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 Post subject: GeoKite Cube
PostPosted: Wed Dec 15, 2010 3:03 pm 
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I drew up a net for a puzzle that I made up in DigiGraph 2, and I wondered if any of you guys could make it real. I also thought it would be cool to have the puzzle have curved cuts, instead of the pieces going over and under each other with straight cuts.


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geokitecube.jpg.JPEG
geokitecube.jpg.JPEG [ 310.17 KiB | Viewed 1546 times ]

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 Post subject: Re: GeoKite Cube
PostPosted: Wed Dec 15, 2010 3:47 pm 
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I really can't tell, is that thing a shape-mod or something else? (it looks FTO to me, but I'm not sure...)
I can't even see how it can possibly turn!
Can you please post more details? :oops:

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 Post subject: Re: GeoKite Cube
PostPosted: Wed Dec 15, 2010 4:24 pm 
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Monopoly wrote:
I really can't tell, is that thing a shape-mod or something else? (it looks FTO to me, but I'm not sure...)
I can't even see how it can possibly turn!
Can you please post more details? :oops:


No, it's not a shape mod. I've highlighted one possible turning plane here:


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geokitecubeturn.jpg
geokitecubeturn.jpg [ 267.83 KiB | Viewed 1495 times ]

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 Post subject: Re: GeoKite Cube
PostPosted: Wed Dec 15, 2010 4:31 pm 
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Oh, I get it now, it's kind of a Skewb with skewed cuts.
Do you have any idea what geometry this is or which pieces would be centers/anchors? I'm still confused.

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 Post subject: Re: GeoKite Cube
PostPosted: Wed Dec 15, 2010 5:00 pm 
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I actually have no idea. It's kinda like Drewseph's Praxis Cube.

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 Post subject: Re: GeoKite Cube
PostPosted: Wed Dec 15, 2010 5:14 pm 
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Let me take a look at it and ill post a pic of what it looks like in 3d :)

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 Post subject: Re: GeoKite Cube
PostPosted: Wed Dec 15, 2010 9:11 pm 
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Okay, I'm back, and i dont see how it is possible but if someone can confirm this that would be awesome :)

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 Post subject: Re: GeoKite Cube
PostPosted: Thu Dec 16, 2010 4:40 pm 
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NXTgen wrote:
Okay, I'm back, and i dont see how it is possible but if someone can confirm this that would be awesome :)


What do you mean by it being "impossible"?

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 Post subject: Re: GeoKite Cube
PostPosted: Thu Dec 16, 2010 6:03 pm 
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Here's an image to help visualise it:

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File comment: 3D image
Screen shot 2010-12-17 at 12.00.52 PM.png
Screen shot 2010-12-17 at 12.00.52 PM.png [ 18.27 KiB | Viewed 1275 times ]


It looks as if it would be possible to turn, but after a single turn I can't see where it would turn next.

If it does have more moves it would be great!

-Mark- :)

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 Post subject: Re: GeoKite Cube
PostPosted: Thu Dec 16, 2010 8:49 pm 
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At first I was gonna post anything on this topic but then I thought, hey, here's an opportunity to show off :wink: :lol:

In all seriousness here is my analysis:

Let me begin by saying that, in general, making random cuts on a solid (even if they are symmetrical or follow some arbitrary pattern) will almost never produce a working puzzle and even if some faces do interact, unless it forms a larger geometry, very few moves can actually be made. So before I even started, I was pretty sure this wouldn't make much of a puzzle. Then again, I would've felt foolish if I didn't examine it and it turned out to be some incredible complex relationship I hadn't thought of, so I broke down the geometry anyway 8-)

You have only described one type of movement and thus one type of cutting plane to us. Although symmetry allows this cut to happen at 4 places, these 4 moves do not account for all of the cuts drawn on your initial picture. Shown below is a 3d rendering of your net
Attachment:
GeoKiteCubeReduction.png
GeoKiteCubeReduction.png [ 65.45 KiB | Viewed 1229 times ]

The red cuts are those accounted for by the movetype you demonstrated. However, none of the dashed green cuts were separated during this movetype. In fact I was unable to find a set of any three coplanar cuts on the puzzle that included a green dashed line so I don't see how they can be used. I therefore eliminated them and analyzed the remaining puzzle show above on the right. (if you can show me how you intended for the puzzle to move along a green line I will update this analysis)

