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 Post subject: Re: Compy/Dino CubiodPosted: Mon Mar 28, 2011 9:23 pm

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
GuiltyBystander wrote:
Assuming A<=B<=C, the invariant should be 2*C. If you make it any larger, those extra pieces will be virtually bandaged to the corners.

Going over this thread again, I think you have a point, but your notation is backwards. You seem to be using AxBxC to refer to the amount of overlap in each of the three directions, which has a very counterintuitive relationship to the physical amount in each direction. Really the notation should refer to the number of yellow pieces in each direction, which has a much more obvious meaning.

In your notation the largest number has its overlap exactly equal to the number of pieces. In my notation the smallest number has the overlap exactly equal to the number of pieces. Adding extra pieces to the outside would just be ornamental.

Some examples are -

Compy Cube - 1x1x1 in either notation.

Simplest example - 2x2x1 in your notation, 2x2x3 in mine.

Most interesting variant - 3x3x2 in your notation, 3x3x4 in mine.

I think I was wrong in saying that the 3x4x4 (my notation) isn't all that interesting, but the 3x3x4 already has plenty of pieces and is a big enough physical challenge to build. Probably the best approach to this is to first build a 2x2x3, then a 3x3x4, then declare victory and let other people go crazy building variants.

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 Post subject: Re: Compy/Dino CubiodPosted: Mon Mar 28, 2011 10:24 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
GuiltyBystander wrote:
It seems like you should be able to calculate the max number of pieces per edge just like we can do to show that a cubic 7x7x7 is impossible. What kind of cuts are you using? Just spheres?
It is... I just hadn't done the math yet. And yes... I'm using spheres. So let's do the math.

From here I define some simple shapes in POV-Ray:

declare s = 3;
declare d = s*(1+sqrt(2));

declare core = difference {
box {<-d,-d,-d>,<d,d,d>}
sphere {<d,d,d>, d+s}
}

declare corner = difference {
intersection {
sphere {<d,d,d>, d+s}
box {<-d,-d,-d>,<d,d,d>}
}
sphere {<-d,d,d>, d+s}
sphere {<d,-d,d>, d+s}
sphere {<d,d,-d>, d+s}
}

declare edge = intersection {
box {<-d,-d,-d>,<d,d,d>}
sphere {<d,d,d>, d+s}
sphere {<-d-(2*s-1),d,d>, d+s}
}

So with 3 edges per corner we have s=1.5 which produces this image:

So with 4 edges per corner we have s=2 which produces this image:

So with 5 edges per corner we have s=2.5 which produces this image:

So with 6 edges per corner we have s=3 which produces this image:

While there is contact between the 5th edge and the core at the half turn position it does work on paper but I'm sure it would never work in practice. You can easily see a 6th edge doesn't even work on paper. So I think the limit of this method is 4 edges in per corner. That said we do have working 7x7x7's today so that doesn't mean another method might not work.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Mon Mar 28, 2011 10:49 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
GuiltyBystander wrote:
Assuming A<=B<=C, the invariant should be 2*C. If you make it any larger, those extra pieces will be virtually bandaged to the corners.
What is "the invariant"? I think I'm missing something interesting here.
GuiltyBystander wrote:
I think I was wrong in saying that the 3x4x4 (my notation) isn't all that interesting, but the 3x3x4 already has plenty of pieces and is a big enough physical challenge to build. Probably the best approach to this is to first build a 2x2x3, then a 3x3x4, then declare victory and let other people go crazy building variants.
I think the plan is to offer a set of pieces which will allow you to build the G1x1x2 (B3x3x2) or the G1x2x2 (B3x2x2) by including two cores. After that assuming the mech is stable in practice, I'd like to offer a set which would allow you to build the G1x2x3 (B5x4x3), the G2x3x3 (B4x3x3), or the G2x2x3 (B4x4x3) by including three cores. G=GuiltyBystander's notation and B=Bram's notation.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Mon Mar 28, 2011 11:06 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
boublez wrote:
Is anyone working towards making a printable file?
I will be but I won't have time to start till after the 15th of April. To that end does any one have a copy of SolidWorks they would be willing to sell or donate to the cause? Or is anyone willing to help me turn my POV-Ray code into the needed stl files? You don't need to know POV-Ray. I've been working with Tanner via Skype on my Thorny Cube and he may be able to help me again... not sure. Who ever helps me I'd plan to offer them the prototype for free.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Tue Mar 29, 2011 12:26 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
GuiltyBystander wrote:
I've been thinking of a new way to draw the pieces and just irrational ratios isn't enough to make an infinite number of pieces. I'll get to this new idea in a later post. For now, all I'll say about the new thing is "iterated function system."
IFSs are a cool way to create fractals. The basic idea is that you take parts of an image and copy them to another part. You repeat the process and you end up with a fractal. See the wiki article for some common examples.
When you rotate a corner, you are kind of copying one overlap onto another. This process is very similar to the steps in using an IFS. So I constructed an IFS diagrammed below. However, instead of copying the entire image into each region, I copy each region mirrored onto itself.
Attachment:

