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 Post subject: Spiral Fracture
PostPosted: Mon Apr 23, 2012 5:31 pm 
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Hi Everyone,

I'm pleased to introduce my second new puzzle today, the Spiral Fracture. As you might guess, Spiral Fracture is part of my series that began with Fracture-10. Spiral Fracture is the same pentagonal dipyramid shape as Fracture-10, but this time curved cuts (just curved, not actually spirals) were used instead of straight cuts. By using just the right curvature, along with a new set of internal parts, I was able to allow edge pieces and face pieces to exchange places. You may recall this feature from my earlier Compound Fracture puzzle, but in that case straight cuts were used. With the new curved cuts, Spiral Fracture retains all five axes of the dipyramid shape, so no irregular geometry is required.

Here is the Spiral Fracture video.

You can buy Spiral Fracture in my Shapeways shop.

It's hard to tell in the photos, but the internal pieces that separate the faces and edges are actually visible due to the curved cuts. This is why the stickers don't form nice straight lines between each other as in most puzzles. That said, here are the pictures of Spiral Fracture:
Attachment:
Spiral Fracture Solved.JPG
Spiral Fracture Solved.JPG [ 202.58 KiB | Viewed 2424 times ]
Attachment:
Spiral Fracture End 2.JPG
Spiral Fracture End 2.JPG [ 218.11 KiB | Viewed 2424 times ]
Attachment:
Spiral Fracture 180 Turn.JPG
Spiral Fracture 180 Turn.JPG [ 174.04 KiB | Viewed 2424 times ]
Attachment:
Spiral Fracture 45 Turn.JPG
Spiral Fracture 45 Turn.JPG [ 179.2 KiB | Viewed 2424 times ]
Attachment:
Spiral Fracture Multiple Turns.JPG
Spiral Fracture Multiple Turns.JPG [ 191.16 KiB | Viewed 2424 times ]
Attachment:
Spiral Fracture Partial Scramble.JPG
Spiral Fracture Partial Scramble.JPG [ 203.81 KiB | Viewed 2424 times ]

Enjoy!
Dave

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 Post subject: Re: Spiral Fracture
PostPosted: Tue Apr 24, 2012 10:24 am 
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David Pitcher wrote:
By using just the right curvature, along with a new set of internal parts, I was able to allow edge pieces and face pieces to exchange places. You may recall this feature from my earlier Compound Fracture puzzle, but in that case straight cuts were used. With the new curved cuts, Spiral Fracture retains all five axes of the dipyramid shape, so no irregular geometry is required.

Nice... its a Fudged Mixup Fracture-10. I like it.

Carl

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 Post subject: Re: Spiral Fracture
PostPosted: Tue Apr 24, 2012 11:36 am 
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wwwmwww wrote:
its a Fudged Mixup Fracture-10
There is no actual fudging involved though. The curved cuts allow this geometry with no additional removal of material required.

Glad you like it!

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 Post subject: Re: Spiral Fracture
PostPosted: Tue Apr 24, 2012 12:32 pm 
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Ouch!
My poor brain.
I am happy that you write your own descriptions.

Edge lengths?


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 Post subject: Re: Spiral Fracture
PostPosted: Tue Apr 24, 2012 4:39 pm 
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David Pitcher wrote:
There is no actual fudging involved though. The curved cuts allow this geometry with no additional removal of material required.
If so I'm impressed and also a bit puzzled. I guess the clue is in your statement.
David Pitcher wrote:
It's hard to tell in the photos, but the internal pieces that separate the faces and edges are actually visible due to the curved cuts. This is why the stickers don't form nice straight lines between each other as in most puzzles.
So if I understand correctly this puzzle is formed from a dipyramid which has been cut with 2D surfaces (are they spherical, conical, or something else?) and that no volume of that dipyramid needs to be removed to allow rotation. Do pieces also stay flush up against each other during rotation? Because I'd consider a rotation which required the pieces being rotated to be pulled out, even if very slightly, also to be a form of fudging.

If this does work out that nice I can't say off the top of my head how one would need to calculate the surface of rotation that would be needed. And it also tells me the pieces you are calling "internal pieces" do have some exposed area on the surface. Granted it appears very thin but I'd really want to turn these pieces into solvable parts of the puzzle. Is each of these thin exposed areas its own part? Hmmm... looking at this picture:

Image

I think the answer is yes. Maybe they could be dyed the color of the face they are on. Just a thought...

I still want to call this a Mixup Fracture-10 as this is to the Fracture-10 what the Mixup Cube is to the 3x3x3. But I'm very curious how you pulled that off without fudging. At the moment I'm guessing the cut surface is spherical and if the radius of that sphere is infinite then you have the Fracture-10 puzzle. Make they radius even smaller and I'm guessing you end up with the Diamond Delight puzzle. Is that the parameter you are playing with? Without sitting down for a while infront of Solidworks to test a few things I could very well be way off base at the moment.

