Message #1036

From: matthewsheerin <damienturtle@hotmail.co.uk>
Subject: Re: definition of a twist
Date: Sat, 17 Jul 2010 23:20:07 -0000

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@yahoogroups.com, Andrew James Gould <agould@…> 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 = 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@…>
> To: "Andrew James Gould" <agould@…>
> Cc: foodiddy@…, "Melinda Green" <melinda@…>
> 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…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
>