IFS.png [ 2.75 KiB | Viewed 8882 times ]
Just repeat the mirroring and you'll end up with what the overlap looks like on the largest overlap.
Attachment:

IFSresult.png [ 3.33 KiB | Viewed 8882 times ]
A*2<B and B*2<C so this is kind of boring. But if you grow B so that B*2>C, the B region with overlap with its mirror in C. I'm dropping A for now but it would get copied in all the mirrors too.
Attachment:

IFS-2BC.png [ 2.9 KiB | Viewed 8882 times ]
I made a java program if you want to play with it at all. The three sliders are for A,B,C. There's no coloring in the program, btw. I had to do that myself in image editing software.
http://landonkryger.com/rubik/DinoCuboid/DinoCuboid.jar

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 Post subject: Re: Compy/Dino CubiodPosted: Thu Mar 31, 2011 2:03 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
Bram wrote:
Going over this thread again, I think you have a point, but your notation is backwards. You seem to be using AxBxC to refer to the amount of overlap in each of the three directions, which has a very counterintuitive relationship to the physical amount in each direction. Really the notation should refer to the number of yellow pieces in each direction, which has a much more obvious meaning.
Both systems have merit. Your system is great for quick identification of the puzzle or if we have multiple cuts. My system is more relevant solve-wise I think because you care about how many pieces move with pairs of turns.

Bram wrote:
I think I was wrong in saying that the 3x4x4 (my notation) isn't all that interesting, but the 3x3x4 already has plenty of pieces and is a big enough physical challenge to build. Probably the best approach to this is to first build a 2x2x3, then a 3x3x4, then declare victory and let other people go crazy building variants.
I was curious about the differences between the B344-G223 and the B334-G233 so I decided to try and look at the differences in orbits again. I'm not sure how useful they are to solving the puzzle, but I think they look neat. I first diagrammed the orbits of the B345-G123 at the bottom of this post.

Before I get to the cool stuff, I thought I start with the Mosaic cube (B222-G222) to show how different and special these "cuboids" are. Green lines are used to indicate that the two points are the same because I can't flatten the 3D wireframe and keep things to scale. I hope you see that these orbits are rather boring. All of the orbits on the Professor's Pyraminx are pretty much the same thing too.
Attachment:

Cuboid-222-FlatOrbits.png [ 4.24 KiB | Viewed 8842 times ]
Next I drew out the orbits of the yellow/blue/orange pieces for two puzzles mentioned above. Like before, they are extremely tangles and hard to decipher from 1 3D render.
Attachment:

Cuboid223-233-WireOrbit.png [ 17.75 KiB | Viewed 8842 times ]
The next step is to flatten them. I didn't color the triangles to show which ones are link, but if that's of interest to someone, I can go back and do that.
Attachment:

Cuboid-223-233-FlatOrbits.png [ 6.86 KiB | Viewed 8842 times ]
Notice the yellow orbit on the B334-G233. The two green connections show that this orbit loop back to itself in two directions (probably describing it wrong but I hope you see what I'm talking about). This is a property normally held by donuts, yet we have it without punching a hole in the puzzle. I think this is a very cool property, even if it doesn't increase difficulty.

Bram mentioned he'd like to see the B223-G122 built early too. If you're curious about those orbits, the yellow/blues of that are the same as the blue/orange of the B334-G233.

