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 Post subject: Hypothesis: Convex Solids & Jumbling
PostPosted: Fri Mar 16, 2012 1:13 pm 
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Hi Everyone,

I've been working on a few different puzzle ideas lately, and I have begun to notice a trend which has lead me to propose the following hypothesis. Unfortunately I'm not sure how to test the hypothesis other than using "brute force" (i.e., building computer models of every proposed solid, which would be an infinite task), and I am hoping that some of the more math-oriented members of the forum might be able to help. That said, here is my hypothesis:

Any convex solid such that all faces are the same shape and size (even if the edges of those faces are not of equal length), and such that the dihedral angles of all edges meeting at a vertex are identical (even if those angles are different from vertex to vertex), can be sliced into a viable face-turning puzzle. If the solid is not one of the five platonic solids, the puzzle will jumble. Certain geometries may be physically viable only through the use of pillowing, truncating, or other means, but all such solids can be sliced into theoretically workable puzzles.

Of course, it's entirely possible that someone has already come across a solid which disproves this. If so, please let me know. So far everything I have looked at works, although I've got to admit my search has not been incredibly extensive (yet). This theory holds true for such shapes as the rhombic dodecahedron, the rhombic triacontahedron, trigonal, pentagonal, and hexagonal dipyramids, the tetragonal trapezohedron, and the triakis tetrahedron.

Hopefully I've explained the hypothesis clearly enough, but please let me know if additional explanation is needed.

Any thoughts?
Dave

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Last edited by David Pitcher on Fri Mar 16, 2012 3:05 pm, edited 1 time in total.

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Fri Mar 16, 2012 1:48 pm 
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Hey Dave,

If I've understood you right (and I may not have) I think the (face turning) icosahedron is a counter example. All of the faces are uniform, all of the edges are uniform, and as far as I can tell, all of the edges meet a vertex at the same angle.

Yet, when cut deep enough the icosahedron jumbles.

If this is not a counter example can you clarify your definition to help me understand why it isn't?


Edit:

Did you mean if and only if for "If the solid is not one of the five platonic solids, the puzzle will jumble. Certain geometries may be physically viable only through the use of pillowing, truncating, or other means, but all such solids can be sliced into theoretically workable puzzles."?. A particularly pedantic reading of that sentence would imply that no platonic solid can be a counter example whether it jumbles or not. Only non-platonic solids can be counter-examples.

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Fri Mar 16, 2012 2:33 pm 
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Can you clarify what you mean by this:

David Pitcher wrote:
all edges at any given vertex converge at the same angle

Are you trying to say that the angles between each edge and it's adjacent edges at a vertex are the same? Or are you saying that the dihedral angles of all edges meeting at a vertex are identical? Or both?

I suspect that if you were referring to dihedral angles, your hypothesis might be correct, because that "should" allow the transfer of pieces between the edges around the corners if the cuts are equidistant from the origin. (I could be wrong)


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Fri Mar 16, 2012 3:13 pm 
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DKwan wrote:
Are you trying to say that the angles between each edge and it's adjacent edges at a vertex are the same? Or are you saying that the dihedral angles of all edges meeting at a vertex are identical? Or both?
I am referring to the dihedral angles, and I think you said it much better than me, so I edited the original post to reflect that language. Thank you!
bmenrigh wrote:
Did you mean if and only if for "If the solid is not one of the five platonic solids, the puzzle will jumble. Certain geometries may be physically viable only through the use of pillowing, truncating, or other means, but all such solids can be sliced into theoretically workable puzzles."?. A particularly pedantic reading of that sentence would imply that no platonic solid can be a counter example whether it jumbles or not. Only non-platonic solids can be counter-examples.
The face-turning icosahedron is an example of an odd platonic solid given that it's the only one that jumbles, but I don't see it as a counter-example to the general hypothesis, which is in regard to non-platonic solids.

