). If t=
hese 2 are solvable, then my rot_B2 twist could be created using each of th=
ese 4 times + a rotation of the entire tesseract and thus I would have intr=
oduced no new states. I now doubt this is the case, however--my guess is t=
hat my twists introduce new states.=20=20

(3) Yeah, when you open up Berglund's program you can choose to allow or d=
isallow my twists. He classifies them as two separate puzzles, which may b=
e the way to go. Another way to go, for example is 2 separate versions of =
MC4D: the current version with only 3D twists allowed vs. a version where =
both 2D and 3D twists allowed.

(4) I was preparing a statement like this...only much worse. I was prepar=
ed for something analogous to Christopher Columbus being laughed off the fl=
at face of the planet for thinking it's round. Of course I was hoping you'=
d phrase it as nicely as you did. Before ever searching and finding your p=
rograms on the internet, I had visualized a 3^4 tesseract in an X, Y, Z, T =
coordinate system as described in my email at the bottom with the center of=
the tesseract being at the origin and having the cubie edge length =3D 1. =
I visualized the seperators t =3D -1/2 and t =3D 1/2 dividing it into 3 "c=
ubes." I simply figured you could twist just the top (z > 1/2) of the t < =
-1/2 "cube." Doing so results in no z nor t coordinate change for any "4D =
atom" of the entire tesseract so no stickers nor cubies will cross these se=
perators during the twist and nothing runs into eachother. This visualizat=
ion method made it difficult to visualize how to twist z > 1/2 and x > 1/2.=
I wasn't sure it was possible, but I realized one can always rotate the e=
ntire tesseract so x ----> -t. That way it's the same as the previous twis=
t, so I knew it was possible. I went back and tried visualizing this twist=
without the rotation, and although it would take a while to describe, I ca=
n tell you, it's neat when you do.=20=20

(5) This I find VERY intriguing. After educating myself with the "Four Di=
mensions" section of http://en.wikipedia.org/wiki/Rotation_%28mathematics%2=
9, it seems a nonsimple rotation is just using multiple rotational planes a=
t once. So I'd say the following example (A) is still a simple rotation: =
in MC4D, if you click on a corner or edge sticker of a face...it's still tw=
isting that face over a 2D plane which is spanned by a line going through t=
hat sticker and the opposite sticker on the face as well as the axis that t=
hat face represents (Y axis if it's the +Y face). The following example (B=
) is nonsimple: 2 completely independent 2D rotations at the same time (th=
e rotational planes are orthogonal). Sure enough, someone made a pic: htt=
p://en.wikipedia.org/wiki/File:Tesseract.gif. In MC4D, this is the equival=
ent of Ctrl + clicking on, say, the top face (repeatedly--so that 4 faces k=
eep moving along the vertical axis) while spinning the entire tesseract abo=
ut that axis (so the other 4 faces go in a circle around that vertical axis=
). This is not a twist, still a rotation of the entire tesseract (surely n=
onsimple).=20=20

I Googled the phrase "non simple rotation"...and the "All rotations of the =
4-cube" section of http://gregegan.customer.netspace.net.au/APPLETS/29/Hype=
rcubeNotes.html has some interesting pics with captions in--difficult to gr=
asp, though. At first my question back was...can the planes of rotation be=
non-orthogonal? Then I remembered taking dynamics classes where spinning =
a top on a flat surface creates non-orthogonal rotational planes--there, th=
e tilted rotational plane follows the rules of the horizontal rotational pl=
ane...but not the other way around. Maybe my question back is: are there =
rotations that cannot be described using combinations of rotational planes?=
At any rate, I'd say it would be a true show for the mind of any of these=
were implemented into one of the programs that displayed the animation.

(6) Yes, I was originally imagining 3 combo boxes, but I could see how 2 c=
olumns and 3 rows...or (in N dimensions), 2 columns and (N - 2) rows would =
be less cumbersome. As I click +Y in the up-left drop down, I'm imagining =
Y disappearing from the options in the boxes below (X, Z, U, V would remain=
) as well as Y buttons graying out as described. I'm imagining the right b=
oxes having lots of options...not just Y < -1/2, Y > 1/2, -1/2 < Y < 1/2...=
but also the combos Y < 1/2, Y > -1/2, Y < -1/2 AND Y > 1/2. This would ma=
ke 2 columns and 3 rows even less cumbersome...relatively...especially for =
4^5, 3^7 etc. In MC4D I told Melinda I was imagining Alt + click for these=
twists (compatible with Alt + # + click). I too don't have great time to =
check out/edit the program codes, but besides that, I only know basics for =
each of html, C, Matlab, and TI-calculator code. I'll leave the major prog=
ramming to the programmers while providing user and geometrical feedback. =
=20

--
Andrew Gould
Masters in Math, UW-Milwaukee
PhD student, UW-Milwaukee

p.s. call me Andy




----- Original Message -----
From: "Roice Nelson"
To: "Andrew James Gould"
Cc: foodiddy@gmail.com, "Melinda Green"
Sent: Friday, July 2, 2010 7:08:59 PM
Subject: Re: rotations missing - 5D cube

Hi Andrew,

Thanks for the email. Nice to learn something new about these
hyperpuzzles after playing with them for 10 years :) Here are my
thoughts:

(1) Many many moons ago, I saw another MC4D implementation by Nate
Berglund which provided moves that may end up being exactly like you've
described. I didn't study them much at the time, and didn't go back and
install his software to verify now, but you'd probably be interested to
check it out. http://people.math.gatech.edu/~berglund/Rubik/index.html

(2) I am curious if the new rotation possibilities are indeed "atomic"
or not. By that I mean that puzzle states using the current twists could
be created from the new ones, but not visa versa. Since the 4D cube
example you provided represented a puzzle state which can be achieved
with the currently supported moves, we know that particular move is not
any "more atomic" in this sense. I very much encourage you to forward
your email to the cubing group at large, perhaps with this question
posed. There are members of the group that understand all the parity
restrictions given the current move set, and they could do an analysis
to see if these new move types lead to new puzzle states (I did not copy
the group on my reply here, but feel free to do so if you reply to
this). If the moves are in fact more atomic, I could see this generating
active discussion since all of the calculations for the number of
permutations in the various puzzles would not apply to extended puzzles.

(3) These new rotation types would make the puzzles easier to solve,
especially if they are not "more atomic" and the size of the state space
hasn't changed. This is just an observation, and not an argument against
them. Still, as an example of the fallout of extending the twist types,
there is an active history of shortest solution competitions which would
be affected. Solutions on extended puzzles would need to fall into a
different category in those competitions, due to the changes in the
nature of solving the puzzles.

(4) An elegance of the current behavior is that a twist moves all
stickers on the twisted face in unison. When I first read your email, I
attempted to formulate a mechanical argument against it for this reason
(something like "well, if you could build a physical MC4D, such twists
would result in colliding stickers.") While it looks like your idea does
not result in any such difficulties, I do still feel there is a tradeoff
in elegance here - you'd both gain and lose by making the change.

(5) You mentioned "after all, any rotation in N dimensions is rotating 2
dimensions about an "N-2"-dimensional object". For completeness, I
thought I'd mention that in 4D and above, there are rotations which
rotate more than 2 dimensions, the rotations you are referring to being
called " simple rotations ". Since twists of faces in MC5D are 4D
rotations, I've had the desire over the years to find a nice way to
support twists in this puzzle that are not simple rotations. It hasn't
happened yet. (This is still to be distinguished from your newly
suggested twists, since the rotations I imagined still moved all
stickers of the twisted face in unison).


(6) I like the direction of your UI suggestion, but are you imagining
that both the restricted axis and the slice (e.g. U, and -1/2 < axis <
1/2) get specified in one combo box? When you first click the +Y face,
it is not clear yet that the other two axes that will be involved are U
and V, since it could be X or Z as well. And specifying the various
restricted axes and slices will need to work on larger puzzles like the
5^7, so a design with only 2 additional combo boxes would get awfully
cumbersome as far as the number of items in the list. There could be 6
combo boxes total though (5 new ones), in 2 columns and 3 rows. The left
column would select the axes to restrict to (with the top combo doubling
as selecting the face to twist). The right column would select the
slices. Things would gray out as you described. Anyway, whatever is
deemed a good specification, I don't think it would be terribly
difficult to implement. However, I'm not able to work on MC5D at this
time, and not sure when I will be able to next. The source code for both
MC4D and MC5D are available online to experiment with though.

Thanks again, and I hope you choose to continue this discussion on the
mailing list.

Cheers,
Roice


On Wed, Jun 30, 2010 at 7:59 PM, Andrew James Gould < agould@uwm.edu >
wrote:


Hello,

I had a similar email conversation with Melinda Green who eventually
gave in. All of your rotations, I would deem "legal," however, her 4D
Magic cube and your 5D Magic cubes are missing possible "atomic
rotations."

Terminology: When I open your program, I can click on the top of the
blue (+Y) face and move that sticker to the back-right of that face
toward the green face. This is the same as making the "Face to Twist"
drop-down menu say +Y and clicking on the X side the "X-Z" button. My
terminology for this rotation would be to restrict Y to the range 1/2 <
Y < 3/2 and rotate the Y face via (X side of X-Z) about the Y-U-V
hyperplane (the hyperplane is all variables except X and Z--after all,
any rotation in N dimensions is rotating 2 dimensions about an
"N-2"-dimensional object). If I hold the '2' key down while doing this
rotation, it restricts Y to the range -1/2 < Y < 1/2, holding '1' AND
'2' during this rotations restricts Y to -1/2 < Y < 3/2, and holding,
'1' and '3' during this rotations restricts Y to -3/2 < Y < -1/2 union
1/2 < Y < 3/2. Note: we only restricted on Y.

Rotations: It seems both Melinda Green's MC4D program and your "atomic
twists" only restrict one variable at a time in this manor, but for a
rotation in N dimensions (N > 1), I find that one can restrict UP TO all
of the N-2 dimensions of the hyperplane being rotated about in similar
manors and independently (just not restricting the 2 dimensions of the
rotation). For example, I can restrict further on my previous "holding
down the '2' key" rotation: if I restrict both variables Y and V to
being between -1/2 and 1/2, and rotate the +Y face via (X side of X-Z
holding '2'), I would get the attached picture 5D_2b (paint-program
edited) where 9 purple and 9 white stickers also rotated (8 of these
purples and 8 of these whites moved). If I restrict all three variables,
Y, U, and V, to being between -1/2 and 1/2 and rotate the +Y face via (X
side of X-Z holding '2'), I would get 5D_2bii where only 12 total
stickers (3 from each: +Z, +X, -Z, -X semi-obscured) even moved--nothing
else would even rotate (except possibly the 0-colored interal piece). I
also attached a similar rotation in MC4D: rot_B2. These additional
restricting choices are unseen in 2D and 3D because rotations there are
about 0-dimensional points and 1-dimensional axes respectively where
there are 0 variables and 1 variable to restrict on
(again...respectively).=20

Melinda says the rot_B2 rotation is possible in
MC4D as is, with macros, which may be the case in your program, but I'm
wondering if these additional restrictions would be possible to
implement into your program as "atomic twists", and if so, how difficult
would that be? I'm imagining them being additional drop-down menus below
the "Face to Twist" drop-down menu, but above the twist buttons. I'm
imagining the following for my triple-restricted example: all the
Y-buttons being greyed out as one clicks +Y for Face to Twist, NO
buttons being greyed out as one restricts Y to -1/2 < Y < 1/2, all the
V-buttons being greyed out as one restricts to -1/2 < V < 1/2, and all
the U-buttons being greyed out as one restricts to -1/2 < U < 1/2. After
those 3 restrictions, one only has the X-Z button left to click on
(number keys at this point would either change only the Y restriction or
give an error sound and not change any restriction).

Stopping at the double-restriction (after restricting -1/2 < V < 1/2,
but before U) would leave 3 buttons to click on: X-Z, X-U, and Z-U.
Clicking the X side of X-Z here gets us to 5D_2b. This is also the
intersection of your 2 rotations: rotating the +Y face via (X side of
X-Z button holding '2') and rotating the +V face via (X side of X-Z
button holding '2'). You probably know that rotating in a positive range
always adds more stickers from another face. The same
double-restriction, but V being restricted to 1/2 < V < 3/2 would rotate
27 stickers in the -1/2 < Y < 1/2 slice of the +V face. I could go on
with possibilities.

-- Andrew Gould
Masters in Math, UW-Milwaukee
PhD student, UW-Milwaukee

Hi Andy,
time for some feedback on those twists :). In order of pics:

A1: Not possible currently. It gives odd permutation of 4C pieces. Combin=
ing any two of these produces a valid state though (assuming 90 degree twis=
ts).