I think we need a better name than yellow/blue/orange pieces to highlight the relationships of their different depths. Any suggestions or should we keep calling them by their colors?

wwwmwww wrote:
GuiltyBystander wrote:
It seems like you should be able to calculate the max number of pieces per edge just like we can do to show that a cubic 7x7x7 is impossible. What kind of cuts are you using? Just spheres?
It is... I just hadn't done the math yet. And yes... I'm using spheres. So let's do the math....
Slightly disappointing that this will at best do a B444-G444, but considering it doesn't work at all for a 222 with planar cuts I guess we should just be grateful if we can get any of these to work. They're also so complicated that I don't know if anyone would want a xx4. Gotta try and get gelatinbrain to add one of these cuboids to his site (if it's possible) and see what those solving gurus think of it.

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 Post subject: Re: Compy/Dino CubiodPosted: Thu Apr 07, 2011 3:08 am

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
GuiltyBystander, I'm fascinated by your illustration of the connectivity graph of the orbits. It allows us to throw away the geometric forms to think about the topology. Although before using it in solving, one has to figure out which node corresponds to which piece on the puzzle, it's a pleasure simply looking at the graph. In the past two hours I wrote some Mathematica script to generate the graph for yellow nodes (edge pieces) for general size {k,m,n}. I specified the connectivity of the graph, and Mathematica automatically placed the nodes in a certain way in 3D space (it can also be 2D).

Let's see some graphs.

G233, is what you have shown. There are two orbits, thus two copies of the same structure. Each structure has two "holes". Those are your findings. This rendering doesn't make it like a donut.
Attachment:
File comment: G233

G233.png [ 13.68 KiB | Viewed 8792 times ]

Other graphs, please see the size in the file name or comment.

Attachment:
File comment: G234

G234.png [ 13.28 KiB | Viewed 8792 times ]

Attachment:
File comment: G235

G235.png [ 11.68 KiB | Viewed 8792 times ]

Attachment:
File comment: G244

G244.png [ 15.38 KiB | Viewed 8792 times ]

Attachment:
File comment: G335

G335.png [ 13.7 KiB | Viewed 8792 times ]

Attachment:
File comment: G344

G344.png [ 15.48 KiB | Viewed 8792 times ]

and for G115, (there was a bug in the script last night)
Attachment:
File comment: G115

G115.png [ 11.82 KiB | Viewed 8727 times ]

Finally, here's the Mathematica code. Sorry for not adding much comments in the code.

Code:
Gsize = {2, 2, 3}; (*GuiltyBystander's notation*)
maxGsize =  Max[Gsize];
Bsize = 2 maxGsize - Gsize;(*Bram's notation*)
cuboidsize =  Bsize + 2;
direction[coordinate_, k_] :=
Table[If[i == k, 1 - 2 coordinate[[i]], 0], {i, 1, 3}]
cornerindex = {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}, {0, 1, 1}, {1, 0,
0}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1}};
cornercoordinate = 1 + cornerindex*Table[cuboidsize - 1, {i, 8}];
(* Generate edges *)
edges = Flatten[
Table[Table[
Table[ToString[{cornercoordinate[[k]] +
j direction[cornerindex[[k]], i]}] ->
ToString[{RotationMatrix[2 \[Pi]/3,
cornerindex[[k]] - {1/2, 1/2, 1/2}].(j direction[
cornerindex[[k]], i]) + cornercoordinate[[k]]}], {i,
3}], {j, 1, maxGsize}], {k, 1, 8}], 2];
GraphPlot3D[edges,
PlotStyle -> {Thick, PointSize[0.02], EdgeForm[{Blue}]}]

Note that in these graphs, we only focus on the positions of pieces not the orientation. In an actual puzzle the pieces are very likely orientation-sensitive. In that way each node should be split into two to distinguish orientations.

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Last edited by schuma on Thu Apr 07, 2011 11:07 am, edited 2 times in total.

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 Post subject: Re: Compy/Dino CubiodPosted: Thu Apr 07, 2011 7:57 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
schuma wrote:
and for G115, all edges are in one weird orbit.
I think I must be missing something. I thought the G115 was basically the same puzzle as the G112 or the G11n for that mater. Aren't most of the edges locked to the pieces below them. That and I thought there were two orbits on the G112.
schuma wrote:
Note that in these graphs, we only focus on the positions of pieces not the orientation. In an actual puzzle the pieces are very likely orientation-sensitive. In that way each node should be split into two to distinguish orientations.
I don't think its possible to get a given edge piece in both orientations in a given position on most (if not all of these). Not sure I can prove that in general but I *think* that is the case.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Thu Apr 07, 2011 11:13 am

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
wwwmwww wrote:
schuma wrote:
and for G115, all edges are in one weird orbit.
I think I must be missing something. I thought the G115 was basically the same puzzle as the G112 or the G11n for that mater. Aren't most of the edges locked to the pieces below them. That and I thought there were two orbits on the G112.

