# Message #1046

From: Andrew James Gould <agould@uwm.edu>

Subject: Re: [MC4D] Re: definition of a twist

Date: Sun, 18 Jul 2010 15:38:07 -0500

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 Roice. Any specific issues you see? In 3D, n = 3 so there’s only one choice 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 stickers nor cubies run into eachother nor do they overlap (I can provide detail here if that appears to be the issue).

I wouldn’t want to mess up your leaderboards. I personally think your leaderboards are a more distinguishing list since only allowing n-1 dimensional 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 vs. 2 up to n-1 dimensional twists.

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 possible. (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 the pic order got messed up, I clicked the MC4D pics first.

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.

–

Andy

—– Original Message —–

From: "Klaus Weidinger" <klaus.weidinger@yahoo.com>

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 <jenelle.levenstein@gmail.com>

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 <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/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" <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…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

>