Carl

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 Post subject: Re: Spiral Fracture
PostPosted: Tue Apr 24, 2012 5:42 pm 
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Great questions Carl! You clearly think very deeply about these puzzles, and it's wonderful to have so much thought put into their analysis. Hopefully I've answered all of your questions below...
wwwmwww wrote:
So if I understand correctly this puzzle is formed from a dipyramid which has been cut with 2D surfaces (are they spherical, conical, or something else?) and that no volume of that dipyramid needs to be removed to allow rotation.
You are correct, the base shape is a dipyramid, and the cutting surfaces are spherical. No volume needs to be removed to allow rotation, although I did remove a bit on the core spider arms to allow them to not rotate if friction doesn't want them to. In a frictionless world this wouldn't be needed.
wwwmwww wrote:
Do pieces also stay flush up against each other during rotation? Because I'd consider a rotation which required the pieces being rotated to be pulled out, even if very slightly, also to be a form of fudging.
The pieces stay flush to each other during rotation, and don't move relative to each other, not even in the slightest bit.
wwwmwww wrote:
And it also tells me the pieces you are calling "internal pieces" do have some exposed area on the surface. Granted it appears very thin but I'd really want to turn these pieces into solvable parts of the puzzle. Is each of these thin exposed areas its own part?
Yes, the thin slivers in between each piece are exposed on the surface. These same pieces exist on Compound fracture, but they are completely hidden due to the straight cuts on that puzzle. I did think about trying to make these parts solvable, but they would be next to impossible to sticker since they are so thin, and I really didn't want to go through 10 different color dye batches to try to make them match. Plus, I figured that the puzzle is hard enough to solve as it is. If someone wants to make one with these parts dyed though, I'd love to see the results!
wwwmwww wrote:
At the moment I'm guessing the cut surface is spherical and if the radius of that sphere is infinite then you have the Fracture-10 puzzle. Make they radius even smaller and I'm guessing you end up with the Diamond Delight puzzle. Is that the parameter you are playing with?
This is exactly correct.
wwwmwww wrote:
If this does work out that nice I can't say off the top of my head how one would need to calculate the surface of rotation that would be needed.
Figuring out how to get just the right curves was the hardest part of designing both puzzles. Various 3D sketches were required.
wwwmwww wrote:
I still want to call this a Mixup Fracture-10 as this is to the Fracture-10 what the Mixup Cube is to the 3x3x3.
True, it is the equivalent of a Mixup Fracture-10, but I felt like taking a bit more "poetic" license with the name.

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 Post subject: Re: Spiral Fracture
PostPosted: Tue Apr 24, 2012 9:56 pm 
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Could this be unbandaged into a doctrinaire puzzle? :o


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 Post subject: Re: Spiral Fracture
PostPosted: Wed Apr 25, 2012 10:59 am 
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David Pitcher wrote:
wwwmwww wrote:
I still want to call this a Mixup Fracture-10 as this is to the Fracture-10 what the Mixup Cube is to the 3x3x3.
True, it is the equivalent of a Mixup Fracture-10, but I felt like taking a bit more "poetic" license with the name.
Oh I wasn't trying to propose a name change. By calling it "a Mixup Fracture-10" that is more like a definition to me. It's still THE Spiral Fracture. Note my Doctor Skewb is a Fudged Slice-turn-only Mixup Master Skewb yet that isn't what I named it.
Jared wrote:
Could this be unbandaged into a doctrinaire puzzle? :o
That is a very good question... and at the moment I'm honestly not sure. If this is a true jumbling puzzle that would require fudging at some point but I'm not even sure this jumbles at the moment. My guess is that if this can be cut up into a doctrinaire puzzle that it would require too many small parts to be able to work well as a physical puzzle. David, have you tried to unbandage this puzzle and seen what happens?

Carl

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 Post subject: Re: Spiral Fracture
PostPosted: Wed Apr 25, 2012 11:23 am 
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Jared wrote:
Could this be unbandaged into a doctrinaire puzzle?
Technically, even the Fracture-10 is not doctrinaire since the slice patterns change with almost every move, despite the fact that it does not change shape. To my knowledge this geometry cannot be unbandaged with straight cuts. The only way (at least that I've seen) to unbandage a vertex-turning pentagonal (or trigonal or hexagonal and higher) dipyramid is by using curved cuts. Looked at this way, the unbandaged (and doctrinaire) form of both Spiral Fracture and Fracture-10 is the Diamond Delight puzzle.