Hi Carl, you are right about it. I just realized my script last night suffers from a "overflow" problem when I tried to convert a string of numbers to a string of characters. Only G115 exhibits the problem. I've corrected the graph and code in my last post. Indeed, there are many small orbits for G115, meaning these pieces cannot go far. The two large orbits are as same as G112. Thank you for catching the bug.

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 Post subject: Re: Compy/Dino CubiodPosted: Thu Apr 07, 2011 12:31 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
wwwmwww wrote:
I don't think its possible to get a given edge piece in both orientations in a given position on most (if not all of these). Not sure I can prove that in general but I *think* that is the case.

I think you are also right about orientations. The reason is just like Dino cube. Whenver an edge piece goes to the original edge (not necessarily the same exact position on that edge), the orientation has to be correct.

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 3:12 am

Joined: Mon Nov 30, 2009 1:03 pm
Hi Twisty Puzzle Friends,

I believe that the offset-axes concept by Boublez can be generalized, see the first picture below.

I am not sure how to describe the offset-axes concept mathematically. Perhaps Bram or Carl can provide a concise definition like for jumbling. It starts from a regular geometry like a cube or cuboctahedron with turning axes that go through the corners (vertex-turning). Then the axes are offset in such a way that
1) all adjacent axes keep going through a single point and
2) they do this in a symmetrical way.
The second picture below shows a simple offset-axes geometry that fails the first criterion. The third picture shows a geometry that satisfies the first criterion, but fails the second, as adjacent access are not crossing to the single point in a symmetrical way. The geometry of the offset-axes cuboctahedron of the first picture satisfies both criteria, so it is a viable Boublez geometry.

Oskar
Attachment:

Offset-axis Cuboctahedron v1a - view 1.jpg [ 130.9 KiB | Viewed 8290 times ]

Attachment:

Offset-Axes Cube geometry.jpg [ 61.65 KiB | Viewed 8670 times ]

Attachment:

Offset-Axes Cube geometry - view 2.jpg [ 57.92 KiB | Viewed 8666 times ]

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Last edited by Oskar on Sun Apr 10, 2011 11:21 am, edited 1 time in total.

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 7:27 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
I believe that the offset-axes concept by Boublez can be generalized, see the first picture below.
You are certainly correct. I was hit with the idea of trying this on a dodecahedron about a week ago but haven't had the chance to work out the geometry. I'm not 100% sure this is possible on a dodecahedron but I was pretty sure it was applicable to other geometries.

Though I can't say I saw this shape coming. The un-Boublezized puzzle of the other puzzles in this thread is the Dino Cube. The un-Boublezized version your your proposed puzzle is a Dino Cuboctahedron isn't it? Has that base puzzle even been built yet?

Ok... a few lights go off... the Dino Cuboctahedron would be a shape mod of a Helicopter Cube minus the corners... wouldn't it? And as such it jumbles and SO does this puzzle proposed by Oskar. The first jumbling Boublezized puzzle. COOL!!!!

By the way, the core should be red, the corners green, the edges yellow, and the other pieces orange. We've sort of been using that to name the pieces in this thread until better names surface. Just being obsessive compulsive there.
Oskar wrote:
I am not sure how to describe the offset-axes concept mathematically. Perhaps Bram or Carl can provide a concise definition like for jumbling.
I think you did a pretty good job. There are probably a few rules about the geometries that this can be applied to and those which it can't. For example I'm pretty sure this can't be done starting with a tetrahedron and just using 4 corners... at least without some fudging. I still think this Boublezation of a puzzle is closely related to siamese puzzles. Each set of corners that shares a common origin can be though of as one puzzle that is siamesed with the others.

