Message #3699

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Physical 4D puzzle V2
Date: Fri, 28 Apr 2017 17:44:04 -0500

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 4:59 PM, Roice Nelson <roice3@gmail.com> wrote:

> I sent my email a bit too fast. Ater 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 :)
>
> Roice
>
>
> On Fri, Apr 28, 2017 at 4:15 PM, Roice Nelson <roice3@gmail.com> 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 (?)
>>
>> 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
>>>>
>>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>
>>
>