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 Post subject: Re: Spiral Fracture
PostPosted: Wed Apr 25, 2012 3:15 pm 
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David Pitcher wrote:
The only way (at least that I've seen) to unbandage a vertex-turning pentagonal (or trigonal or hexagonal and higher) dipyramid is by using curved cuts. Looked at this way, the unbandaged (and doctrinaire) form of both Spiral Fracture and Fracture-10 is the Diamond Delight puzzle.
I don't consider the Diamond Delight puzzle to be an unbandaged Spiral Fracture or Fracture-10. To me its a different puzzle. The construction technique is very similiar but the types of pieces and their properties are very different. The way to construct the unbandaged Spiral Fracture or Fracture-10 puzzle would be to start with the original puzzle as is. Turn the puzzle such that a rotation is blocked and then extend the cut such that the puzzle is no longer blocked. Do this as many times as needed such that the puzzle is never blocked. One of two things will happen.

(1) After some finite number of cuts are added the puzzle will be doctrinaire and you'll have the puzzle you are after. This would mean the original puzzle doesn't jumble and that its just a bandaged puzzle.

(2) You will find that an infinite number of cuts are needed to totally unbandage the puzzle and in this case the original puzzle jumbles. The cutting process if allowed to continue would keep producing smaller and smaller pieces, and at this point you again have two options.

(a) You could say the puzzle has no doctrinaire form.
(b) Or you could at some point in the cutting process stop and simply throw out all the smallest pieces and see if its possible to fudge the remaining pieces to work without them. You could consider this a fudged doctrinaire form of the original puzzle. Note how Fracture-6 can be fudged to form the Constellation Six puzzle.

I'm pretty sure that the Fracture-10 puzzle was proven to jumble at one point. Maybe it was in this thread but all the pics seem to be gone so I'm not sure at the moment. My guess is that the Spiral Fracture puzzle also jumbles but short if trying to unbandage it and seeing what happens (and no I'm not talking about changing the radius of your cut surface) I can not say I'm certain that is the case.

Carl

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 Post subject: Re: Spiral Fracture
PostPosted: Wed Apr 25, 2012 3:40 pm 
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David Pitcher wrote:
The only way (at least that I've seen) to unbandage a vertex-turning pentagonal (or trigonal or hexagonal and higher) dipyramid is by using curved cuts.
I was mistaken with this comment, since making the (straight) cuts shallower will have the same effect. This can be seen in the Sunrise Sunset puzzle. If the polar axes and their associated cuts are ignored, you end up with a doctrinaire puzzle with the same turning axes as Fracture-6 or Constellation Six, but without the bandaging or jumbling. The same can be done with the pentagonal dipyramid (I actually almost made this puzzle before hitting upon the curved cuts that produced Diamond Delight).
wwwmwww wrote:
I don't consider the Diamond Delight puzzle to be an unbandaged Spiral Fracture or Fracture-10. To me its a different puzzle. The construction technique is very similiar but the types of pieces and their properties are very different.
You're right, changing the cuts to either make them shallower or curved fundamentally changes the puzzle.
wwwmwww wrote:
(2) You will find that an infinite number of cuts are needed to totally unbandage the puzzle and in this case the original puzzle jumbles. The cutting process if allowed to continue would keep producing smaller and smaller pieces, and at this point you again have two options.
This is what happens if you keep extending the cuts on Fracture-6, Fracture-10, or any higher order dipyramids. I believe this was shown in the thread discussing Constellation Six (there might have even been an animation in there). So the puzzles do jumble per the definition above, even though they do not change shape. Is this a unique property of the vertex turning dipyramid group of puzzles?
wwwmwww wrote:
(b) Or you could at some point in the cutting process stop and simply throw out all the smallest pieces and see if its possible to fudge the remaining pieces to work without them. You could consider this a fudged doctrinaire form of the original puzzle. Note how Fracture-6 can be fudged to form the Constellation Six puzzle.
The same can be done with Fracture-10, but you end up with a lot more "blank" space between the pieces. The edge pieces turn into thin diamond shapes, and the centers turn into circles. Imagine the edges of Fracture-10 cut in half the short way, then throw away the large end and mirror the small end on the cut plane. This is the unbandaged edge piece. The center pieces are circles that are tangent to the centers of those new edge pieces. For the Spiral Fracture the same thing will happen as with Fracture-10, so it should then be considered to jumble, although the Spiral Fracture can change shape where the Fracture-10 cannot.

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 Post subject: Re: Spiral Fracture
PostPosted: Wed Apr 25, 2012 3:45 pm 
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My intuition also says that this puzzle jumbles. The corners have fourfold rotational symmetry, but by allowing face and edge pieces to mix up (making them the same piece type) you give the corners eightfold rotation symmetry, in a way.

The faceturning cube jumbles when given the same ability to turn faces 45 degrees, and Timur's Skyglobe, a tetrahedron given the ability to rotate corners 60 degrees, has to be fudged to be doctrinaire (as in it would jumble without fudging), so I'd expect the trend to continue here.

Attachment:
File comment: (URU'R')2
45turncube.png
45turncube.png [ 92.5 KiB | Viewed 1841 times ]


Can any of you think of a counterexample where such a puzzle is doctrinaire? (I think Jewel 45 wouldn't do for a counterexample as it's pretty heavily fudged to only leave only 1 type of piece)

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