So here is a challenge for you Oskar... can you make a Boublezized tetrahedron? I'm pretty sure fudging will be required but as it will have fewer pieces I would think it would make a much more afordable puzzle on shapeways. Not that I don't want to see the Boublezized Dino Cuboctahedron there too. Just that money is tight at the moment and I'm pretty sure I couldn't aford it at the moment.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 8:04 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
... The first jumbling Boublezized puzzle. COOL!!!! ... can you make a Boublezized tetrahedron? ...
Carl,

Thanks. Can you find a geometry for Boublezizing a deep-cut jumbling Olzed puzzle that requires fudging?

Oskar

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 9:43 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Thanks. Can you find a geometry for Boublezizing a deep-cut jumbling Olzed puzzle that requires fudging?
The simplest deep-cut jumbling puzzle is the 24-Cube I believe. Its an edge turing cube so it has the same symmetry as the Dino Cuboctahedron. So what happens when you take the cuts in such that they pass through the origin of their respective corners and are planner on your Boublezized Dino Cuboctahedron? Deep cut is only well defined with planar cuts. It should be easy enough to make the picture. The puzzle on the other hand is another story. The 24-Cube by itself has a mech that I still don't fully understand and to Boublezize that mech is going to be insane.

As for Olzing... do we have a good definition of that term? To me its simply allowing the cuts that allow the puzzle to function to be visible on the surface of the puzzle. Could the version of the Mosaic Cube made with spherical cuts visable on the surface be considered an Olzed Mosaic Cube? If so then your Boublezized Dino Cuboctahedron above is already Olzed.

As for requiring fudging, I think you are back to the tetrahedron. So start with a tetrahedron... grab two corners with one hand and the other two corners with the other hand and pull slightly. The edges connecting the two corners held in the same hand should now have 2 edge pieces on them and the others should have 3 edge pieces. This is your Boublezized Olzed puzzle that requires fudging. Replace the spherical cuts with planar cuts such that each planar cut about a corner goes through the point its axis crosses the axis of its nearest neighbor. You are in effect making a siamese version of two Halpern-Meier Tetrahedrons. If this can be made I'd say you have everything you want except the jumbling.

You want jumbling than you are back to doing the same to the Boublezized Dino Cuboctahedron but that shouldn't require fudging. Infact I'm not even sure you can mix fudging and jumbling in the way you are thinking. See the definition of fudging that I proposed here:

http://twistypuzzles.com/forum/viewtopic.php?p=245885#p245885
wwwmwww wrote:
Fudged Puzzle = a jumbling puzzle which has been made doctrinaire via the removal of an infinite number of tiny pieces created through an iterative planar cutting process. Pieces which remain that have gained an infinite number of orientation states must either remain circular thus having NO orientation to remain doctrinaire OR they may be shape altered using the empty space left by the removed pieces to give them a finite number of orientation states. The consequence of the later is that a turn of the puzzle that contains one of these such pieces doesn't move as one unified whole. The pieces are loosely held together allowing the pieces within the layer to change their relative position/orientation slightly during a turn.
If correct ALL fudged puzzles are doctrinaire. Jumbling puzzles are not. So you can't really have a jumbling fudged puzzle if my definition is valid... not 100% sure that it is. Let's assume you start with a jumbling puzzle and start this process of making it doctrinaire but stop short. What do you have? A partially fudged jumbling puzzle?

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 11:13 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
boublez wrote:
So I was sitting in a conference when this thought crossed my mind. I drew up a quick sketch, then transferred the thought to google sketch up.
Wow!! So much has now grown from this idea through such a collaborative effort that I have to ask... what was this conference you were sitting in about? Just more curious then anything...

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 11:29 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
wwwmwww wrote:
Though I can't say I saw this shape coming. The un-Boublezized puzzle of the other puzzles in this thread is the Dino Cube. The un-Boublezized version your your proposed puzzle is a Dino Cuboctahedron isn't it? Has that base puzzle even been built yet?
Is that what a DaYan Gem Cube is? A shape mode of a Dino Cuboctahedron?

Oskar, what would your Boublezized Dino Cuboctahedron look like with planar cuts? Can you make that image?

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 5:37 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
GuiltyBystander or schuma,

Could you take Oskar's Boublezized Dino Cuboctahedron above and plot out the orbits as you have done for the Cuboids? First limit the puzzle to 180 degree turns, then allow the jumbling turns, how does that change the orbits? I'm curious if that allows edges to flip orientation or jump from one orbit to another.