This puzzle is deepcut (opposite moves cut along the same line), and has 4 axes of rotation. The geometry is similair to that of a Skewb. The mathematical core is infintessimal, but assuming this is a legitimate twisty puzzle, the limit of the core should have the following geometry: (this will tell us ALOT about how the puzzle could possibly function)
Attachment:
ReducedGeoKiteCubeCore.png
ReducedGeoKiteCubeCore.png [ 33.37 KiB | Viewed 1229 times ]

The blue lines are the axes of rotation. There are perpendicular to the faces of the core and all pass through the center. This is the shape of an irregular octahedron. Specifically it has the symmetry of a square dipyramid (two identical square pyramids fused on their bases). Now then, given ANY convex solid, the pieces of the standard twisty puzzle puzzle based on its geometry are able to mix if and only if
1) the distances from the center of rotation of a face to two adjacent faces are equal (in which case any piece moved by one of these adjacent faces can be moved by the other face once it has been rotated by the intermediate face in question)
AND 2) the dihedral angles between the above faces are equal

property 1) is being demonstrated in the above picture. Notice how the pink circles of the four faces in one square pyramid all touch by no pink circles touch between the two square pyramids. This means that the pieces on or below the equator of any puzzle based on this geometry can never interact with pieces on or above the equator. They may be able to move into the other regions, but then the only available move that affect these pieces again will undo the last move that put them in the other region. The second property is examined below:
Attachment:
ReducedGeoKiteCubeCoreDetail.png
ReducedGeoKiteCubeCoreDetail.png [ 52.49 KiB | Viewed 1229 times ]

Here all adjacent dihedral angles are marked at the same radius. The teal arcs mark the dihedral angles between faces within the same hemisphere and the purple arcs mark the dihedral angles between the two hemispheres. I didn't bother calculating the exact angles but all the teal dihedral angles are about 61.9275130641 degrees while all the purple dihedral angles are about 86.6277133166 degrees. Note that above every face of the core two teal arcs meet. These represent identical dihedral angles of two faces adjacent to the same face. This means the pieces shared by two of these faces can interact with the pieces shared by the other two. The angle the middle face must rotate between these two dihedral angles is approximately 111.1001960241 degrees (marked in yellow above). As this same angle does not match the angle between other adjacent faces, this move is a jumbling move.

So there you have it, can the reduced form of this function as a puzzle? Yes in the sense that multiple interacting moves are possible. The driving cause behind this is the fact that the puzzle is rotational symmetric. However any given face can ONLY interact with 2 out of 3 adjacent faces and in fact, many pieces can NEVER interact with eachother. Because of the very limited moveset, I suspect that the puzzle will not be able to scramble beyond 2 or 3 moves, but the theory shows that a partial jumbling unbandage will free up some interactive moves.... just not very good ones.

Does that shed any light? This geomety does work a little, but not enough to have make a puzzle that can really scramble. Trying to create a system of cuts that form a workable puzzle from scratch and then figuring out the mechanism is borderline impossible (except in the simplest of cases). It is better to build off existing puzzles or come up with a mechanism first and then determine what the exterior of the puzzle will look like. For example I suspect Oskar did not just magically draw up a Meteor Madness one day. He discovered the jumbling properties of the geometry of a triangular prism and then determined what a puzzle based off it would like.

Peace,
Matt Galla

Edit: After posting this, I went ahead and performed a rotation of about 111.1001960241 degrees. Here is the result:
Attachment:
ReducedGeoKiteMove1.png
ReducedGeoKiteMove1.png [ 20.55 KiB | Viewed 1206 times ]

As you can see, the red lines are all coplanar and this move is available! However, this next move can only stop at one point, at which point the only new potential move is undoing the one shown here. After that move, the only new potential you can make is undoing move in 2 in this sequence, resetting the internal state of the mechanism rotations to 0 degrees. I'm not sure that this entire sequence is available, but what I can tell you is if this sequence is not possible, the puzzle cannot do any interesting scramble. The Cayley graph of the puzzle would simply be a node tree because the theory tells us no other interaction can take place (mostly due to the fact that only 2 of 3 adjacent faces have the same dihedral angle).

PS interestingly enough, the two faces marked with green X's are exactly coplanar, making the picture look very odd and distorted... but it's an accurate rendering! :P


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 Post subject: Re: GeoKite Cube
PostPosted: Fri Dec 17, 2010 4:51 pm 
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Firstly, wow. :shock: I can't believe that you went through all that work to see if my puzzle would work. And I'm also sorry that I put you through all of that. Maybe I should actually TRY to make a puzzle using geometry... but it's still cool how those those faces in the last picture were coplanar. Maybe I should just try to make a FULL twisty puzzle that could do that...