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sat Mar 17, 2012 12:03 am 
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The simplest way to do this is to make a solid whose faces are all scalene triangles. You can do that by taking an octahedron and translating the up, right, and left vertices upwards in space a bit.


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sat Mar 17, 2012 12:29 am 
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Bram wrote:
The simplest way to do this is to make a solid whose faces are all scalene triangles. You can do that by taking an octahedron and translating the up, right, and left vertices upwards in space a bit.

I had been about to suggest a scalene tetrahedron or a snub disphenoid as counter-examples the first time I read his hypothesis, but neither of those meet the now specified characteristic of identical dihedral angles. I believe the polyhedron you described also does not meet this condition (if I correctly understood your description).

EDIT: David, I "think" a list of polyhedra matching your conditions would consist of ONLY the duals of all archimedean solids as well as the trivially infinite set of duals of prisms and antiprisms (dipyramids and trapezohedra). (I will probably think about this more and realize I am wrong)

I think to open up the list to be more all-encompassing of workable puzzle-geometries, you should remove the condition about all faces having to be identical. I'm starting to think that just identical dihedral angles is enough to make a functioning puzzle. One example of such a puzzle is Eric Vergo's Rhomdo.


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sat Mar 17, 2012 9:17 am 
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DKwan wrote:
Bram wrote:
The simplest way to do this is to make a solid whose faces are all scalene triangles. You can do that by taking an octahedron and translating the up, right, and left vertices upwards in space a bit.

I had been about to suggest a scalene tetrahedron or a snub disphenoid as counter-examples the first time I read his hypothesis, but neither of those meet the now specified characteristic of identical dihedral angles. I believe the polyhedron you described also does not meet this condition (if I correctly understood your description).
I'm also fairly certain that none of the convex deltahedra outside of the trigonal and pentagonal dipyramids (excepting of course the tetrahedron, octahedron, and icosahedron) can be modified to meet the criteria.
DKwan wrote:
EDIT: David, I "think" a list of polyhedra matching your conditions would consist of ONLY the duals of all archimedean solids as well as the trivially infinite set of duals of prisms and antiprisms (dipyramids and trapezohedra). (I will probably think about this more and realize I am wrong)
I am beginning to think this is correct as well. There are 13 archimedean solid duals, and of them four have already been tested. I sliced both the triakis tetrahedron and tetrakis hexahedron yesterday and found them to work, and of course we already know that both of the rhombic solids work. That leaves nine of the duals left to try, although I'm hoping someone has a good idea of how to do this other than the "brute force" method I've been using. Creating each one in Solidworks, slicing it up, and then turning it to look for further alignments will take quite a bit of time with the more complex solids that remain.
DKwan wrote:
I think to open up the list to be more all-encompassing of workable puzzle-geometries, you should remove the condition about all faces having to be identical. I'm starting to think that just identical dihedral angles is enough to make a functioning puzzle. One example of such a puzzle is Eric Vergo's Rhomdo.
This is a very interesting possibility, and you may well be correct. Any idea how many more solids this change would allow?

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Last edited by David Pitcher on Sat Mar 17, 2012 11:55 am, edited 1 time in total.

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sat Mar 17, 2012 10:59 am 
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Of the archimedean duals, I count 5 that have been made into physical puzzles already:

Rhombic Dodecahedron --> Made by various designers
Rhombic Triacontahedron --> Big Kahuna by Eitan (also the same cut geometry as any edge-turning dodecas/icosas)
Deltoidal Icositetrahedron --> Face-Turning Deltoidal Icositetrahedron by Eitan
Pentagonal Icositetrahedron --> Small Boulder by Ryan (Cubedude76)
Pentagonal Hexecontahedron --> Big Boulder by Oskar

These 5 plus the 2 others that you tested leaves only 6 more.

David Pitcher wrote:
This is a very interesting possibility, and you may well be correct. Any idea how many more solids this change would allow?