Oskar,

How many different types of edges does the Cuboctahedron geometry allow. For example with the Cuboids we can have 3 different types of edges. The G1x2x3 has 5 pieces along one set of edges, 4 pieces along another, and 3 along the remaining edges. On your Cuboctahedron I see edges with 2 pieces and edges with 3 pieces, but I'm curious if that's a limitation of the geometry or can you have more then two types? If yes, how many?

Thanks,
Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Fri Apr 08, 2011 7:10 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
wwwmwww wrote:
GuiltyBystander or schuma,

Could you take Oskar's Boublezized Dino Cuboctahedron above and plot out the orbits as you have done for the Cuboids? First limit the puzzle to 180 degree turns, then allow the jumbling turns, how does that change the orbits? I'm curious if that allows edges to flip orientation or jump from one orbit to another.

Orbits for only 180 degree turns:

Each cyan piece can only be in an orbit of three positions. The topology is of the orbit is like:

a -- b -- c

There are 12 such orbits.

Each red piece can only be in an orbit of five positions. The topology is like:

a -- b -- c -- d -- e

There are also 12 such orbits.

It's not possible to flip an edge in place.

With jumble: (I hope my understanding of jumble for this puzzle is correct)

All red pieces are in a huge orbit. It is possible move a piece to its original position with orientation flipped. The topology of the orbit is unknown.

Each cyan piece can only be in an orbit of nine positions. The topology of the orbits is a triangle graph (3 nodes pairwise connected) + two extra nodes connected to each node on the triangle (6 extra nodes in total). The extra nodes are not connected to anything else.

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 12:23 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
So here is a challenge for you Oskar... can you make a Boublezized tetrahedron?
Carl,

Here you are. The first picture and this 3D PDF show the full puzzle on a Jaap's Sphere. The second and third picture and this 3D PDF illustrate its geometry.
Yes, it jumbles.
No, it is not fudged.
As for classification, this would be a T112233 (T=Tetrahedron).

Oskar

Note that you can click and drag the 3D PDF to get views from all sides.

TetraBoublez:
Attachment:

Tetraboublez v2 - view 1.jpg [ 83.12 KiB | Viewed 8502 times ]

Geometry:
Attachment:

Tetraboublez - view 1.jpg [ 82.82 KiB | Viewed 8506 times ]

Attachment:

Tetraboublez - view 2.jpg [ 68.55 KiB | Viewed 8506 times ]

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Last edited by Oskar on Sun Apr 10, 2011 11:13 am, edited 3 times in total.
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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 1:11 am

Joined: Mon Nov 30, 2009 1:03 pm
Oh, and here is FlatBoublez. With a bit of fantasy, one can wrap this around a sphere.
As for classification, how about F53342 (F=Flat)?

Oskar
Attachment:

FlatBoublez - view 1.jpg [ 114.92 KiB | Viewed 8498 times ]

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 2:51 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Here you are. The first picture and this 3D PDF show the full puzzle on a Jaap's Sphere. The second and third picture and this 3D PDF illustrate its geometry.
Yes, it jumbles.
No, it is not fudged.
As for classification, this would be a T112233 (T=Tetrahedron).

WOW!!! Color me impressed. I see one corner moves with 2 edges in all directions and the others move with 3. So much for my idea of order to these puzzles. Does that also mean all 4 axis of rotation cross at a single point? If so it also throws my notion of thinking of these as siamese puzzles out of the window. And still NOT fudged... TOO COOL!!! My mind is already officially blown... and you add to this that it jumbles!!!! This only has 4 axis of rotation and only 1 cut plane per axis. This HAS to be a record for the minimum number of cut planes needed to make a jumbling 3D twisty puzzle. The next simpliest I can think of is the 24-Cube which has 6 axis of rotation with 1 cut plane per axis... which actually is anything but simple to make.

Is it possible to make a jumbling 3D twisty puzzle with just 3 cut planes? Some weird cousin of the 2x2x2?

Oh, and the obsessive compulsive in me noticed you even got the colors correct!!! LOL!!!

This MUST be on your list for Shapeways and considerings Uwe Meffert's love of the Pyraminx/Tetrahedron I can't see the world's first Jumbling Pyraminx not making it to mass production status at some point.

WOW!!!
Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 3:15 am

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Oskar wrote:
Oh, and here is FlatBoublez. With a bit of fantasy, one can wrap this around a sphere.
As for classification, how about F53342 (F=Flat)?

