# Message #3701

From: Joel Karlsson <joelkarlsson97@gmail.com>

Subject: Re: [MC4D] Physical 4D puzzle V2

Date: Sun, 30 Apr 2017 22:51:41 +0200

I am no expert on group theory, so to better understand what twists are

legal I read through the part of Kamack and Keane’s *The Rubik Tesseract *about

orienting the corners. Since all even permutations are allowed the easiest

way to check if a twist is legal might be to:

- Check that the twist is an even permutation, that is: the same twist can

be done by performing an even number of piece swaps (2-cycles). - Check the periodicity of the twist. If A^k=I (A^k meaning performing the

twist k times and I (the identity) representing the permutation of doing

nothing) and k is not divisible by 3 the twist A definitely doesn’t violate

the restriction of the orientations since kx mod 3 = 0 and k mod 3 != 0

implies x mod 3 = 0 meaning that the change of the total orientation x for

the twist A mod 3 is 0 (which precisely is the restriction of legal twists;

that they must preserve the orientation mod 3).

For instance, this implies that the restacking moves are legal 2x2x2x2

moves since both are composed of 8 2-cycles and both can be performed twice

(note that 2 is not divisible by 3) to obtain the identity.

Note that 1 and 2 are sufficient to check if a twist is legal but only 1 is

necessary; there can indeed exist a twist violating 2 that still is legal

and in that case, I believe that we might have to study the orientation

changes for that specific twist in more detail. However, if a twist can be

composed by other legal twists it is, of course, legal as well.

Best regards,

Joel

2017-04-29 1:04 GMT+02:00 Melinda Green melinda@superliminal.com

[4D_Cubing] <4D_Cubing@yahoogroups.com>:

>

>

> First off, thanks everyone for the helpful and encouraging feedback!

> Thanks Joel for showing us that there are 6 orbits in the 2^4 and for your

> rederivation of the state count. And thanks Matt and Roice for pointing out

> the importance of the inverted views. It looks so strange in that

> configuration that I always want to get back to a normal view as quickly as

> possible, but it does seem equally valid, and as you’ve shown, it can be

> helpful for more than just finding short sequences.

>

> I don’t understand Matt’s "pinwheel" configuration, but I will point out

> that all that is needed to create your twin interior octahedra is a single

> half-rotation like I showed in the video at 5:29

> <https://www.youtube.com/watch?v=zqftZ8kJKLo&t=5m29s>. The two main

> halves do end up being mirror images of each other on the visible outside

> like he described. Whether it’s the pinwheel or the half-rotated version

> that’s correct, I’m not sure that it’s a bummer that the solved state is

> not at all obvious, so long as we can operate it in my original

> configuration and ignore the fact that the outer faces touch. That would

> just mean that the "correct" view is evidence that that the more

> understandable view is legitimate.

>

> I’m going to try to make a snapable V3 which should allow the pieces to be

> more easily taken apart and reassembled into other forms. Shapeways does

> offer a single, clear translucent plastic that they call "Frosted Detail",

> and another called "Transparent Acrylic", but I don’t think that any sort

> of transparent stickers will help us, especially since this thing is chock

> full of magnets. The easiest way to let you see into the two hemispheres

> would be to simply truncate the pointy tips of the stickers. That already

> happens a little bit due to the way I’ve rounded the edges. Here is a

> close-up <http://superliminal.com/cube/inverted1.jpg> of a half-rotation

> in which you can see that the inner yellow and white faces are solved. Your

> suggestion of little mapping dots on the corners also works, but just

> opening the existing window further would work more directly.

>

> -Melinda

>

> On 4/28/2017 2:15 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:

>

> I agree with Don’s arguments about adjacent sticker colors needing to be

> next to each other. I think this can be turned into an accurate 2^4 with

> coloring changes, so I agree with Joel too :)

>

> To help me think about it, I started adding a new projection option for

> spherical puzzles to MagicTile, which takes the two hemispheres of a puzzle

> and maps them to two disks with identified boundaries connected at a point,

> just like a physical "global chess

> <http://www.pa-network.com/global-chess/indexf.html>" game I have.

> Melinda’s puzzle is a lot like this up a dimension, so think about two

> disjoint balls, each representing a hemisphere of the 2^4, each a "subcube"

> of Melinda’s puzzle. The two boundaries of the balls are identified with

> each other and as you roll one around, the other half rolls around so that

> identified points connect up. We need to have the same restriction on

> Melinda’s puzzle.