B1: Certainly possible. One of the 'parity' cases Klaus presented. Not su=
re I would class it as parity though. 5 move solution which I'm sure I upl=
oaded somewhere around here ...

bii: First, what's with the change of notation? Second, can't see what's h=
appening there, I need a better pic.

A2: Seems to be the same as B1.

b: This order is confusing! So is the picture again. Try showing only the=
necessary faces, otherwise the screen becomes too cluttered in 5D.

B2: This isn't immediately obvious ... got it. Possible in 6 twists.


That seems to be the lot of them. Personally, I prefer the current twist s=
ystem, it seems to be the most natural. Also, I might upload log files of =
these to the folder the pics are in, or at least the possible ones (and may=
be re-upload the Klaus cases). However, nice to hear about a different app=
roach to these puzzles.

Matt

--- In 4D_Cubing@yahoogroups.com, Andrew James Gould wrote:
>
> This is my first email to the group so hello group,
>=20
> I read that I don't need to apologize for length...fewf. I've been ha=
ving conversations with both Melinda and Roice on what appears to be (from =
my perspective) all 3 programs, MC4D, MC5D, and MC7D, using too strict of a=
definition of a "twist." More specifically, the twists in these programs =
twist an n-1 dimensional face, the definition of a twist that I can see is =
that the face being twisted has to be anywhere from 2 dimensional up to n-1=
dimensional. This would allow some 2-d atomic twists in MC4D and MC5D whi=
ch I've edited and posted in Photos > More possible twists. You'll note th=
at my twists in 4D make your famous Klaus parity errors quite simple--possi=
bly too simple for your liking (see 2-4 below). Going with Roice's suggest=
ion, I will continue our conversation with the whole group.=20=20
>=20
> (1) I installed Nate Berglund's program (http://people.math.gatech.edu/~=
berglund/Rubik/index.html), and yes indeed those are exactly the missing mo=
ves for 4D.
>=20
> (2) Great question: do these twists make for more possible states? We =
do need the group's help here. For the 3^4 cube, Matthew Sheerin, or Klaus=
may be able to help answer. I'm referring to messages #695, 772, 778, and=
Photos > "parity problems" by Klaus: http://groups.yahoo.com/group/4D_Cub=
ing/photos/album/565962423/pic/list. Matt says he posted a solution to the=
se, but I can't find it. I notice Klaus's Oct. 13 parity error has the col=
ors across from each other. Therefore we still need to know if my rot_A1 t=
wist is solvable using current MC4D twists as well as Klaus's Nov. 14 parit=
y error (same as my rot_A2 twist...same as my rot_B1 twist in a sense). If=
these 2 are solvable, then my rot_B2 twist could be created using each of =
these 4 times + a rotation of the entire tesseract and thus I would have in=
troduced no new states. I now doubt this is the case, however--my guess is=
that my twists introduce new states.=20=20
>=20
> (3) Yeah, when you open up Berglund's program you can choose to allow or=
disallow my twists. He classifies them as two separate puzzles, which may=
be the way to go. Another way to go, for example is 2 separate versions o=
f MC4D: the current version with only 3D twists allowed vs. a version wher=
e both 2D and 3D twists allowed.
>=20
> (4) I was preparing a statement like this...only much worse. I was prep=
ared for something analogous to Christopher Columbus being laughed off the =
flat face of the planet for thinking it's round. Of course I was hoping yo=
u'd phrase it as nicely as you did. Before ever searching and finding your=
programs on the internet, I had visualized a 3^4 tesseract in an X, Y, Z, =
T coordinate system as described in my email at the bottom with the center =
of the tesseract being at the origin and having the cubie edge length =3D 1=
. I visualized the seperators t =3D -1/2 and t =3D 1/2 dividing it into 3 =
"cubes." I simply figured you could twist just the top (z > 1/2) of the t =
< -1/2 "cube." Doing so results in no z nor t coordinate change for any "4=
D atom" of the entire tesseract so no stickers nor cubies will cross these =
seperators during the twist and nothing runs into eachother. This visualiz=
ation method made it difficult to visualize how to twist z > 1/2 and x > 1/=
2. I wasn't sure it was possible, but I realized one can always rotate the=
entire tesseract so x ----> -t. That way it's the same as the previous tw=
ist, so I knew it was possible. I went back and tried visualizing this twi=
st without the rotation, and although it would take a while to describe, I =
can tell you, it's neat when you do.=20=20
>=20
> (5) This I find VERY intriguing. After educating myself with the "Four =
Dimensions" section of http://en.wikipedia.org/wiki/Rotation_%28mathematics=
%29, it seems a nonsimple rotation is just using multiple rotational planes=
at once. So I'd say the following example (A) is still a simple rotation:=
in MC4D, if you click on a corner or edge sticker of a face...it's still =
twisting that face over a 2D plane which is spanned by a line going through=
that sticker and the opposite sticker on the face as well as the axis that=
that face represents (Y axis if it's the +Y face). The following example =
(B) is nonsimple: 2 completely independent 2D rotations at the same time (=
the rotational planes are orthogonal). Sure enough, someone made a pic: h=
ttp://en.wikipedia.org/wiki/File:Tesseract.gif. In MC4D, this is the equiv=
alent of Ctrl + clicking on, say, the top face (repeatedly--so that 4 faces=
keep moving along the vertical axis) while spinning the entire tesseract a=
bout that axis (so the other 4 faces go in a circle around that vertical ax=
is). This is not a twist, still a rotation of the entire tesseract (surely=
nonsimple).=20=20
>=20
> I Googled the phrase "non simple rotation"...and the "All rotations of th=
e 4-cube" section of http://gregegan.customer.netspace.net.au/APPLETS/29/Hy=
percubeNotes.html has some interesting pics with captions in--difficult to =
grasp, though. At first my question back was...can the planes of rotation =
be non-orthogonal? Then I remembered taking dynamics classes where spinnin=
g a top on a flat surface creates non-orthogonal rotational planes--there, =
the tilted rotational plane follows the rules of the horizontal rotational =
plane...but not the other way around. Maybe my question back is: are ther=
e rotations that cannot be described using combinations of rotational plane=
s? At any rate, I'd say it would be a true show for the mind of any of the=
se were implemented into one of the programs that displayed the animation.
>=20
> (6) Yes, I was originally imagining 3 combo boxes, but I could see how 2=
columns and 3 rows...or (in N dimensions), 2 columns and (N - 2) rows woul=
d be less cumbersome. As I click +Y in the up-left drop down, I'm imaginin=
g Y disappearing from the options in the boxes below (X, Z, U, V would rema=
in) as well as Y buttons graying out as described. I'm imagining the right=
boxes having lots of options...not just Y < -1/2, Y > 1/2, -1/2 < Y < 1/2.=
..but also the combos Y < 1/2, Y > -1/2, Y < -1/2 AND Y > 1/2. This would =
make 2 columns and 3 rows even less cumbersome...relatively...especially fo=
r 4^5, 3^7 etc. In MC4D I told Melinda I was imagining Alt + click for the=
se twists (compatible with Alt + # + click). I too don't have great time t=
o check out/edit the program codes, but besides that, I only know basics fo=
r each of html, C, Matlab, and TI-calculator code. I'll leave the major pr=
ogramming to the programmers while providing user and geometrical feedback.=
=20=20
>=20
> --
> Andrew Gould
> Masters in Math, UW-Milwaukee
> PhD student, UW-Milwaukee
>=20
> p.s. call me Andy
>=20
>=20
>=20
>=20
> ----- Original Message -----
> From: "Roice Nelson"
> To: "Andrew James Gould"
> Cc: foodiddy@..., "Melinda Green"
> Sent: Friday, July 2, 2010 7:08:59 PM
> Subject: Re: rotations missing - 5D cube
>=20
> Hi Andrew,
>=20
> Thanks for the email. Nice to learn something new about these
> hyperpuzzles after playing with them for 10 years :) Here are my
> thoughts:
>=20
> (1) Many many moons ago, I saw another MC4D implementation by Nate
> Berglund which provided moves that may end up being exactly like you've
> described. I didn't study them much at the time, and didn't go back and
> install his software to verify now, but you'd probably be interested to
> check it out. http://people.math.gatech.edu/~berglund/Rubik/index.html
>=20
> (2) I am curious if the new rotation possibilities are indeed "atomic"
> or not. By that I mean that puzzle states using the current twists could
> be created from the new ones, but not visa versa. Since the 4D cube
> example you provided represented a puzzle state which can be achieved
> with the currently supported moves, we know that particular move is not
> any "more atomic" in this sense. I very much encourage you to forward
> your email to the cubing group at large, perhaps with this question
> posed. There are members of the group that understand all the parity
> restrictions given the current move set, and they could do an analysis
> to see if these new move types lead to new puzzle states (I did not copy
> the group on my reply here, but feel free to do so if you reply to
> this). If the moves are in fact more atomic, I could see this generating
> active discussion since all of the calculations for the number of
> permutations in the various puzzles would not apply to extended puzzles.
>=20
> (3) These new rotation types would make the puzzles easier to solve,
> especially if they are not "more atomic" and the size of the state space
> hasn't changed. This is just an observation, and not an argument against
> them. Still, as an example of the fallout of extending the twist types,
> there is an active history of shortest solution competitions which would
> be affected. Solutions on extended puzzles would need to fall into a
> different category in those competitions, due to the changes in the
> nature of solving the puzzles.
>=20
> (4) An elegance of the current behavior is that a twist moves all
> stickers on the twisted face in unison. When I first read your email, I
> attempted to formulate a mechanical argument against it for this reason
> (something like "well, if you could build a physical MC4D, such twists
> would result in colliding stickers.") While it looks like your idea does
> not result in any such difficulties, I do still feel there is a tradeoff
> in elegance here - you'd both gain and lose by making the change.
>=20
> (5) You mentioned "after all, any rotation in N dimensions is rotating 2
> dimensions about an "N-2"-dimensional object". For completeness, I
> thought I'd mention that in 4D and above, there are rotations which
> rotate more than 2 dimensions, the rotations you are referring to being
> called " simple rotations ". Since twists of faces in MC5D are 4D
> rotations, I've had the desire over the years to find a nice way to
> support twists in this puzzle that are not simple rotations. It hasn't
> happened yet. (This is still to be distinguished from your newly
> suggested twists, since the rotations I imagined still moved all
> stickers of the twisted face in unison).
>=20
>=20
> (6) I like the direction of your UI suggestion, but are you imagining
> that both the restricted axis and the slice (e.g. U, and -1/2 < axis <
> 1/2) get specified in one combo box? When you first click the +Y face,
> it is not clear yet that the other two axes that will be involved are U
> and V, since it could be X or Z as well. And specifying the various
> restricted axes and slices will need to work on larger puzzles like the
> 5^7, so a design with only 2 additional combo boxes would get awfully
> cumbersome as far as the number of items in the list. There could be 6
> combo boxes total though (5 new ones), in 2 columns and 3 rows. The left
> column would select the axes to restrict to (with the top combo doubling
> as selecting the face to twist). The right column would select the
> slices. Things would gray out as you described. Anyway, whatever is
> deemed a good specification, I don't think it would be terribly
> difficult to implement. However, I'm not able to work on MC5D at this
> time, and not sure when I will be able to next. The source code for both
> MC4D and MC5D are available online to experiment with though.
>=20
> Thanks again, and I hope you choose to continue this discussion on the
> mailing list.
>=20
> Cheers,
> Roice
>=20
>=20
> On Wed, Jun 30, 2010 at 7:59 PM, Andrew James Gould < agould@... >
> wrote:
>=20
>=20
> Hello,
>=20
> I had a similar email conversation with Melinda Green who eventually
> gave in. All of your rotations, I would deem "legal," however, her 4D
> Magic cube and your 5D Magic cubes are missing possible "atomic
> rotations."
>=20
> Terminology: When I open your program, I can click on the top of the
> blue (+Y) face and move that sticker to the back-right of that face
> toward the green face. This is the same as making the "Face to Twist"
> drop-down menu say +Y and clicking on the X side the "X-Z" button. My
> terminology for this rotation would be to restrict Y to the range 1/2 <
> Y < 3/2 and rotate the Y face via (X side of X-Z) about the Y-U-V
> hyperplane (the hyperplane is all variables except X and Z--after all,
> any rotation in N dimensions is rotating 2 dimensions about an
> "N-2"-dimensional object). If I hold the '2' key down while doing this
> rotation, it restricts Y to the range -1/2 < Y < 1/2, holding '1' AND
> '2' during this rotations restricts Y to -1/2 < Y < 3/2, and holding,
> '1' and '3' during this rotations restricts Y to -3/2 < Y < -1/2 union
> 1/2 < Y < 3/2. Note: we only restricted on Y.
>=20
> Rotations: It seems both Melinda Green's MC4D program and your "atomic
> twists" only restrict one variable at a time in this manor, but for a
> rotation in N dimensions (N > 1), I find that one can restrict UP TO all
> of the N-2 dimensions of the hyperplane being rotated about in similar
> manors and independently (just not restricting the 2 dimensions of the
> rotation). For example, I can restrict further on my previous "holding
> down the '2' key" rotation: if I restrict both variables Y and V to
> being between -1/2 and 1/2, and rotate the +Y face via (X side of X-Z
> holding '2'), I would get the attached picture 5D_2b (paint-program
> edited) where 9 purple and 9 white stickers also rotated (8 of these
> purples and 8 of these whites moved). If I restrict all three variables,
> Y, U, and V, to being between -1/2 and 1/2 and rotate the +Y face via (X
> side of X-Z holding '2'), I would get 5D_2bii where only 12 total
> stickers (3 from each: +Z, +X, -Z, -X semi-obscured) even moved--nothing
> else would even rotate (except possibly the 0-colored interal piece). I
> also attached a similar rotation in MC4D: rot_B2. These additional
> restricting choices are unseen in 2D and 3D because rotations there are
> about 0-dimensional points and 1-dimensional axes respectively where
> there are 0 variables and 1 variable to restrict on
> (again...respectively).=20
>=20
> Melinda says the rot_B2 rotation is possible in
> MC4D as is, with macros, which may be the case in your program, but I'm
> wondering if these additional restrictions would be possible to
> implement into your program as "atomic twists", and if so, how difficult
> would that be? I'm imagining them being additional drop-down menus below
> the "Face to Twist" drop-down menu, but above the twist buttons. I'm
> imagining the following for my triple-restricted example: all the
> Y-buttons being greyed out as one clicks +Y for Face to Twist, NO
> buttons being greyed out as one restricts Y to -1/2 < Y < 1/2, all the
> V-buttons being greyed out as one restricts to -1/2 < V < 1/2, and all
> the U-buttons being greyed out as one restricts to -1/2 < U < 1/2. After
> those 3 restrictions, one only has the X-Z button left to click on
> (number keys at this point would either change only the Y restriction or
> give an error sound and not change any restriction).
>=20
> Stopping at the double-restriction (after restricting -1/2 < V < 1/2,
> but before U) would leave 3 buttons to click on: X-Z, X-U, and Z-U.
> Clicking the X side of X-Z here gets us to 5D_2b. This is also the
> intersection of your 2 rotations: rotating the +Y face via (X side of
> X-Z button holding '2') and rotating the +V face via (X side of X-Z
> button holding '2'). You probably know that rotating in a positive range
> always adds more stickers from another face. The same
> double-restriction, but V being restricted to 1/2 < V < 3/2 would rotate
> 27 stickers in the -1/2 < Y < 1/2 slice of the +V face. I could go on
> with possibilities.
>=20
> -- Andrew Gould
> Masters in Math, UW-Milwaukee
> PhD student, UW-Milwaukee
>