Oh, here's that picture you wanted:


Attachments:
geokitecubeturn2.jpg
geokitecubeturn2.jpg [ 267.66 KiB | Viewed 1137 times ]

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 Post subject: Re: GeoKite Cube
PostPosted: Fri Dec 17, 2010 9:25 pm 
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Jorbs3210 wrote:
Firstly, wow. :shock: I can't believe that you went through all that work to see if my puzzle would work. And I'm also sorry that I put you through all of that.

It was fun :mrgreen: I've never actually gotten the opportunity before to analyze something that isn't already a working puzzle :wink: And don't be sorry, it was actually one of the most interesting looking edeavours I've ever taken from someone else on TP :)
Jorbs3210 wrote:
Maybe I should actually TRY to make a puzzle using geometry... but it's still cool how those those faces in the last picture were coplanar. Maybe I should just try to make a FULL twisty puzzle that could do that...

If you want my honest opinion, I don't think you should pursue this by drawing up cuts at convenient points on the cube because this has an almost 0% chance of working. While you mey get some partially doable moves like you did here, you will probably never get a fully unbandaged puzzle UNLESS you end up drawing a shapemod of an existing puzzle (like Axis Cube or Pyracue). When building some of the coolest looking puzzles: Anything shape-shifting by Tony, Drew's Praxis Cube, Helicopter Cube, Axis Cube, etc.... the original designers didn't know exactly what the puzzle was going to look like until they built it. They didn't design the exterior look of the puzzle, they designed the mech of the puzzle, which is heavily dictated by the geometry of the core. About 85% of all twisty puzzles follow roughly the same design. Most start with a core in the shape of a Platonic solid which consists most often of either a set of cylinders intersecting at the correct angles to eachother or a sphere with holes in the correct orientation to one another. From there, you can lay out the rotation planes to cut up the 3d space around the core into different cells (think about cutting up a piece of paper and think taking that analogy to 3d space, you cut up space) Now by simply "filling in" the empty space around the core with a single (usually convex) solid, you can make a design with compatible cuts. The rotation planes that cut up 3d space before will now cut up the solid you have just placed in the region around the core (The analogy to 2d here is to draw a shape on a piece of paper before you cut it up, then using the same cuts you did before, cut it up again. Since the cuts are being kept the same, you aren't REALLY in control of where the shape gets cut at. It is just a result of how the shape falls in the cuts. Does that make sense?) Sorry if all of this is obvious to you. In that case, there is a reason people will typically do it this way (design mechanism -> exterior patter is a result) and not the way you are trying (design the exterior pattern -> mechanism is a result).

So if you want a puzzle that works, I think you SHOULD try starting with a core that already works (tetrahedron and octahedron are easiest and don't think the resulting puzzle will look simple! (Golden Cube has a simple tetrahedral Core!). Granted this is MUCH easier if you have a CAD program. You should look into getting one. The pictures I have been making are from Cabri 3d, which won't do a lot of good fro designing the actual mechanism of any pieces, but it does great with geometry, although it is not free. I believe many people use Solid Works here (again not free). For a simple free CAD look into Google Sketchup and I'm sure some others can give more suggestions. Now then let's look at this supposed move you seem to think will work.... :wink:
Jorbs3210 wrote:
Oh, here's that picture you wanted:


Oh really? You think that should work huh? :wink:
Have a look at a 3d rendering of just one of these sets of cuts:
Attachment:
GeoKiteCubeNonPlanarCut.png
GeoKiteCubeNonPlanarCut.png [ 23.61 KiB | Viewed 1109 times ]

I'd LOVE to see something that can rotate on the red line! I hope you can see that there is no way to make this puzzle spin here: the cuts don't line up! Although I will admit it is very hard to deduce this just by looking at a 2d net. I'm pretty sure there's no way to find a cut that includes any of the lines I marked green in my first post. And if I haven't said it enough already, it is NEARLY impossible to come up with a set of cuts on a cube that form a set of moves without basing it off of some predesigned core. Even if a 3d program is used to make sure all cuts line up, there are many more properties that must be met internally before a puzzle can actually scramble :scrambled:

Good luck with your next design! :D

Peace,
Matt Galla


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 Post subject: Re: GeoKite Cube
PostPosted: Sat Dec 18, 2010 3:51 pm 
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Ow. :shock:

The idea wasn't entirely random, though. I wanted to see if kite-shaped faces would work, and that happened. Well, it's time to take a look at a design I KNOW will work because it doesn't jumble.

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