I'm not even sure if going off only the identical dihedral angles rule might generate an infinite set. Also, I think that condition alone allows for a puzzle to turn, but doesn't guarantee it can be scrambled. That's where the condition of identical faces came into play. I'm not sure what you might replace that condition with... maybe allowing for only a set number of different faces or edge lengths might make sense? Or maybe it has to do with the edges being equidistant from the center?


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sat Mar 17, 2012 1:41 pm 
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If a single face has two different edges which form the same angle to the adjacent faces then the puzzle can be made to jumble if the cut depths are set to the right amount. If they're set to different depths it won't necessarily be, for example you could make a pyraminx with all four cuts at different depths then it couldn't be scrambled at all, but setting the depths to the right amounts can always work.


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sat Mar 17, 2012 3:57 pm 
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DKwan wrote:
Of the archimedean duals, I count 5 that have been made into physical puzzles already:

Rhombic Dodecahedron --> Made by various designers
Rhombic Triacontahedron --> Big Kahuna by Eitan (also the same cut geometry as any edge-turning dodecas/icosas)
Deltoidal Icositetrahedron --> Face-Turning Deltoidal Icositetrahedron by Eitan
Pentagonal Icositetrahedron --> Small Boulder by Ryan (Cubedude76)
Pentagonal Hexecontahedron --> Big Boulder by Oskar
:oops: You are absolutely right, I should have remembered the last three fantastic puzzles on your list. I also tested the triakis octahedron today, and that one works as well. This leaves five of the archimedean duals to try. The two "disdyakis" solids are interesting cases in that they are made of scalene triangles. Since there are no equal sides on any given face, I'm wondering if these will be an exception. On to figuring out how to construct those...

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Sun Mar 18, 2012 9:37 pm 
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I have just completed testing the disdyakis dodecahedron, and despite the fact all three edges of every face are unequal, it can (theoretically) be made into a functional face-turning jumbling puzzle. The remaining four archimedean duals are all larger variants of the smaller solids already tested, so I would expect them to work as well. If this is true, then there are 13 of these types of jumbling axis systems, plus the icosahedral axis system, and the two infinite sets of the dipyramids and trapezohedrons.

Of course, if we remove the constraint for identical faces, as DKwan and Bram have suggested, there may be many more possibilities. However, I am not sure we've seen any examples of puzzles that use an axis system other than one of these. The Rhomdo, despite having non-identical faces, still uses the rhombic dodecahedron axis system. Also, I believe that Oskar and Bram's Jumble Prism actually uses the axis system from the trigonal dipyramid, making it the functional equivalent of More Madness (Oskar or Bram, please let me know if this is correct). Can anyone point out an example of a functional jumbling puzzle that does not use one of the already defined axis systems?

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Tue Mar 20, 2012 4:36 pm 
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David Pitcher wrote:
Also, I believe that Oskar and Bram's Jumble Prism actually uses the axis system from the trigonal dipyramid, making it the functional equivalent of More Madness (Oskar or Bram, please let me know if this is correct).

Yup. More Madness has every piece the jumble prism has.

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Tue Mar 20, 2012 5:51 pm 
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David Pitcher wrote:
Can anyone point out an example of a functional jumbling puzzle that does not use one of the already defined axis systems?
This jumbles:

http://www.shapeways.com/model/247067/bubble-block.html

Though I still have yet to completely understand the geometry. Its got 4 axes but are those the same at the tetrahedron/skewb? I'm not sure.

