Message #1014

From: Andrew James Gould <agould@uwm.edu>
Subject: Re: definition of a twist
Date: Fri, 16 Jul 2010 20:05:18 -0500

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 = 1. I visualized the seperators t = -1/2 and t = 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/HypercubeNotes.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" <roice@gravitation3d.com>
To: "Andrew James Gould" <agould@uwm.edu>
Cc: foodiddy@gmail.com, "Melinda Green" <melinda@superliminal.com>
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).

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