>

> In the pristine state then, I think it’d be nice to have an internal

> (hidden), solid colored octahedron on each half. The other 6 faces should

> all have equal colors split between each hemisphere, 4 stickers on each

> half. You should be able to reorient the two subcubes to make a half

> octahedron of any color on each subcube. I just saw Matt’s email and

> picture, and it looks like we were going down the same thought path. I

> think with recoloring (mirroring some of the current piece colorings)

> though, the windmill’s can be avoided (?)

>

> […] After staring/thinking a bit more, the coloring Matt came up with is

> right-on if you want to put a solid color at the center of each

> hemisphere. His comment about the "mirrored" pieces on each side helped me

> understand better. 3 of the stickers are mirrored and the 4th is the

> hidden color (different on each side for a given pair of "mirrored"

> pieces). All faces behave identically as well, as they should. It’s a

> little bit of a bummer that it doesn’t look very pristine in the pristine

> state, but it does look like it should work as a 2^4.

>

> I wonder if there might be some adjustments to be made when shapeways

> allows printing translucent as a color :)

>

> […] Sorry for all the streaming, but I wanted to share one more

> thought. I now completely agree with Joel/Matt about it behaving as a 2^4,

> even with the original coloring. You just need to consider the corner

> colors of the two subcubes (pink/purple near the end of the video) as being

> a window into the interior of the piece. The other colors match up as

> desired. (Sorry if folks already understood this after their emails and

> I’m just catching up!)

>

> In fact, you could alter the coloring of the pieces slightly so that the

> behavior was similar with the inverted coloring. At the corners where 3

> colors meet on each piece, you could put a little circle of color of the

> opposite 4th color. In Matt’s windmill coloring then, you’d be able to see

> all four colors of a piece, like you can with some of the pieces on

> Melinda’s original coloring. And again you’d consider the color circles a

> window to the interior that did not require the same matching constraints

> between the subcubes.

>

> I’m looking forward to having one of these :)

>

> Happy Friday everyone,

> Roice

>

> On Fri, Apr 28, 2017 at 1:14 AM, Joel Karlsson joelkarlsson97@gmail.com

> [4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:

>

>>

>>

>> Seems like there was a slight misunderstanding. I meant that you need to

>> be able to twist one of the faces and in MC4D the most natural choice is

>> the center face. In your physical puzzle you can achieve this type of twist

>> by twisting the two subcubes although this is indeed a twist of the

>> subcubes themselves and not the center face, however, this is still the

>> same type of twist just around another face.

>>

>> If the magnets are that allowing the 2x2x2x2 is obviously a subgroup of

>> this puzzle. Hopefully the restrictions will be quite natural and only some

>> "strange" moves would be illegal. Regarding the "families of states" (aka

>> orbits), the 2x2x2x2 has 6 orbits. As I mentioned earlier all allowed

>> twists preserves the parity of the pieces, meaning that only half of the

>> permutations you can achieve by disassembling and reassembling can be

>> reached through legal moves. Because of some geometrical properties of the

>> 2x2x2x2 and its twists, which would take some time to discuss in detail

>> here, the orientation of the stickers mod 3 are preserved, meaning that the

>> last corner only can be oriented in one third of the number of orientations

>> for the other corners. This gives a total number of orbits of 2x3=6. To

>> check this result let’s use this information to calculate all the possible

>> states of the 2x2x2x2; if there were no restrictions we would have 16! for

>> permuting the pieces (16 pieces) and 12^16 for orienting them (12

>> orientations for each corner). If we now take into account that there are 6

>> equally sized orbits this gets us to 12!16^12/6. However, we should also

>> note that the orientation of the puzzle as a hole is not set by some kind

>> of centerpieces and thus we need to devide with the number of orientations

>> of a 4D cube if we want all our states to be separated with twists and not

>> only rotations of the hole thing. The number of ways to orient a 4D cube in

>> space (only allowing rotations and not mirroring) is 8x6x4=192 giving a

>> total of 12!16^12/(6*192) states which is indeed the same number that

>> for example David Smith arrived at during his calculations. Therefore,

>> when determining whether or not a twist on your puzzle is legal or not it

>> is sufficient and necessary to confirm that the twist is an even

>> permutation of the pieces and preserves the orientation of stickers mod 3.