--00c09fa2180386929d048ba1c3d2
Content-Type: text/plain; charset=windows-1252
Content-Transfer-Encoding: quoted-printable

I haven't posted on this list that often but I find the discussion about
redefining what a twist is in a N dimensional puzzle interesting. It never
accured to me that you could define moves in any way other than the way it
was done in the MC4D. The new definition of moves would definitely make the
puzzle easier but that's not necessarily a reason to rule out the
possibility. I think that implementing a puzzle that allowed two dimensiona=
l
twists would make the puzzle accessable to people it wouldn't already be
accessible to. This would be good if you are trying to increase the
visibility of the program but may be bad for people who want all the
individuals who have solved the puzzle to form an elite group. The new move=
s
could be implemented in a very similar way to the way moves are implemented
in the current puzzle but when you click on a 2C piece instead of the entir=
e
cube rotating around that access only a 2D face would rotate around that
access. If you wanted to allow both 2D rotations and 3D rotations then you
would need to use another control character.

I don't have the patients to try to solve the puzzle linked to earlier in
this post due to the bad graphics, and only allowing moves on the center
face, but allowing a person to rotate 2D faces would dramatically change ho=
w
the puzzle is solved. Allowing these moves will allow individuals to use
some 3D logic in order to solve the puzzle because once you get all the
pieces of one color on a cube you can move them around on that cube without
messing up any pieces on the entire rest of the cube. You would still need
to take the fourth dimension into account when locating pieces but there
would be fewer dependencies to worry about when placing the piece. There ar=
e
ways to move pieces around independently in the current puzzle obviously,
but they are more involved and often require long sequences of moves that
can be difficult to keep track of.

On Sat, Jul 17, 2010 at 6:20 PM, matthewsheerin
wrote:

>
>
> Hi Andy,
> time for some feedback on those twists :). In order of pics:
>
> A1: Not possible currently. It gives odd permutation of 4C pieces.
> Combining any two of these produces a valid state though (assuming 90 deg=
ree
> twists).
>
> B1: Certainly possible. One of the 'parity' cases Klaus presented. Not su=
re
> I would class it as parity though. 5 move solution which I'm sure I uploa=
ded
> somewhere around here ...
>
> bii: First, what's with the change of notation? Second, can't see what's
> happening there, I need a better pic.
>
> A2: Seems to be the same as B1.
>
> b: This order is confusing! So is the picture again. Try showing only the
> necessary faces, otherwise the screen becomes too cluttered in 5D.
>
> B2: This isn't immediately obvious ... got it. Possible in 6 twists.
>
> That seems to be the lot of them. Personally, I prefer the current twist
> system, it seems to be the most natural. Also, I might upload log files o=
f
> these to the folder the pics are in, or at least the possible ones (and
> maybe re-upload the Klaus cases). However, nice to hear about a different
> approach to these puzzles.
>
> Matt
>
>
> --- In 4D_Cubing@yahoogroups.com <4D_Cubing%40yahoogroups.com>, Andrew
> James Gould wrote:
> >
> > This is my first email to the group so hello group,
> >
> > I read that I don't need to apologize for length...fewf. I've been havi=
ng
> conversations with both Melinda and Roice on what appears to be (from my
> perspective) all 3 programs, MC4D, MC5D, and MC7D, using too strict of a
> definition of a "twist." More specifically, the twists in these programs
> twist an n-1 dimensional face, the definition of a twist that I can see i=
s
> that the face being twisted has to be anywhere from 2 dimensional up to n=
-1
> dimensional. This would allow some 2-d atomic twists in MC4D and MC5D whi=
ch
> I've edited and posted in Photos > More possible twists. You'll note that=
my
> twists in 4D make your famous Klaus parity errors quite simple--possibly =
too
> simple for your liking (see 2-4 below). Going with Roice's suggestion, I
> will continue our conversation with the whole group.
> >
> > (1) I installed Nate Berglund's program (
> http://people.math.gatech.edu/~berglund/Rubik/index.htmlth.gatech.edu/%7Eberglund/Rubik/index.html>),
> and yes indeed those are exactly the missing moves for 4D.
> >
> > (2) Great question: do these twists make for more possible states? We d=
o
> need the group's help here. For the 3^4 cube, Matthew Sheerin, or Klaus m=
ay
> be able to help answer. I'm referring to messages #695, 772, 778, and Pho=
tos
> > "parity problems" by Klaus:
> http://groups.yahoo.com/group/4D_Cubing/photos/album/565962423/pic/list.M=
att says he posted a solution to these, but I can't find it. I notice
> Klaus's Oct. 13 parity error has the colors across from each other.
> Therefore we still need to know if my rot_A1 twist is solvable using curr=
ent
> MC4D twists as well as Klaus's Nov. 14 parity error (same as my rot_A2
> twist...same as my rot_B1 twist in a sense). If these 2 are solvable, the=
n
> my rot_B2 twist could be created using each of these 4 times + a rotation=
of
> the entire tesseract and thus I would have introduced no new states. I no=
w
> doubt this is the case, however--my guess is that my twists introduce new
> states.
> >
> > (3) Yeah, when you open up Berglund's program you can choose to allow o=
r
> disallow my twists. He classifies them as two separate puzzles, which may=
be
> the way to go. Another way to go, for example is 2 separate versions of
> MC4D: the current version with only 3D twists allowed vs. a version where
> both 2D and 3D twists allowed.
> >
> > (4) I was preparing a statement like this...only much worse. I was
> prepared for something analogous to Christopher Columbus being laughed of=
f
> the flat face of the planet for thinking it's round. Of course I was hopi=
ng
> you'd phrase it as nicely as you did. Before ever searching and finding y=
our
> programs on the internet, I had visualized a 3^4 tesseract in an X, Y, Z,=
T
> coordinate system as described in my email at the bottom with the center =
of
> the tesseract being at the origin and having the cubie edge length =3D 1.=
I
> visualized the seperators t =3D -1/2 and t =3D 1/2 dividing it into 3 "cu=
bes." I
> simply figured you could twist just the top (z > 1/2) of the t < -1/2
> "cube." Doing so results in no z nor t coordinate change for any "4D atom=
"
> of the entire tesseract so no stickers nor cubies will cross these
> seperators during the twist and nothing runs into eachother. This
> visualization method made it difficult to visualize how to twist z > 1/2 =
and
> x > 1/2. I wasn't sure it was possible, but I realized one can always rot=
ate
> the entire tesseract so x ----> -t. That way it's the same as the previou=
s
> twist, so I knew it was possible. I went back and tried visualizing this
> twist without the rotation, and although it would take a while to describ=
e,
> I can tell you, it's neat when you do.
> >
> > (5) This I find VERY intriguing. After educating myself with the "Four
> Dimensions" section of
> http://en.wikipedia.org/wiki/Rotation_%28mathematics%29, it seems a
> nonsimple rotation is just using multiple rotational planes at once. So I=
'd
> say the following example (A) is still a simple rotation: in MC4D, if you
> click on a corner or edge sticker of a face...it's still twisting that fa=
ce
> over a 2D plane which is spanned by a line going through that sticker and
> the opposite sticker on the face as well as the axis that that face
> represents (Y axis if it's the +Y face). The following example (B) is
> nonsimple: 2 completely independent 2D rotations at the same time (the
> rotational planes are orthogonal). Sure enough, someone made a pic:
> http://en.wikipedia.org/wiki/File:Tesseract.gif. In MC4D, this is the
> equivalent of Ctrl + clicking on, say, the top face (repeatedly--so that =
4
> faces keep moving along the vertical axis) while spinning the entire
> tesseract about that axis (so the other 4 faces go in a circle around tha=
t
> vertical axis). This is not a twist, still a rotation of the entire
> tesseract (surely nonsimple).
> >
> > I Googled the phrase "non simple rotation"...and the "All rotations of
> the 4-cube" section of
> http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.htmlha=
s some interesting pics with captions in--difficult to grasp, though. At
> first my question back was...can the planes of rotation be non-orthogonal=
?
> Then I remembered taking dynamics classes where spinning a top on a flat
> surface creates non-orthogonal rotational planes--there, the tilted
> rotational plane follows the rules of the horizontal rotational plane...b=
ut
> not the other way around. Maybe my question back is: are there rotations
> that cannot be described using combinations of rotational planes? At any
> rate, I'd say it would be a true show for the mind of any of these were
> implemented into one of the programs that displayed the animation.
> >
> > (6) Yes, I was originally imagining 3 combo boxes, but I could see how =
2
> columns and 3 rows...or (in N dimensions), 2 columns and (N - 2) rows wou=
ld
> be less cumbersome. As I click +Y in the up-left drop down, I'm imagining=
Y
> disappearing from the options in the boxes below (X, Z, U, V would remain=
)
> as well as Y buttons graying out as described. I'm imagining the right bo=
xes
> having lots of options...not just Y < -1/2, Y > 1/2, -1/2 < Y < 1/2...but
> also the combos Y < 1/2, Y > -1/2, Y < -1/2 AND Y > 1/2. This would make =
2
> columns and 3 rows even less cumbersome...relatively...especially for 4^5=
,
> 3^7 etc. In MC4D I told Melinda I was imagining Alt + click for these twi=
sts
> (compatible with Alt + # + click). I too don't have great time to check
> out/edit the program codes, but besides that, I only know basics for each=
of
> html, C, Matlab, and TI-calculator code. I'll leave the major programming=
to
> the programmers while providing user and geometrical feedback.
> >
> > --
> > Andrew Gould
> > Masters in Math, UW-Milwaukee
> > PhD student, UW-Milwaukee
> >
> > p.s. call me Andy
> >
> >
> >
> >
> > ----- Original Message -----
> > From: "Roice Nelson"
> > To: "Andrew James Gould"
> > Cc: foodiddy@..., "Melinda Green"
> > Sent: Friday, July 2, 2010 7:08:59 PM
> > Subject: Re: rotations missing - 5D cube
> >
> > Hi Andrew,
> >
> > Thanks for the email. Nice to learn something new about these
> > hyperpuzzles after playing with them for 10 years :) Here are my
> > thoughts:
> >
> > (1) Many many moons ago, I saw another MC4D implementation by Nate
> > Berglund which provided moves that may end up being exactly like you've
> > described. I didn't study them much at the time, and didn't go back and
> > install his software to verify now, but you'd probably be interested to
> > check it out. http://people.math.gatech.edu/~berglund/Rubik/index.html<=
http://people.math.gatech.edu/%7Eberglund/Rubik/index.html>
> >
> > (2) I am curious if the new rotation possibilities are indeed "atomic"
> > or not. By that I mean that puzzle states using the current twists coul=
d
> > be created from the new ones, but not visa versa. Since the 4D cube
> > example you provided represented a puzzle state which can be achieved
> > with the currently supported moves, we know that particular move is not
> > any "more atomic" in this sense. I very much encourage you to forward
> > your email to the cubing group at large, perhaps with this question
> > posed. There are members of the group that understand all the parity
> > restrictions given the current move set, and they could do an analysis
> > to see if these new move types lead to new puzzle states (I did not cop=
y
> > the group on my reply here, but feel free to do so if you reply to
> > this). If the moves are in fact more atomic, I could see this generatin=
g
> > active discussion since all of the calculations for the number of
> > permutations in the various puzzles would not apply to extended puzzles=
.
> >
> > (3) These new rotation types would make the puzzles easier to solve,
> > especially if they are not "more atomic" and the size of the state spac=
e
> > hasn't changed. This is just an observation, and not an argument agains=
t
> > them. Still, as an example of the fallout of extending the twist types,
> > there is an active history of shortest solution competitions which woul=
d
> > be affected. Solutions on extended puzzles would need to fall into a
> > different category in those competitions, due to the changes in the
> > nature of solving the puzzles.
> >
> > (4) An elegance of the current behavior is that a twist moves all
> > stickers on the twisted face in unison. When I first read your email, I
> > attempted to formulate a mechanical argument against it for this reason
> > (something like "well, if you could build a physical MC4D, such twists
> > would result in colliding stickers.") While it looks like your idea doe=
s
> > not result in any such difficulties, I do still feel there is a tradeof=
f
> > in elegance here - you'd both gain and lose by making the change.
> >
> > (5) You mentioned "after all, any rotation in N dimensions is rotating =
2
> > dimensions about an "N-2"-dimensional object". For completeness, I
> > thought I'd mention that in 4D and above, there are rotations which
> > rotate more than 2 dimensions, the rotations you are referring to being
> > called " simple rotations ". Since twists of faces in MC5D are 4D
> > rotations, I've had the desire over the years to find a nice way to
> > support twists in this puzzle that are not simple rotations. It hasn't
> > happened yet. (This is still to be distinguished from your newly
> > suggested twists, since the rotations I imagined still moved all
> > stickers of the twisted face in unison).
> >
> >
> > (6) I like the direction of your UI suggestion, but are you imagining
> > that both the restricted axis and the slice (e.g. U, and -1/2 < axis <
> > 1/2) get specified in one combo box? When you first click the +Y face,
> > it is not clear yet that the other two axes that will be involved are U
> > and V, since it could be X or Z as well. And specifying the various
> > restricted axes and slices will need to work on larger puzzles like the
> > 5^7, so a design with only 2 additional combo boxes would get awfully
> > cumbersome as far as the number of items in the list. There could be 6
> > combo boxes total though (5 new ones), in 2 columns and 3 rows. The lef=
t
> > column would select the axes to restrict to (with the top combo doublin=
g
> > as selecting the face to twist). The right column would select the
> > slices. Things would gray out as you described. Anyway, whatever is
> > deemed a good specification, I don't think it would be terribly
> > difficult to implement. However, I'm not able to work on MC5D at this
> > time, and not sure when I will be able to next. The source code for bot=
h
> > MC4D and MC5D are available online to experiment with though.
> >
> > Thanks again, and I hope you choose to continue this discussion on the
> > mailing list.
> >
> > Cheers,
> > Roice
> >
> >
> > On Wed, Jun 30, 2010 at 7:59 PM, Andrew James Gould < agould@... >
>
> > wrote:
> >
> >
> > Hello,
> >
> > I had a similar email conversation with Melinda Green who eventually
> > gave in. All of your rotations, I would deem "legal," however, her 4D
> > Magic cube and your 5D Magic cubes are missing possible "atomic
> > rotations."
> >
> > Terminology: When I open your program, I can click on the top of the
> > blue (+Y) face and move that sticker to the back-right of that face
> > toward the green face. This is the same as making the "Face to Twist"
> > drop-down menu say +Y and clicking on the X side the "X-Z" button. My
> > terminology for this rotation would be to restrict Y to the range 1/2 <
> > Y < 3/2 and rotate the Y face via (X side of X-Z) about the Y-U-V
> > hyperplane (the hyperplane is all variables except X and Z--after all,
> > any rotation in N dimensions is rotating 2 dimensions about an
> > "N-2"-dimensional object). If I hold the '2' key down while doing this
> > rotation, it restricts Y to the range -1/2 < Y < 1/2, holding '1' AND
> > '2' during this rotations restricts Y to -1/2 < Y < 3/2, and holding,
> > '1' and '3' during this rotations restricts Y to -3/2 < Y < -1/2 union
> > 1/2 < Y < 3/2. Note: we only restricted on Y.
> >
> > Rotations: It seems both Melinda Green's MC4D program and your "atomic
> > twists" only restrict one variable at a time in this manor, but for a
> > rotation in N dimensions (N > 1), I find that one can restrict UP TO al=
l
> > of the N-2 dimensions of the hyperplane being rotated about in similar
> > manors and independently (just not restricting the 2 dimensions of the
> > rotation). For example, I can restrict further on my previous "holding
> > down the '2' key" rotation: if I restrict both variables Y and V to
> > being between -1/2 and 1/2, and rotate the +Y face via (X side of X-Z
> > holding '2'), I would get the attached picture 5D_2b (paint-program
> > edited) where 9 purple and 9 white stickers also rotated (8 of these
> > purples and 8 of these whites moved). If I restrict all three variables=
,
> > Y, U, and V, to being between -1/2 and 1/2 and rotate the +Y face via (=
X
> > side of X-Z holding '2'), I would get 5D_2bii where only 12 total
> > stickers (3 from each: +Z, +X, -Z, -X semi-obscured) even moved--nothin=
g
> > else would even rotate (except possibly the 0-colored interal piece). I
> > also attached a similar rotation in MC4D: rot_B2. These additional
> > restricting choices are unseen in 2D and 3D because rotations there are
> > about 0-dimensional points and 1-dimensional axes respectively where
> > there are 0 variables and 1 variable to restrict on
> > (again...respectively).
> >
> > Melinda says the rot_B2 rotation is possible in
> > MC4D as is, with macros, which may be the case in your program, but I'm
> > wondering if these additional restrictions would be possible to
> > implement into your program as "atomic twists", and if so, how difficul=
t
> > would that be? I'm imagining them being additional drop-down menus belo=
w
> > the "Face to Twist" drop-down menu, but above the twist buttons. I'm
> > imagining the following for my triple-restricted example: all the
> > Y-buttons being greyed out as one clicks +Y for Face to Twist, NO
> > buttons being greyed out as one restricts Y to -1/2 < Y < 1/2, all the
> > V-buttons being greyed out as one restricts to -1/2 < V < 1/2, and all
> > the U-buttons being greyed out as one restricts to -1/2 < U < 1/2. Afte=
r
> > those 3 restrictions, one only has the X-Z button left to click on
> > (number keys at this point would either change only the Y restriction o=
r
> > give an error sound and not change any restriction).
> >
> > Stopping at the double-restriction (after restricting -1/2 < V < 1/2,
> > but before U) would leave 3 buttons to click on: X-Z, X-U, and Z-U.
> > Clicking the X side of X-Z here gets us to 5D_2b. This is also the
> > intersection of your 2 rotations: rotating the +Y face via (X side of
> > X-Z button holding '2') and rotating the +V face via (X side of X-Z
> > button holding '2'). You probably know that rotating in a positive rang=
e
> > always adds more stickers from another face. The same
> > double-restriction, but V being restricted to 1/2 < V < 3/2 would rotat=
e
> > 27 stickers in the -1/2 < Y < 1/2 slice of the +V face. I could go on
> > with possibilities.
> >
> > -- Andrew Gould
> > Masters in Math, UW-Milwaukee
> > PhD student, UW-Milwaukee
> >
>
>=20=20
>

--00c09fa2180386929d048ba1c3d2
Content-Type: text/html; charset=windows-1252
Content-Transfer-Encoding: quoted-printable

I haven't posted on this list that often but I find the discussion abou=
t redefining what a twist is in a N dimensional puzzle interesting. It neve=
r accured to me that you could define moves in any way other than the way i=
t was done in the MC4D. The new definition of moves would definitely make t=
he puzzle easier but that's not necessarily a reason to rule out the po=
ssibility. I think that implementing a puzzle that allowed two dimensional =
twists would make the puzzle accessable to people it wouldn't already b=
e accessible to. This would be good if you are trying to increase the visib=
ility of the program but may be bad for people who want all the individuals=
who have solved the puzzle to form an elite group. The new moves could be =
implemented in a very similar way to the way moves are implemented in the c=
urrent puzzle but when you click on a 2C piece instead of the entire cube r=
otating around that access only a 2D face would rotate around that access. =
If you wanted to allow both 2D rotations and 3D rotations then you would ne=
ed to use another control character.


I don't have the patients to try to solve the puzzle linked to earl=
ier in this post due to the bad graphics, and only allowing moves on the ce=
nter face, but allowing a person to rotate 2D faces would dramatically chan=
ge how the puzzle is solved. Allowing these moves will allow individuals t=
o use some 3D logic in=20
order to solve the puzzle because once you get all the pieces of one=20
color on a cube you can move them around on that cube without messing up
any pieces on the entire rest of the cube.=A0 You would still need to take=
the fourth dimension into account when locating pieces but there would be =
fewer dependencies to worry about when placing the piece. There are ways to=
move pieces around independently in the current puzzle obviously,=A0 but t=
hey are more involved and often require long sequences of moves that can be=
difficult to keep track of.

=A0

--0-1530771421-1279448205=:52288
Content-Type: text/plain; charset=utf-8
Content-Transfer-Encoding: quoted-printable

It is an interesting idea to allow lower-dimensional twists, but I don't=20
consider this feature necessary enough
to be implemented. First of all Matthew showed, that some of these twists a=
re=20
possible to be obtained by
short algs, and more important, these twists would not be possible on a rea=
l 4D=20
cube in 4D space.=20


Happy Hypercubing,
Klaus




________________________________
From: Jenelle Levenstein
To: 4D_Cubing@yahoogroups.com
Sent: Sun, July 18, 2010 6:19:53 AM
Subject: Re: [MC4D] Re: definition of a twist

=20=20
I haven't posted on this list that often but I find the discussion about=20
redefining what a twist is in a N dimensional puzzle interesting. It never=
=20
accured to me that you could define moves in any way other than the way it =
was=20
done in the MC4D. The new definition of moves would definitely make the puz=
zle=20
easier but that's not necessarily a reason to rule out the possibility. I t=
hink=20
that implementing a puzzle that allowed two dimensional twists would make t=
he=20
puzzle accessable to people it wouldn't already be accessible to. This woul=
d be=20
good if you are trying to increase the visibility of the program but may be=
bad=20
for people who want all the individuals who have solved the puzzle to form =
an=20
elite group. The new moves could be implemented in a very similar way to th=
e way=20
moves are implemented in the current puzzle but when you click on a 2C piec=
e=20
instead of the entire cube rotating around that access only a 2D face would=
=20
rotate around that access. If you wanted to allow both 2D rotations and 3D=
=20
rotations then you would need to use another control character.=20


I don't have the patients to try to solve the puzzle linked to earlier in t=
his=20
post due to the bad graphics, and only allowing moves on the center face, b=
ut=20
allowing a person to rotate 2D faces would dramatically change how the puzz=
le is=20
solved. Allowing these moves will allow individuals to use some 3D logic i=
n=20=20
order to solve the puzzle because once you get all the pieces of one color=
on a=20
cube you can move them around on that cube without messing up any pieces o=
n the=20
entire rest of the cube. You would still need to take the fourth dimension=
into=20
account when locating pieces but there would be fewer dependencies to worry=
=20
about when placing the piece. There are ways to move pieces around independ=
ently=20
in the current puzzle obviously, but they are more involved and often requ=
ire=20
long sequences of moves that can be difficult to keep track of.=20

=20

On Sat, Jul 17, 2010 at 6:20 PM, matthewsheerin uk>=20
wrote:

=20=20
>Hi Andy,
>time for some feedback on those twists :). In order of pics:
>
>A1: Not possible currently. It gives odd permutation of 4C pieces. Combi=
ning=20
>any two of these produces a valid state though (assuming 90 degree twists)=
.
>
>B1: Certainly possible. One of the 'parity' cases Klaus presented. Not s=
ure I=20
>would class it as parity though. 5 move solution which I'm sure I uploade=
d=20
>somewhere around here ...
>
>bii: First, what's with the change of notation? Second, can't see what's=
=20
>happening there, I need a better pic.
>
>A2: Seems to be the same as B1.
>
>b: This order is confusing! So is the picture again. Try showing only th=
e=20
>necessary faces, otherwise the screen becomes too cluttered in 5D.
>
>B2: This isn't immediately obvious ... got it. Possible in 6 twists.
>
>That seems to be the lot of them. Personally, I prefer the current twist=
=20
>system, it seems to be the most natural. Also, I might upload log files o=
f=20
>these to the folder the pics are in, or at least the possible ones (and ma=
ybe=20
>re-upload the Klaus cases). However, nice to hear about a different appro=
ach to=20
>these puzzles.
>
>Matt
>
>
>--- In 4D_Cubing@yahoogrou ps.com, Andrew James Gould wrote:
>>
>> This is my first email to the group so hello group,
>>=20
>> I read that I don't need to apologize for length...fewf. I've been h=
aving=20
>>conversations with both Melinda and Roice on what appears to be (from my=
=20
>>perspective) all 3 programs, MC4D, MC5D, and MC7D, using too strict of a=
=20
>>definition of a "twist." More specifically, the twists in these programs=
twist=20
>>an n-1 dimensional face, the definition of a twist that I can see is that=
the=20
>>face being twisted has to be anywhere from 2 dimensional up to n-1 dimens=
ional.=20=20
>>This would allow some 2-d atomic twists in MC4D and MC5D which I've edite=
d and=20
>>posted in Photos > More possible twists. You'll note that my twists in 4=
D make=20
>>your famous Klaus parity errors quite simple--possibly too simple for you=
r=20
>>liking (see 2-4 below). Going with Roice's suggestion, I will continue o=
ur=20
>>conversation with the whole group.=20
>>
>>=20
>> (1) I installed Nate Berglund's program (http://people. math.gatech.=20
>>edu/~berglund/ Rubik/index. html), and yes indeed those are exactly the m=
issing=20
>>moves for 4D.
>>=20
>> (2) Great question: do these twists make for more possible states? We=
do=20
>>need the group's help here. For the 3^4 cube, Matthew Sheerin, or Klaus =
may be=20
>>able to help answer. I'm referring to messages #695, 772, 778, and Photo=
s >=20
>>"parity problems" by Klaus: http://groups. yahoo.com/ group/4D_ Cubing/ph=
otos/=20
>>album/565962423/ pic/list. Matt says he posted a solution to these, but I=
can't=20
>>find it. I notice Klaus's Oct. 13 parity error has the colors across fro=
m each=20
>>other. Therefore we still need to know if my rot_A1 twist is solvable us=
ing=20
>>current MC4D twists as well as Klaus's Nov. 14 parity error (same as my r=
ot_A2=20
>>twist...same as my rot_B1 twist in a sense). If these 2 are solvable, th=
en my=20
>>rot_B2 twist could be created using each of these 4 times + a rotation of=
the=20
>>entire tesseract and thus I would have introduced no new states. I now d=
oubt=20
>>this is the case, however--my guess is that my twists introduce new state=
s.=20
>>
>>=20
>> (3) Yeah, when you open up Berglund's program you can choose to allow o=
r=20
>>disallow my twists. He classifies them as two separate puzzles, which ma=
y be=20
>>the way to go. Another way to go, for example is 2 separate versions of =
MC4D:=20=20
>>the current version with only 3D twists allowed vs. a version where both =
2D and=20
>>3D twists allowed.
>>=20
>> (4) I was preparing a statement like this...only much worse. I was pre=
pared=20
>>for something analogous to Christopher Columbus being laughed off the fla=
t face=20
>>of the planet for thinking it's round. Of course I was hoping you'd phra=
se it=20
>>as nicely as you did. Before ever searching and finding your programs on=
the=20
>>internet, I had visualized a 3^4 tesseract in an X, Y, Z, T coordinate sy=
stem as=20
>>described in my email at the bottom with the center of the tesseract bein=
g at=20
>>the origin and having the cubie edge length =3D 1. I visualized the sepe=
rators t=20
>>=3D -1/2 and t =3D 1/2 dividing it into 3 "cubes." I simply figured you =
could twist=20
>>just the top (z > 1/2) of the t < -1/2 "cube." Doing so results in no z =
nor t=20
>>coordinate change for any "4D atom" of the entire tesseract so no sticker=
s nor=20
>>cubies will cross these seperators during the twist and nothing runs into=
=20
>>eachother. This visualization method made it difficult to visualize how =
to=20
>>twist z > 1/2 and x > 1/2. I wasn't sure it was possible, but I realized=
one=20
>>can always rotate the entire tesseract so x ----> -t. That way it's the =
same as=20
>>the previous twist, so I knew it was possible. I went back and tried=20
>>visualizing this twist without the rotation, and although it would take a=
while=20
>>to describe, I can tell you, it's neat when you do.=20
>>
>>=20
>> (5) This I find VERY intriguing. After educating myself with the "Four=
=20
>>Dimensions" section of http://en.wikipedia.org/wiki/Rotation_%28mathemati=
cs%29,=20
>>it seems a nonsimple rotation is just using multiple rotational planes at=
once.=20=20
>>So I'd say the following example (A) is still a simple rotation: in MC4D=
, if=20
>>you click on a corner or edge sticker of a face...it's still twisting tha=
t face=20
>>over a 2D plane which is spanned by a line going through that sticker and=
the=20
>>opposite sticker on the face as well as the axis that that face represent=
s (Y=20
>>axis if it's the +Y face). The following example (B) is nonsimple: 2=20
>>completely independent 2D rotations at the same time (the rotational plan=
es are=20
>>orthogonal). Sure enough, someone made a pic: http://en.wikipedia .org/w=
iki/=20
>>File:Tesseract. gif. In MC4D, this is the equivalent of Ctrl + clicking o=
n, say,=20
>>the top face (repeatedly- -so that 4 faces keep moving along the vertical=
axis)=20
>>while spinning the entire tesseract about that axis (so the other 4 faces=
go in=20
>>a circle around that vertical axis). This is not a twist, still a rotati=
on of=20
>>the entire tesseract (surely nonsimple).=20
>>
>>=20
>> I Googled the phrase "non simple rotation"...and the "All rotations of t=
he=20
>>4-cube" section of http://gregegan. customer. netspace. net.au/APPLETS/=20
>>29/HypercubeNote s.html has some interesting pics with captions in--diffi=
cult to=20
>>grasp, though. At first my question back was...can the planes of rotatio=
n be=20
>>non-orthogonal? Then I remembered taking dynamics classes where spinning=
a top=20
>>on a flat surface creates non-orthogonal rotational planes--there, the ti=
lted=20
>>rotational plane follows the rules of the horizontal rotational plane...b=
ut not=20
>>the other way around. Maybe my question back is: are there rotations th=
at=20
>>cannot be described using combinations of rotational planes? At any rate=
, I'd=20
>>say it would be a true show for the mind of any of these were implemented=
into=20
>>one of the programs that displayed the animation.
>>=20
>> (6) Yes, I was originally imagining 3 combo boxes, but I could see how =
2=20
>>columns and 3 rows...or (in N dimensions), 2 columns and (N - 2) rows wou=
ld be=20
>>less cumbersome. As I click +Y in the up-left drop down, I'm imagining Y=
=20
>>disappearing from the options in the boxes below (X, Z, U, V would remain=
) as=20
>>well as Y buttons graying out as described. I'm imagining the right boxe=
s=20
>>having lots of options...not just Y < -1/2, Y > 1/2, -1/2 < Y < 1/2...but=
also=20
>>the combos Y < 1/2, Y > -1/2, Y < -1/2 AND Y > 1/2. This would make 2 co=
lumns=20
>>and 3 rows even less cumbersome.. .relatively. ..especially for 4^5, 3^7 =
etc.=20=20
>>In MC4D I told Melinda I was imagining Alt + click for these twists (comp=
atible=20
>>with Alt + # + click). I too don't have great time to check out/edit the=
=20
>>program codes, but besides that, I only know basics for each of html, C, =
Matlab,=20
>>and TI-calculator code. I'll leave the major programming to the programm=
ers=20
>>while providing user and geometrical feedback.=20
>>
>>=20
>> --
>> Andrew Gould
>> Masters in Math, UW-Milwaukee
>> PhD student, UW-Milwaukee
>>=20
>> p.s. call me Andy
>>=20
>>=20
>>=20
>>=20
>> ----- Original Message -----
>
>> From: "Roice Nelson"
>> To: "Andrew James Gould"
>> Cc: foodiddy@... , "Melinda Green"
>> Sent: Friday, July 2, 2010 7:08:59 PM
>> Subject: Re: rotations missing - 5D cube
>>=20
>> Hi Andrew,
>>=20
>> Thanks for the email. Nice to learn something new about these
>> hyperpuzzles after playing with them for 10 years :) Here are my
>> thoughts:
>>=20
>> (1) Many many moons ago, I saw another MC4D implementation by Nate
>> Berglund which provided moves that may end up being exactly like you've
>> described. I didn't study them much at the time, and didn't go back and
>> install his software to verify now, but you'd probably be interested to
>> check it out. http://people. math.gatech. edu/~berglund/ Rubik/index. ht=
ml
>>=20
>> (2) I am curious if the new rotation possibilities are indeed "atomic"
>> or not. By that I mean that puzzle states using the current twists could
>> be created from the new ones, but not visa versa. Since the 4D cube
>> example you provided represented a puzzle state which can be achieved
>> with the currently supported moves, we know that particular move is not
>> any "more atomic" in this sense. I very much encourage you to forward
>> your email to the cubing group at large, perhaps with this question
>> posed. There are members of the group that understand all the parity
>> restrictions given the current move set, and they could do an analysis
>> to see if these new move types lead to new puzzle states (I did not copy
>> the group on my reply here, but feel free to do so if you reply to
>> this). If the moves are in fact more atomic, I could see this generating
>> active discussion since all of the calculations for the number of
>> permutations in the various puzzles would not apply to extended puzzles.
>>=20
>> (3) These new rotation types would make the puzzles easier to solve,
>> especially if they are not "more atomic" and the size of the state space
>> hasn't changed. This is just an observation, and not an argument against
>> them. Still, as an example of the fallout of extending the twist types,
>> there is an active history of shortest solution competitions which would
>> be affected. Solutions on extended puzzles would need to fall into a
>> different category in those competitions, due to the changes in the
>> nature of solving the puzzles.
>>=20
>> (4) An elegance of the current behavior is that a twist moves all
>> stickers on the twisted face in unison. When I first read your email, I
>> attempted to formulate a mechanical argument against it for this reason
>> (something like "well, if you could build a physical MC4D, such twists
>> would result in colliding stickers.") While it looks like your idea does
>> not result in any such difficulties, I do still feel there is a tradeoff
>> in elegance here - you'd both gain and lose by making the change.
>>=20
>> (5) You mentioned "after all, any rotation in N dimensions is rotating 2
>> dimensions about an "N-2"-dimensional object". For completeness, I
>> thought I'd mention that in 4D and above, there are rotations which
>> rotate more than 2 dimensions, the rotations you are referring to being
>> called " simple rotations ". Since twists of faces in MC5D are 4D
>> rotations, I've had the desire over the years to find a nice way to
>> support twists in this puzzle that are not simple rotations. It hasn't
>> happened yet. (This is still to be distinguished from your newly
>> suggested twists, since the rotations I imagined still moved all
>> stickers of the twisted face in unison).
>>=20
>>=20
>> (6) I like the direction of your UI suggestion, but are you imagining
>> that both the restricted axis and the slice (e.g. U, and -1/2 < axis <
>> 1/2) get specified in one combo box? When you first click the +Y face,
>> it is not clear yet that the other two axes that will be involved are U
>> and V, since it could be X or Z as well. And specifying the various
>> restricted axes and slices will need to work on larger puzzles like the
>> 5^7, so a design with only 2 additional combo boxes would get awfully
>> cumbersome as far as the number of items in the list. There could be 6
>> combo boxes total though (5 new ones), in 2 columns and 3 rows. The left
>> column would select the axes to restrict to (with the top combo doubling
>> as selecting the face to twist). The right column would select the
>> slices. Things would gray out as you described. Anyway, whatever is
>> deemed a good specification, I don't think it would be terribly
>> difficult to implement. However, I'm not able to work on MC5D at this
>> time, and not sure when I will be able to next. The source code for both
>> MC4D and MC5D are available online to experiment with though.
>>=20
>> Thanks again, and I hope you choose to continue this discussion on the
>> mailing list.
>>=20
>> Cheers,
>> Roice
>>=20
>>=20
>
> On Wed, Jun 30, 2010 at 7:59 PM, Andrew James Gould < agould@... >
>
>> wrote:
>>=20
>>=20
>> Hello,
>>=20
>> I had a similar email conversation with Melinda Green who eventually
>> gave in. All of your rotations, I would deem "legal," however, her 4D
>> Magic cube and your 5D Magic cubes are missing possible "atomic
>> rotations."
>>=20
>> Terminology: When I open your program, I can click on the top of the
>> blue (+Y) face and move that sticker to the back-right of that face
>> toward the green face. This is the same as making the "Face to Twist"
>> drop-down menu say +Y and clicking on the X side the "X-Z" button. My
>> terminology for this rotation would be to restrict Y to the range 1/2 <
>> Y < 3/2 and rotate the Y face via (X side of X-Z) about the Y-U-V
>> hyperplane (the hyperplane is all variables except X and Z--after all,
>> any rotation in N dimensions is rotating 2 dimensions about an
>> "N-2"-dimensional object). If I hold the '2' key down while doing this
>> rotation, it restricts Y to the range -1/2 < Y < 1/2, holding '1' AND
>> '2' during this rotations restricts Y to -1/2 < Y < 3/2, and holding,
>> '1' and '3' during this rotations restricts Y to -3/2 < Y < -1/2 union
>> 1/2 < Y < 3/2. Note: we only restricted on Y.
>>=20
>> Rotations: It seems both Melinda Green's MC4D program and your "atomic
>> twists" only restrict one variable at a time in this manor, but for a
>> rotation in N dimensions (N > 1), I find that one can restrict UP TO all
>> of the N-2 dimensions of the hyperplane being rotated about in similar
>> manors and independently (just not restricting the 2 dimensions of the
>> rotation). For example, I can restrict further on my previous "holding
>> down the '2' key" rotation: if I restrict both variables Y and V to
>> being between -1/2 and 1/2, and rotate the +Y face via (X side of X-Z
>> holding '2'), I would get the attached picture 5D_2b (paint-program
>> edited) where 9 purple and 9 white stickers also rotated (8 of these
>> purples and 8 of these whites moved). If I restrict all three variables,
>> Y, U, and V, to being between -1/2 and 1/2 and rotate the +Y face via (X
>> side of X-Z holding '2'), I would get 5D_2bii where only 12 total
>> stickers (3 from each: +Z, +X, -Z, -X semi-obscured) even moved--nothing
>> else would even rotate (except possibly the 0-colored interal piece). I
>> also attached a similar rotation in MC4D: rot_B2. These additional
>> restricting choices are unseen in 2D and 3D because rotations there are
>> about 0-dimensional points and 1-dimensional axes respectively where
>> there are 0 variables and 1 variable to restrict on
>> (again...respective ly).=20
>>=20
>> Melinda says the rot_B2 rotation is possible in
>> MC4D as is, with macros, which may be the case in your program, but I'm
>> wondering if these additional restrictions would be possible to
>> implement into your program as "atomic twists", and if so, how difficult
>> would that be? I'm imagining them being additional drop-down menus below
>> the "Face to Twist" drop-down menu, but above the twist buttons. I'm
>> imagining the following for my triple-restricted example: all the
>> Y-buttons being greyed out as one clicks +Y for Face to Twist, NO
>> buttons being greyed out as one restricts Y to -1/2 < Y < 1/2, all the
>> V-buttons being greyed out as one restricts to -1/2 < V < 1/2, and all
>> the U-buttons being greyed out as one restricts to -1/2 < U < 1/2. After
>> those 3 restrictions, one only has the X-Z button left to click on
>> (number keys at this point would either change only the Y restriction or
>> give an error sound and not change any restriction) .
>>=20
>> Stopping at the double-restriction (after restricting -1/2 < V < 1/2,
>> but before U) would leave 3 buttons to click on: X-Z, X-U, and Z-U.
>> Clicking the X side of X-Z here gets us to 5D_2b. This is also the
>> intersection of your 2 rotations: rotating the +Y face via (X side of
>> X-Z button holding '2') and rotating the +V face via (X side of X-Z
>> button holding '2'). You probably know that rotating in a positive range
>> always adds more stickers from another face. The same
>> double-restriction, but V being restricted to 1/2 < V < 3/2 would rotate
>> 27 stickers in the -1/2 < Y < 1/2 slice of the +V face. I could go on
>> with possibilities.
>>=20
>> -- Andrew Gould
>> Masters in Math, UW-Milwaukee
>> PhD student, UW-Milwaukee
>>
>
>