Carl

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Thu Mar 22, 2012 8:18 am 
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wwwmwww wrote:
This jumbles:

http://www.shapeways.com/model/247067/bubble-block.html

Though I still have yet to completely understand the geometry. Its got 4 axes but are those the same at the tetrahedron/skewb? I'm not sure.
The "Bubbles" puzzles are indeed interesting cases, although I'm thinking these might be a fundamentally different type of jumbling. "Bubble Block" and "Bubbles in a Plane" (http://www.shapeways.com/model/245918/bubbles-in-a-plane.html) depend on circular or spherical cuts in order to function. The other "classical" jumbling geometries use straight cuts. (Some puzzles have used curved cuts, but this is only to make the puzzle more user-friendly and to reduce the number of piece types. They could all in theory use straight cuts). So while circular and spherical slices may allow an infinite set of jumbling geometries, I'm still thinking that we may have a finite set (excepting the trivially infinite dipyramid and trapezohedra sets) of "classical" or straight-cut jumbling geometries, or axis systems.

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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Tue Apr 17, 2012 9:27 pm 
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Sorry for the bump here, but I somehow must have missed this thread....

David Pitcher wrote:
... there are 13 of these types of jumbling axis systems, plus the icosahedral axis system, and the two infinite sets of the dipyramids and trapezohedrons.
.
.
.
Can anyone point out an example of a functional jumbling puzzle that does not use one of the already defined axis systems?


Perhaps I misunderstood something, but yes, you can easily construct infinitely many more... :?
What about Oskar's Fairly Twisted, Arrow Planet, and Distorted Cube? Do those count? Or are you excluding them for some reason? There was also a puzzle made about a year ago (?) based on the shape you get when you turn a Rhombic Dodecahedron Skewb puzzle 60 degrees (sorry no time to find picture, creator, or shape-name :oops: hopefully someone will know what I'm talking about). While very similar to the Rhombic Dodecahedron geometry, it isn't identical.

The only requirements to make a jumbling puzzle is to have dihedral angles between a given face and other faces to be equal and cut planes between those two faces to occur at the same depth. A move occurs whenever the direction vector between the rotating face and one face with a given dihedral angle lines up with the original location of the direction vector between the rotating face and another face with the same dihedral angle. If this same rotation causes all vectors to align simultaneously, the puzzle does not jumble. If only some of the vectors align simultaneously, the puzzle jumbles. Note this definition doesn't even require universal symmetry (as shown by Oskar's Fairly Twisted). The systems you have mentioned are only the "cleanest" ones.

Peace,
Matt Galla


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Wed Apr 18, 2012 10:21 am 
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Allagem wrote:
There was also a puzzle made about a year ago (?) based on the shape you get when you turn a Rhombic Dodecahedron Skewb puzzle 60 degrees (sorry no time to find picture, creator, or shape-name :oops: hopefully someone will know what I'm talking about).
You are talking about these two:
http://www.twistypuzzles.com/cgi-bin/pu ... ?pkey=1983
http://www.twistypuzzles.com/cgi-bin/pu ... ?pkey=3117


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 Post subject: Re: Hypothesis: Convex Solids & Jumbling
PostPosted: Wed Apr 18, 2012 11:55 am 
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Allagem wrote:
Perhaps I misunderstood something, but yes, you can easily construct infinitely many more...
What about Oskar's Fairly Twisted, Arrow Planet, and Distorted Cube? Do those count?
Yes, you are quite correct. Looking at Oskar's shop again, there are even more examples, such as the Squashed Cube, and the PentaJumble.

It seems that what I at first thought was revelatory was really just the light turning on in my own head :roll:. If nothing else, at least this thread helped me to understand jumbling much better. Hopefully a few others got a similar benefit as well. I even made a couple of new puzzles based on the ideas discussed here! And as a bonus, Matt has provided an excellent explanation of the phenomenon, which could be quite helpful in developing future puzzles:
Allagem wrote:
The only requirements to make a jumbling puzzle is to have dihedral angles between a given face and other faces to be equal and cut planes between those two faces to occur at the same depth. A move occurs whenever the direction vector between the rotating face and one face with a given dihedral angle lines up with the original location of the direction vector between the rotating face and another face with the same dihedral angle. If this same rotation causes all vectors to align simultaneously, the puzzle does not jumble. If only some of the vectors align simultaneously, the puzzle jumbles.

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