>>

>> Best regards,

>> Joel

>>

>> Den 28 apr. 2017 3:02 fm skrev "Melinda Green melinda@superliminal.com

>> [4D_Cubing]" <4D_Cubing@yahoogroups.com>:

>>

>>

>>

>> The new arrangement of magnets allows every valid orientation of pieces.

>> The only invalid ones are those where the diagonal lines cutting each

>> cube’s face cross each other rather than coincide. In other words, you can

>> assemble the puzzle in all ways that preserve the overall diamond/harlequin

>> pattern. Just about every move you can think of on the whole puzzle is

>> valid though there are definitely invalid moves that the magnets allow. The

>> most obvious invalid move is twisting of a single end cap.

>>

>> I think your description of the center face is not correct though. Twists

>> of the outer faces cause twists "through" the center face, not "of" that

>> face. Twists of the outer faces are twists of those faces themselves

>> because they are the ones not changing, just like the center and outer

>> faces of MC4D when you twist the center face. The only direct twist of the

>> center face that this puzzle allows is a 90 degree twist about the outer

>> axis. That happens when you simultaneously twist both end caps in the same

>> direction.

>>

>> Yes, it’s quite straightforward reorienting the whole puzzle to put any

>> of the four axes on the outside. This is a very nice improvement over the

>> first version and should make it much easier to solve. You may be right

>> that we just need to find the right way to think about the outside faces.

>> I’ll leave it to the math geniuses on the list to figure that out.

>>

>> -Melinda

>>

>>

>> On 4/27/2017 10:31 AM, Joel Karlsson joelkarlsson97@gmail.com

>> [4D_Cubing] wrote:

>>

>>

>> Hi Melinda,

>>

>> I do not agree with the criticism regarding the white and yellow stickers

>> touching each other, this could simply be an effect of the different

>> representations of the puzzle. To really figure out if this indeed is a

>> representation of a 2x2x2x2 we need to look at the possible moves (twists

>> and rotations) and figure out the equivalent moves in the MC4D software.

>> From the MC4D software, it’s easy to understand that the only moves

>> required are free twists of one of the faces (that is, only twisting the

>> center face in the standard perspective projection in MC4D) and 4D

>> rotations swapping which face is in the center (ctrl-clicking in MC4D). The

>> first is possible in your physical puzzle by rotating the white and yellow

>> subcubes (from here on I use subcube to refer to the two halves of the

>> puzzle and the colours of the subcubes to refer to the "outer colours").

>> The second is possible if it’s possible to reach a solved state with any

>> two colours on the subcubes that still allow you to perform the previously

>> mentioned twists. This seems to be the case from your demonstration and is

>> indeed true if the magnets allow the simple twists regardless of the

>> colours of the subcubes. Thus, it is possible to let your puzzle be a

>> representation of a 2x2x2x2, however, it might require that some moves that

>> the magnets allow aren’t used.

>>

>> Best regards,

>> Joel

>>

>> 2017-04-27 3:09 GMT+02:00 Melinda Green melinda@superliminal.com

>> [4D_Cubing] <4D_Cubing@yahoogroups.com>:

>>

>>>

>>>

>>> Dear Cubists,

>>>

>>> I’ve finished version 2 of my physical puzzle and uploaded a video of it

>>> here:

>>> https://www.youtube.com/watch?v=zqftZ8kJKLo

>>> Again, please don’t share these videos outside this group as their

>>> purpose is just to get your feedback. I’ll eventually replace them with a

>>> public video.

>>>

>>> Here is an extra math puzzle that I bet you folks can answer: How many

>>> families of states does this puzzle have? In other words, if disassembled

>>> and reassembled in any random configuration the magnets allow, what are the

>>> odds that it can be solved? This has practical implications if all such

>>> configurations are solvable because it would provide a very easy way to

>>> fully scramble the puzzle.

>>>

>>> And finally, a bit of fun: A relatively new friend of mine and new list

>>> member, Marc Ringuette, got excited enough to make his own version. He

>>> built it from EPP foam and colored tape, and used honey instead of magnets

>>> to hold it together. Check it out here: http://superliminal.com/cube/d

>>> essert_cube.jpg I don’t know how practical a solution this is but it

>>> sure looks delicious! Welcome Marc!

>>>

>>> -Melinda

>>>

>>>

>>

>>

>>

>>

>>

>

>

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