=20


=20=20=20=20=20=20
--0-1530771421-1279448205=:52288
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable

ad>

Hello,
Actually Klaus, I don't see why they wouldn't be possible on a 4D cube i=
n a 4D space. That's why I got into the conversations with Melinda and Roi=
ce. Any specific issues you see? In 3D, n =3D 3 so there's only one choic=
e when twisting anywhere from a 2 up to an n-1 dimensional section, but in =
upper dimensions, 2 < n-1 so there's a choice. During these twists, no sti=
ckers nor cubies run into eachother nor do they overlap (I can provide deta=
il here if that appears to be the issue).=20=20

I wouldn't want to mess up your leaderboards. I personally think your l=
eaderboards are a more distinguishing list since only allowing n-1 dimensio=
nal twists is more restrictive. If the new twists were added, I think the =
way to go is separate leaderboards: only allowing n-1 dimensional twists v=
s. 2 up to n-1 dimensional twists.=20=20

Matt, very neat that A1 is not possible, but that B1, A2, and B2 are. I=
didn't imagine that being a possible set of outcomes, but it makes clear s=
ense to me when you described an even number of A1 twists would be possible=
. (I'll get caught up with the lingo--i.e. 4C pieces.) Sorry about the ch=
ange in notation, it changed when I took the conversation from Melinda to R=
oice. In the 5D pics, the notation went: each new character (2, b, and ii=
) meant another restriction to -1/2 < variable < 1/2. I don't know how the=
pic order got messed up, I clicked the MC4D pics first.=20=20

Anyway, I uploaded the 5D pics again but without the faces that aren't of=
mixed color after the twist--less obscuring and with labels: 5D_2D_2bii a=
nd 5D_3D_2b.=20=20

--
Andy

----- Original Message -----
From: "Klaus Weidinger"
To: "4D Cubing" <4D_Cubing@yahoogroups.com>
Sent: Sunday, July 18, 2010 5:16:45 AM
Subject: Re: [MC4D] Re: definition of a twist

It is an interesting idea to allow lower-dimensional twists, but I don't
consider this feature necessary enough
to be implemented. First of all Matthew showed, that some of these
twists are possible to be obtained by
short algs, and more important, these twists would not be possible on a
real 4D cube in 4D space.

Happy Hypercubing,
Klaus




From: Jenelle Levenstein
To: 4D_Cubing@yahoogroups.com
Sent: Sun, July 18, 2010 6:19:53 AM
Subject: Re: [MC4D] Re: definition of a twist




I haven't posted on this list that often but I find the discussion about
redefining what a twist is in a N dimensional puzzle interesting. It
never accured to me that you could define moves in any way other than
the way it was done in the MC4D. The new definition of moves would
definitely make the puzzle easier but that's not necessarily a reason to
rule out the possibility. I think that implementing a puzzle that
allowed two dimensional twists would make the puzzle accessable to
people it wouldn't already be accessible to. This would be good if you
are trying to increase the visibility of the program but may be bad for
people who want all the individuals who have solved the puzzle to form
an elite group. The new moves could be implemented in a very similar way
to the way moves are implemented in the current puzzle but when you
click on a 2C piece instead of the entire cube rotating around that
access only a 2D face would rotate around that access. If you wanted to
allow both 2D rotations and 3D rotations then you would need to use
another control character.

I don't have the patients to try to solve the puzzle linked to earlier
in this post due to the bad graphics, and only allowing moves on the
center face, but allowing a person to rotate 2D faces would dramatically
change how the puzzle is solved. Allowing these moves will allow
individuals to use some 3D logic in order to solve the puzzle because
once you get all the pieces of one color on a cube you can move them
around on that cube without messing up any pieces on the entire rest of
the cube. You would still need to take the fourth dimension into account
when locating pieces but there would be fewer dependencies to worry
about when placing the piece. There are ways to move pieces around
independently in the current puzzle obviously, but they are more
involved and often require long sequences of moves that can be difficult
to keep track of.


On Sat, Jul 17, 2010 at 6:20 PM, matthewsheerin < damienturtle@
hotmail.co. uk > wrote:








Hi Andy,
time for some feedback on those twists :). In order of pics:

A1: Not possible currently. It gives odd permutation of 4C pieces.
Combining any two of these produces a valid state though (assuming 90
degree twists).

B1: Certainly possible. One of the 'parity' cases Klaus presented. Not
sure I would class it as parity though. 5 move solution which I'm sure I
uploaded somewhere around here ...

bii: First, what's with the change of notation? Second, can't see what's
happening there, I need a better pic.

A2: Seems to be the same as B1.

b: This order is confusing! So is the picture again. Try showing only
the necessary faces, otherwise the screen becomes too cluttered in 5D.

B2: This isn't immediately obvious ... got it. Possible in 6 twists.

That seems to be the lot of them. Personally, I prefer the current twist
system, it seems to be the most natural. Also, I might upload log files
of these to the folder the pics are in, or at least the possible ones
(and maybe re-upload the Klaus cases). However, nice to hear about a
different approach to these puzzles.

Matt


--- In 4D_Cubing@yahoogrou ps.com , Andrew James Gould
wrote:
>
> This is my first email to the group so hello group,
>
> I read that I don't need to apologize for length...fewf. I've been
> having conversations with both Melinda and Roice on what appears to be
> (from my perspective) all 3 programs, MC4D, MC5D, and MC7D, using too
> strict of a definition of a "twist." More specifically, the twists in
> these programs twist an n-1 dimensional face, the definition of a
> twist that I can see is that the face being twisted has to be anywhere
> from 2 dimensional up to n-1 dimensional. This would allow some 2-d
> atomic twists in MC4D and MC5D which I've edited and posted in Photos
> > More possible twists. You'll note that my twists in 4D make your
> famous Klaus parity errors quite simple--possibly too simple for your
> liking (see 2-4 below). Going with Roice's suggestion, I will continue
> our conversation with the whole group.
>
> (1) I installed Nate Berglund's program ( http://people. math.gatech.
> edu/~berglund/ Rubik/index. html ), and yes indeed those are exactly
> the missing moves for 4D.
>
> (2) Great question: do these twists make for more possible states? We
> do need the group's help here. For the 3^4 cube, Matthew Sheerin, or
> Klaus may be able to help answer. I'm referring to messages #695, 772,
> 778, and Photos > "parity problems" by Klaus: http://groups.
> yahoo.com/ group/4D_ Cubing/photos/ album/565962423/ pic/list. Matt
> says he posted a solution to these, but I can't find it. I notice
> Klaus's Oct. 13 parity error has the colors across from each other.
> Therefore we still need to know if my rot_A1 twist is solvable using
> current MC4D twists as well as Klaus's Nov. 14 parity error (same as
> my rot_A2 twist...same as my rot_B1 twist in a sense). If these 2 are
> solvable, then my rot_B2 twist could be created using each of these 4
> times + a rotation of the entire tesseract and thus I would have
> introduced no new states. I now doubt this is the case, however--my
> guess is that my twists introduce new states.
>
> (3) Yeah, when you open up Berglund's program you can choose to allow
> or disallow my twists. He classifies them as two separate puzzles,
> which may be the way to go. Another way to go, for example is 2
> separate versions of MC4D: the current version with only 3D twists
> allowed vs. a version where both 2D and 3D twists allowed.
>
> (4) I was preparing a statement like this...only much worse. I was
> prepared for something analogous to Christopher Columbus being laughed
> off the flat face of the planet for thinking it's round. Of course I
> was hoping you'd phrase it as nicely as you did. Before ever searching
> and finding your programs on the internet, I had visualized a 3^4
> tesseract in an X, Y, Z, T coordinate system as described in my email
> at the bottom with the center of the tesseract being at the origin and
> having the cubie edge length =3D 1. I visualized the seperators t =3D -1/=
2
> and t =3D 1/2 dividing it into 3 "cubes." I simply figured you could
> twist just the top (z > 1/2) of the t < -1/2 "cube." Doing so results
> in no z nor t coordinate change for any "4D atom" of the entire
> tesseract so no stickers nor cubies will cross these seperators during
> the twist and nothing runs into eachother. This visualization method
> made it difficult to visualize how to twist z > 1/2 and x > 1/2. I
> wasn't sure it was possible, but I realized one can always rotate the
> entire tesseract so x ----> -t. That way it's the same as the previous
> twist, so I knew it was possible. I went back and tried visualizing
> this twist without the rotation, and although it would take a while to
> describe, I can tell you, it's neat when you do.
>
> (5) This I find VERY intriguing. After educating myself with the "Four
> Dimensions" section of
> http://en.wikipedia.org/wiki/Rotation_%28mathematics%29 , it seems a
> nonsimple rotation is just using multiple rotational planes at once.
> So I'd say the following example (A) is still a simple rotation: in
> MC4D, if you click on a corner or edge sticker of a face...it's still
> twisting that face over a 2D plane which is spanned by a line going
> through that sticker and the opposite sticker on the face as well as
> the axis that that face represents (Y axis if it's the +Y face). The
> following example (B) is nonsimple: 2 completely independent 2D
> rotations at the same time (the rotational planes are orthogonal).
> Sure enough, someone made a pic: http://en.wikipedia .org/wiki/
> File:Tesseract. gif. In MC4D, this is the equivalent of Ctrl +
> clicking on, say, the top face (repeatedly- -so that 4 faces keep
> moving along the vertical axis) while spinning the entire tesseract
> about that axis (so the other 4 faces go in a circle around that
> vertical axis). This is not a twist, still a rotation of the entire
> tesseract (surely nonsimple).
>
> I Googled the phrase "non simple rotation"...and the "All rotations of
> the 4-cube" section of http://gregegan. customer. netspace.
> net.au/APPLETS/ 29/HypercubeNote s.html has some interesting pics with
> captions in--difficult to grasp, though. At first my question back
> was...can the planes of rotation be non-orthogonal? Then I remembered
> taking dynamics classes where spinning a top on a flat surface creates
> non-orthogonal rotational planes--there, the tilted rotational plane
> follows the rules of the horizontal rotational plane...but not the
> other way around. Maybe my question back is: are there rotations that
> cannot be described using combinations of rotational planes? At any
> rate, I'd say it would be a true show for the mind of any of these
> were implemented into one of the programs that displayed the
> animation.
>
> (6) Yes, I was originally imagining 3 combo boxes, but I could see how
> 2 columns and 3 rows...or (in N dimensions), 2 columns and (N - 2)
> rows would be less cumbersome. As I click +Y in the up-left drop down,
> I'm imagining Y disappearing from the options in the boxes below (X,
> Z, U, V would remain) as well as Y buttons graying out as described.
> I'm imagining the right boxes having lots of options...not just Y <
> -1/2, Y > 1/2, -1/2 < Y < 1/2...but also the combos Y < 1/2, Y > -1/2,
> Y < -1/2 AND Y > 1/2. This would make 2 columns and 3 rows even less
> cumbersome.. .relatively. ..especially for 4^5, 3^7 etc. In MC4D I
> told Melinda I was imagining Alt + click for these twists (compatible
> with Alt + # + click). I too don't have great time to check out/edit
> the program codes, but besides that, I only know basics for each of
> html, C, Matlab, and TI-calculator code. I'll leave the major
> programming to the programmers while providing user and geometrical
> feedback.
>
> -- Andrew Gould
> Masters in Math, UW-Milwaukee
> PhD student, UW-Milwaukee
>
> p.s. call me Andy
>
>
>
>
> ----- Original Message -----



> From: "Roice Nelson"
> To: "Andrew James Gould"
> Cc: foodiddy@... , "Melinda Green"
> Sent: Friday, July 2, 2010 7:08:59 PM
> Subject: Re: rotations missing - 5D cube
>
> Hi Andrew,
>
> Thanks for the email. Nice to learn something new about these
> hyperpuzzles after playing with them for 10 years :) Here are my
> thoughts:
>
> (1) Many many moons ago, I saw another MC4D implementation by Nate
> Berglund which provided moves that may end up being exactly like
> you've described. I didn't study them much at the time, and didn't go
> back and
> install his software to verify now, but you'd probably be interested
> to check it out. http://people. math.gatech. edu/~berglund/
> Rubik/index. html
>
> (2) I am curious if the new rotation possibilities are indeed "atomic"
> or not. By that I mean that puzzle states using the current twists
> could be created from the new ones, but not visa versa. Since the 4D
> cube example you provided represented a puzzle state which can be
> achieved with the currently supported moves, we know that particular
> move is not
> any "more atomic" in this sense. I very much encourage you to forward
> your email to the cubing group at large, perhaps with this question
> posed. There are members of the group that understand all the parity
> restrictions given the current move set, and they could do an analysis
> to see if these new move types lead to new puzzle states (I did not
> copy the group on my reply here, but feel free to do so if you reply
> to this). If the moves are in fact more atomic, I could see this
> generating active discussion since all of the calculations for the
> number of
> permutations in the various puzzles would not apply to extended
> puzzles.
>
> (3) These new rotation types would make the puzzles easier to solve,
> especially if they are not "more atomic" and the size of the state
> space hasn't changed. This is just an observation, and not an argument
> against them. Still, as an example of the fallout of extending the
> twist types,
> there is an active history of shortest solution competitions which
> would be affected. Solutions on extended puzzles would need to fall
> into a
> different category in those competitions, due to the changes in the
> nature of solving the puzzles.
>
> (4) An elegance of the current behavior is that a twist moves all
> stickers on the twisted face in unison. When I first read your email,
> I attempted to formulate a mechanical argument against it for this
> reason (something like "well, if you could build a physical MC4D, such
> twists would result in colliding stickers.") While it looks like your
> idea does
> not result in any such difficulties, I do still feel there is a
> tradeoff in elegance here - you'd both gain and lose by making the
> change.
>
> (5) You mentioned "after all, any rotation in N dimensions is rotating
> 2 dimensions about an "N-2"-dimensional object". For completeness, I
> thought I'd mention that in 4D and above, there are rotations which
> rotate more than 2 dimensions, the rotations you are referring to
> being called " simple rotations ". Since twists of faces in MC5D are
> 4D rotations, I've had the desire over the years to find a nice way to
> support twists in this puzzle that are not simple rotations. It hasn't
> happened yet. (This is still to be distinguished from your newly
> suggested twists, since the rotations I imagined still moved all
> stickers of the twisted face in unison).
>
>
> (6) I like the direction of your UI suggestion, but are you imagining
> that both the restricted axis and the slice (e.g. U, and -1/2 < axis <
> 1/2) get specified in one combo box? When you first click the +Y face,
> it is not clear yet that the other two axes that will be involved are
> U and V, since it could be X or Z as well. And specifying the various
> restricted axes and slices will need to work on larger puzzles like
> the 5^7, so a design with only 2 additional combo boxes would get
> awfully cumbersome as far as the number of items in the list. There
> could be 6
> combo boxes total though (5 new ones), in 2 columns and 3 rows. The
> left column would select the axes to restrict to (with the top combo
> doubling as selecting the face to twist). The right column would
> select the
> slices. Things would gray out as you described. Anyway, whatever is
> deemed a good specification, I don't think it would be terribly
> difficult to implement. However, I'm not able to work on MC5D at this
> time, and not sure when I will be able to next. The source code for
> both MC4D and MC5D are available online to experiment with though.
>
> Thanks again, and I hope you choose to continue this discussion on the
> mailing list.
>
> Cheers,
> Roice
>
>
> On Wed, Jun 30, 2010 at 7:59 PM, Andrew James Gould < agould@... >



> wrote:
>
>
> Hello,
>
> I had a similar email conversation with Melinda Green who eventually
> gave in. All of your rotations, I would deem "legal," however, her 4D
> Magic cube and your 5D Magic cubes are missing possible "atomic
> rotations."
>
> Terminology: When I open your program, I can click on the top of the
> blue (+Y) face and move that sticker to the back-right of that face
> toward the green face. This is the same as making the "Face to Twist"
> drop-down menu say +Y and clicking on the X side the "X-Z" button. My
> terminology for this rotation would be to restrict Y to the range 1/2
> < Y < 3/2 and rotate the Y face via (X side of X-Z) about the Y-U-V
> hyperplane (the hyperplane is all variables except X and Z--after all,
> any rotation in N dimensions is rotating 2 dimensions about an
> "N-2"-dimensional object). If I hold the '2' key down while doing this
> rotation, it restricts Y to the range -1/2 < Y < 1/2, holding '1' AND
> '2' during this rotations restricts Y to -1/2 < Y < 3/2, and holding,
> '1' and '3' during this rotations restricts Y to -3/2 < Y < -1/2 union
> 1/2 < Y < 3/2. Note: we only restricted on Y.
>
> Rotations: It seems both Melinda Green's MC4D program and your "atomic
> twists" only restrict one variable at a time in this manor, but for a
> rotation in N dimensions (N > 1), I find that one can restrict UP TO
> all of the N-2 dimensions of the hyperplane being rotated about in
> similar manors and independently (just not restricting the 2
> dimensions of the
> rotation). For example, I can restrict further on my previous "holding
> down the '2' key" rotation: if I restrict both variables Y and V to
> being between -1/2 and 1/2, and rotate the +Y face via (X side of X-Z
> holding '2'), I would get the attached picture 5D_2b (paint-program
> edited) where 9 purple and 9 white stickers also rotated (8 of these
> purples and 8 of these whites moved). If I restrict all three
> variables, Y, U, and V, to being between -1/2 and 1/2 and rotate the
> +Y face via (X
> side of X-Z holding '2'), I would get 5D_2bii where only 12 total
> stickers (3 from each: +Z, +X, -Z, -X semi-obscured) even
> moved--nothing else would even rotate (except possibly the 0-colored
> interal piece). I
> also attached a similar rotation in MC4D: rot_B2. These additional
> restricting choices are unseen in 2D and 3D because rotations there
> are about 0-dimensional points and 1-dimensional axes respectively
> where there are 0 variables and 1 variable to restrict on
> (again...respective ly).
>
> Melinda says the rot_B2 rotation is possible in
> MC4D as is, with macros, which may be the case in your program, but
> I'm wondering if these additional restrictions would be possible to
> implement into your program as "atomic twists", and if so, how
> difficult would that be? I'm imagining them being additional drop-down
> menus below
> the "Face to Twist" drop-down menu, but above the twist buttons. I'm
> imagining the following for my triple-restricted example: all the
> Y-buttons being greyed out as one clicks +Y for Face to Twist, NO
> buttons being greyed out as one restricts Y to -1/2 < Y < 1/2, all the
> V-buttons being greyed out as one restricts to -1/2 < V < 1/2, and all
> the U-buttons being greyed out as one restricts to -1/2 < U < 1/2.
> After those 3 restrictions, one only has the X-Z button left to click
> on (number keys at this point would either change only the Y
> restriction or
> give an error sound and not change any restriction) .
>
> Stopping at the double-restriction (after restricting -1/2 < V < 1/2,
> but before U) would leave 3 buttons to click on: X-Z, X-U, and Z-U.
> Clicking the X side of X-Z here gets us to 5D_2b. This is also the
> intersection of your 2 rotations: rotating the +Y face via (X side of
> X-Z button holding '2') and rotating the +V face via (X side of X-Z
> button holding '2'). You probably know that rotating in a positive
> range always adds more stickers from another face. The same
> double-restriction, but V being restricted to 1/2 < V < 3/2 would
> rotate 27 stickers in the -1/2 < Y < 1/2 slice of the +V face. I could
> go on
> with possibilities.
>
> -- Andrew Gould
> Masters in Math, UW-Milwaukee
> PhD student, UW-Milwaukee
>

I think I see how these twists are sort of possible, since nothing overlaps=
during the rotations. I'm not sure whether a person in 4D with a physical=
n^4 would be able to perform these twists though. I think it would take s=
omeone who knows more theory about 4D then I do to answer that one. I also=
think it could be interesting to implement, especially since it might gave=
beginners an easier stepping stone into solving 4D puzzles.
To go into more detail about the A1 twists (I was waiting for clearer scree=
nshots of the 5D cases, but more on those later), I'll give a quick overvie=
w of parity restrictions. I'm sure these are in previous posts somewhere a=
nyway, but just to mention them while they are relevant again. Permutation=
of 4C pieces must be even (a 90 degree face twist performs 2 4-cycles, and=
edge and corner twists can be constructed from these so parity must be the=
same for those too). Permutation of 2C and 3C pieces is linked, just like=
in 3D. Either both permutations are even, or both are odd. This is true =
(I'm sure) in all dimensions. 2C and 3C parities are linked, and pieces wi=
th more colours have even parity.
I've looked at the new 5D cases. Both are possible in the current interfac=
e, and I've already uploaded the log files. The first one I constructed th=
e same way as B2. The second one built on the first one, and the setup mov=
es for the commutators to make the position were a little tricky and took a=
little while to sort out, though I could tell it was possible.
While on the discussion of twists, I feel that it's worth mentioning the ol=
d problem in MC4D. A face can be moved to most its possible positions in o=
ne move, but the three positions reached by two 90 degree twists require tw=
o moves. I'm sure it has been discussed before but a quick look didn't fin=
d it.

Matt

--- In 4D_Cubing@yahoogroups.com, Andrew James Gould wrote:
>
> Hello,
> Actually Klaus, I don't see why they wouldn't be possible on a 4D cube=
in a 4D space. That's why I got into the conversations with Melinda and R=
oice. Any specific issues you see? In 3D, n =3D 3 so there's only one cho=
ice when twisting anywhere from a 2 up to an n-1 dimensional section, but i=
n upper dimensions, 2 < n-1 so there's a choice. During these twists, no s=
tickers nor cubies run into eachother nor do they overlap (I can provide de=
tail here if that appears to be the issue).=20=20
>=20
> I wouldn't want to mess up your leaderboards. I personally think your=
leaderboards are a more distinguishing list since only allowing n-1 dimens=
ional twists is more restrictive. If the new twists were added, I think th=
e way to go is separate leaderboards: only allowing n-1 dimensional twists=
vs. 2 up to n-1 dimensional twists.=20=20
>=20
> Matt, very neat that A1 is not possible, but that B1, A2, and B2 are. =
I didn't imagine that being a possible set of outcomes, but it makes clear=
sense to me when you described an even number of A1 twists would be possib=
le. (I'll get caught up with the lingo--i.e. 4C pieces.) Sorry about the =
change in notation, it changed when I took the conversation from Melinda to=
Roice. In the 5D pics, the notation went: each new character (2, b, and =
ii) meant another restriction to -1/2 < variable < 1/2. I don't know how t=
he pic order got messed up, I clicked the MC4D pics first.=20=20
>=20
> Anyway, I uploaded the 5D pics again but without the faces that aren't =
of mixed color after the twist--less obscuring and with labels: 5D_2D_2bii=
and 5D_3D_2b.=20=20
>=20
> --
> Andy

Matthew,

I know it's been discussed too but I'll just give my current opinion
which is that I like the idea of defining a twist as any combination of
currently supported twists on a single face that can be reduced to a
single one with a single slice mask. So a twist with one slice mask
followed by the same twist with a different mask should only count as a
single twist and represented in the log file with their combined mask.
Likewise three 90 degree twists should only count as one -90 degree
twist. And of course any combination that leaves the puzzle unchanged
shouldn't count at all. I currently cancel pairs of twists that are the
inverse of each other, so long as they don't cross macro boundaries. I'd
love to support more such cases.

The one case that I'm not currently ready to accept are combinations of
face twists that can be represented by a single transformation but which
are not reachable by a single existing twist. I'm talking about moves
like double 90 degree twists. At first blush that seems reasonable to
represent as an atomic move but look at the Onehundredagonal Duoprism.
It's not at all clear to me that 37 single twists around the cylinder
should count as a single twist. maybe, maybe not.

Even though these sorts of changes will affect the twist counts of
previous solutions, there is the possibility of adding more twist
compression logic to MC4D or to a standalone program that will factor
out any redundancies so that comparisons will always be reasonably fair.
At the moment I don't have plans to do any of this work but I think it's
great to have these discussions in the hope that we can come up with the
prefect definitions to base future work on.

-Melinda

matthewsheerin wrote:
> [...] I feel that it's worth mentioning the old problem in MC4D. A face can be moved to most its possible positions in one move, but the three positions reached by two 90 degree twists require two moves. I'm sure it has been discussed before but a quick look didn't find it.

Just my first thought, but I'd be with Matt on counting 180 degree twists a=
s one twist--it's what I do on a 3D cube. The 37 twist example you mention=
ed, Melina, I'd also count as 1.=20=20

Matt,
I play enough video games to keep the visual part of my brain in shape to a=
nswer this. One way a 4D person could do the A1 twist is to put his right =
hand on the 3 brown stickers and the 3 opposing green stickers that will mo=
ve, and his left hand on the 24 brown stickers and the 24 opposing green st=
ickers that don't move. He can then twist. Anything you can do to a 3^3 M=
agic cube, a 4D person could do to the -t face of the 3^4 Magic Cube. That=
includes all x-y, x-z, and y-z twists, and he would never have to touch th=
e stickers on the -t face to do those twists.=20=20

Note: of those 54 green and brown stickers, he wouldn't just be touching t=
he surfaces that we see in MC4D, he'd be touching every molecule inside. (=
Here, I use the 'going from 2D to 3D' analogy to explain going from 3D to 4=
D.) If you're a stick figure in 2D, you can't get inside a solid 2D square=
sticker to cover the whole thing without a 3rd dimension. But on a 3^3 Ma=
gic cube in 3D with its 9 yellow stickers at z =3D 1.5, placing your hand o=
n top of the 9 yellow stickers is like placing your hand at z =3D 1.50001 a=
t the same x & y coordinates of the cube--touching all the yellow stickers =
on their +z side. Similarly in 3D, you can't get inside a solid 3D cube st=
icker to "cover" the whole thing in 3D. But on a 3^4 Magic Cube in 4D with=
its 27 blue-gray stickers located at t =3D -1.5 in an (x,y,z,t) coordinate=
system, he could "cover" all molecules of those stickers by placing his 4D=
hand at t =3D -1.50001 at the same x, y, & z coordinates of the tesseract.=
His hand is therefore touching all the blue-gray stickers everywhere on t=
heir -t side. Now THAT's what I call...an up-grayed.

--
Andy


----- Original Message -----
From: "Melinda Green"
To: "4D Cubing" <4D_Cubing@yahoogroups.com>
Sent: Monday, July 19, 2010 3:11:59 PM
Subject: Re: [MC4D] Re: definition of a twist

Matthew,

I know it's been discussed too but I'll just give my current opinion
which is that I like the idea of defining a twist as any combination of
currently supported twists on a single face that can be reduced to a
single one with a single slice mask. So a twist with one slice mask
followed by the same twist with a different mask should only count as a
single twist and represented in the log file with their combined mask.
Likewise three 90 degree twists should only count as one -90 degree
twist. And of course any combination that leaves the puzzle unchanged
shouldn't count at all. I currently cancel pairs of twists that are the
inverse of each other, so long as they don't cross macro boundaries. I'd
love to support more such cases.

The one case that I'm not currently ready to accept are combinations of
face twists that can be represented by a single transformation but which
are not reachable by a single existing twist. I'm talking about moves
like double 90 degree twists. At first blush that seems reasonable to
represent as an atomic move but look at the Onehundredagonal Duoprism.
It's not at all clear to me that 37 single twists around the cylinder
should count as a single twist. maybe, maybe not.

Even though these sorts of changes will affect the twist counts of
previous solutions, there is the possibility of adding more twist
compression logic to MC4D or to a standalone program that will factor
out any redundancies so that comparisons will always be reasonably fair.
At the moment I don't have plans to do any of this work but I think it's
great to have these discussions in the hope that we can come up with the
prefect definitions to base future work on.

-Melinda

matthewsheerin wrote:
> [...] I feel that it's worth mentioning the old problem in MC4D. A
> face can be moved to most its possible positions in one move, but the
> three positions reached by two 90 degree twists require two moves. I'm
> sure it has been discussed before but a quick look didn't find it.

If we consider two sides of 4d cube then we can rotate each of them around =
the axis that goes in direction of another side (click of two stickers of t=
he same 2C in MC4D). In both cases block 3x3x1x1 will be rotated around its=
axis but it will to that as a part of different 3^3 sides - so it can free=
ly twist relative to both of them! Looks like it's really possible to twist=
this 2D facet in any construction of 3^4. It's more difficult to imagine t=
he rotation of middle layer of 3^3, but probably it's possible as well.
But I think that without such twists puzzle it a little more interesting =
)))

Andrey

Melinda,
Situation with the sequence of twists of the same side with different mas=
ks is a little more difficult. It looks like we should check the rotation o=
f each layer after this sequence, count the number C of different rotations=
and say that actial number of twists is C-1. E.g. if we rotate two opposit=
e sides of 3^3 clockwise, it counts as two moves in any metrics that I know=
(save for 3-colored nonoriented 3^3 - in that puzzle it's one - and only p=
ossible - twist :) ).

Andrey


--- In 4D_Cubing@yahoogroups.com, Melinda Green wrote:
>
> Matthew,
>=20
> I know it's been discussed too but I'll just give my current opinion=20
> which is that I like the idea of defining a twist as any combination of=20
> currently supported twists on a single face that can be reduced to a=20
> single one with a single slice mask. So a twist with one slice mask=20
> followed by the same twist with a different mask should only count as a=20
> single twist and represented in the log file with their combined mask.=20
> Likewise three 90 degree twists should only count as one -90 degree=20
> twist. And of course any combination that leaves the puzzle unchanged=20
> shouldn't count at all. I currently cancel pairs of twists that are the=20
> inverse of each other, so long as they don't cross macro boundaries. I'd=
=20
> love to support more such cases.
>=20
> The one case that I'm not currently ready to accept are combinations of=20
> face twists that can be represented by a single transformation but which=
=20
> are not reachable by a single existing twist. I'm talking about moves=20
> like double 90 degree twists. At first blush that seems reasonable to=20
> represent as an atomic move but look at the Onehundredagonal Duoprism.=20
> It's not at all clear to me that 37 single twists around the cylinder=20
> should count as a single twist. maybe, maybe not.
>=20
> Even though these sorts of changes will affect the twist counts of=20
> previous solutions, there is the possibility of adding more twist=20
> compression logic to MC4D or to a standalone program that will factor=20
> out any redundancies so that comparisons will always be reasonably fair.=
=20
> At the moment I don't have plans to do any of this work but I think it's=
=20
> great to have these discussions in the hope that we can come up with the=
=20
> prefect definitions to base future work on.
>=20
> -Melinda
>=20
> matthewsheerin wrote:
> > [...] I feel that it's worth mentioning the old problem in MC4D. A fac=
e can be moved to most its possible positions in one move, but the three po=
sitions reached by two 90 degree twists require two moves. I'm sure it has=
been discussed before but a quick look didn't find it.
>

Interesting construction, which seems to show that these twists are indeed =
possible. Following on from this idea, I think I can clarify for the other=
twists too. The construction you gave describes (unless I have got mixed =
up somewhere) and A1 twist: essentially taking one 3x3x3 face and turning o=
ne face. It follows that if you can rotate one face clockwise and the oppo=
site face anticlockwise, you have the essence of turning a middle layer of =
the 3x3x3 face. In other words, if all outside layers can rotate by themse=
lves, the inner ones can too, which gives a B1 twist. Since a B2 twist can=
be constructed from two B1 twists and two normal face twists, and none of =
these seem to overlap, it stand to reason that they can somehow be performe=
d at the same time, which gives a B2 twist. I'm still not 100% convinced p=
ersonally, only because I have difficulty thinking of a 4D hand interacting=
with a physical 4D cube. A question I can't help but think of, is one of =
mechanism. If a 4D person made a 4D cube, which of these twists would be s=
tandard? Would these extra twists require a more sophisticated mechanism, =
or would they be inherently possible? I know that this is slightly off the=
point, not necessary to have answered, and I have no idea how to answer it=
anyway, but I decided to put the question out here to see what reaction it=
gets.

In response to Melinda: just to say that I would also consider the 37 twist=
s as 1. I would consider moving any face (and/or any layer(s) parallel to =
that face) from any valid position of that face, to any other valid positio=
n of that face. To clarify, if more than one layer is moved, then all the =
layers must move in the same way, ie. doing R L on a 3x3x3 would be 2 twist=
s since layers are going in different directions. Obviously this is an amb=
iguous issue as it is possible to make a valid argument for more than one s=
ystem, but that is the most natural definition for me.

Matt

--- In 4D_Cubing@yahoogroups.com, "Andrey" wrote:
>
> If we consider two sides of 4d cube then we can rotate each of them aroun=
d the axis that goes in direction of another side (click of two stickers of=
the same 2C in MC4D). In both cases block 3x3x1x1 will be rotated around i=
ts axis but it will to that as a part of different 3^3 sides - so it can fr=
eely twist relative to both of them! Looks like it's really possible to twi=
st this 2D facet in any construction of 3^4. It's more difficult to imagine=
the rotation of middle layer of 3^3, but probably it's possible as well.
> But I think that without such twists puzzle it a little more interestin=
g )))
>=20
> Andrey
>