Took me a while to see it with that bit of fantasy but this isn't a tetrahedron. This I think has six axes of rotation all in the same plane and which all cross at the center of the sphere. The yellow edge pieces are 360/17 degrees wide on the equator... correct? The last 2 yellow pieces in the pic overlap with the first 2 I believe.

This makes me think of your Number Planet puzzle for some reason. Didn't you once have a second Number Planet like puzzle in your shop? I could only find one.

I think if you add a cut plane through the equator it would turn this into a jumbling puzzle too. Another very interesting find.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 11:47 am

Joined: Wed May 13, 2009 4:58 pm
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schuma wrote:
GuiltyBystander, I'm fascinated by your illustration of the connectivity graph of the orbits. It allows us to throw away the geometric forms to think about the topology. Although before using it in solving, one has to figure out which node corresponds to which piece on the puzzle, it's a pleasure simply looking at the graph. In the past two hours I wrote some Mathematica script to generate the graph for yellow nodes (edge pieces) for general size {k,m,n}. I specified the connectivity of the graph, and Mathematica automatically placed the nodes in a certain way in 3D space (it can also be 2D).

Let's see some graphs.

G233, is what you have shown. There are two orbits, thus two copies of the same structure. Each structure has two "holes". Those are your findings. This rendering doesn't make it like a donut.
Great pics. And yeah, I notices that the "donutness" disappear on some of my later rearrangements too. I'm kind of confused why. I'm not crazy about initially thinking it looked like a donut right?
I'm guessing it has something to do with it only having 1 point loop back in each direction. Or maybe it's because it's a finite set of points and not a continuous surface. Are there any topology experts on these forums? I was going to say Pantazis, but I just found a quote of him saying "I am no expert in topology."

wwwmwww wrote:
How many different types of edges does the Cuboctahedron geometry allow. For example with the Cuboids we can have 3 different types of edges. The G1x2x3 has 5 pieces along one set of edges, 4 pieces along another, and 3 along the remaining edges. On your Cuboctahedron I see edges with 2 pieces and edges with 3 pieces, but I'm curious if that's a limitation of the geometry or can you have more then two types? If yes, how many?
Good question. It's quite easy to see how to stretch a cube in 3 directions. I'm still confused how the cuboctohedron was squished but I trust that it works. Oskar, can you show us a rendering of each of the axis and where the pairs intersect?

I've been thinking back to my first post in this topic and Oskar's proposed definition/properties.
GuiltyBystander wrote:
Cool idea. Similar to wwwmwww's Uniaxial 3x3x3 and Timur's F-Skewb except you unbalanced the cuts in very cool different way.
Oskar wrote:
1) all adjacent axes keep going through a single point and
2) they do this in a symmetrical way.
I see another big differences between the puzzles I mention and the initial Dino Cuboids. On the Dino Cuboid, pairs of axes have their cuts equidistant from the intersection point. Carl's Uniaxial 3x3x3 doesn't keep this property but it maintains Oskar's points 1 & 2. Do you have this property on the Cuboctohedron and Tetrahedron? Can you unbalance the cut depths to achieve the same effects as moving the axis?

One last thought in this post. So far, all of the pieces we've looked at are essentially edges. At any one time, they can only be move by 2 different axes. Can we ever see corners (pieces that can be move by 3 different axes) on these cuboids? Looking back at Oskar's point #1, would we have to modify that to be that "adjacent triplets of axes keep going through a single point?" It seems like that would just force all axes to go through the same point. Perhaps a bit of fudging is necessary to get away from that requirement.

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 11:55 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
wwwmwww wrote:
GuiltyBystander or schuma,

Could you take Oskar's Boublezized Dino Cuboctahedron above and plot out the orbits as you have done for the Cuboids? First limit the puzzle to 180 degree turns, then allow the jumbling turns, how does that change the orbits? I'm curious if that allows edges to flip orientation or jump from one orbit to another.
schuma wrote:
Orbits for only 180 degree turns:...
Found a cool way to show them. Here's a skewb like cut showing some of the orbit overlap. If you did this on a helicopter cube, you'd see those center piece orbits as a hexagon. There are 4 of these orbit sets and each set has 3 orbits in them.
Attachment:

NoJumble.png [ 3.75 KiB | Viewed 8455 times ]
schuma wrote:
Each cyan piece can only be in an orbit of nine positions. The topology of the orbits is a triangle graph (3 nodes pairwise connected) + two extra nodes connected to each node on the triangle (6 extra nodes in total). The extra nodes are not connected to anything else.
With jumbling, our blue, Oskar's cyan, looks like this. Pretty simple. Should be 4 of them I think.
Attachment:

Jumble-Blue.png [ 9.5 KiB | Viewed 8455 times ]
schuma wrote:
All red pieces are in a huge orbit. It is possible move a piece to its original position with orientation flipped. The topology of the orbit is unknown.
Yes, it is indeed a huge orbit. Not sure if we still call them orbits if there's only one, but oh well. With jumbling, our yellow, Oskar's red, looks like this.
Attachment:

Jumble-Yellow.png [ 8.95 KiB | Viewed 8455 times ]
And flattening attempt #1
Attachment:

Jumble-Yellow-Flat.png [ 8.09 KiB | Viewed 8455 times ]
Looks more like a plate of spaghetti that I'd rather not mess with. So to make it easier, I mess around with it's dual instead.
Attachment:

Jumble-Yellow-Dual.png [ 6.96 KiB | Viewed 8455 times ]
Ah, much better. Now we can take the dual of the dual and see it better. I've added 3 red lines to show some non-jumbling orbits.
Attachment:

Jumble-Yellow-Flat2.png [ 8.35 KiB | Viewed 8455 times ]

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 12:00 pm

Joined: Mon Nov 30, 2009 1:03 pm
And here if the F123 (F=Flat) geometry wrapped around a sphere, see picture below and 3D PDF. Note that you can click and drag a 3D PDF to get views from all sides.

Oskar
Attachment:

BubbleTriangle.jpg [ 83.08 KiB | Viewed 8452 times ]

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 12:30 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
And here if the F123 (F=Flat) geometry wrapped around a sphere, see picture below and 3D PDF.

Ok... just just broke the record you set yesterday. A jumbling puzzle with only 3 cut planes. I want to call this one the Jumbling Cheese. Looking at this I think ALL its moves are jumbling so I'm not sure how much this could actually be scrambled.

Dare I say that I think its impossible to make a jumbling puzzle with just 2 cut planes. I now half expect that to be proven wrong tomorrow.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 12:39 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
GuiltyBystander wrote:
schuma wrote:
Each cyan piece can only be in an orbit of nine positions. The topology of the orbits is a triangle graph (3 nodes pairwise connected) + two extra nodes connected to each node on the triangle (6 extra nodes in total). The extra nodes are not connected to anything else.

schuma wrote:
All red pieces are in a huge orbit. It is possible move a piece to its original position with orientation flipped. The topology of the orbit is unknown.

Yes, it is indeed a huge orbit.
<SNIP>
I've added 3 red lines to show some non-jumbling orbits.

Very very cool!! Now a few questions...
(1) Where are these quotes of schuma coming from? I feel like I'm missing part of the conversation.
(2) Does that one huge orbit show two spots for each position, one for each orientation?

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 12:56 pm

Joined: Wed May 13, 2009 4:58 pm
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wwwmwww wrote:
Very very cool!! Now a few questions...
(1) Where are these quotes of schuma coming from? I feel like I'm missing part of the conversation.
(2) Does that one huge orbit show two spots for each position, one for each orientation?
1. viewtopic.php?p=253715#p253715
2. The huge orbit I drew shows 1 spot for each position. This means that if you're trying to draw a graph for all states (position + orientation), it'd be twice as big

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 1:12 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
GuiltyBystander wrote:
My bad... looks like I missed that post somehow.
GuiltyBystander wrote:
2. The huge orbit I drew shows 1 spot for each position. This means that if you're trying to draw a graph for all states (position + orientation), it'd be twice as big
Could you draw the graph for all states? Just curious... Not trying to ask for alot of work. It's just I don't have a clue how you are making these and I find them fasinating.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 1:15 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Dare I say that I think its impossible to make a jumbling puzzle with just 2 cut planes. I now half expect that to be proven wrong tomorrow.
Carl,

You had already been proven wrong when Bram and I published our paper "Puzzles that Jumble" in Cubism For Fun and at the Twisty Puzzles Forum last year. The drawing below illustrates a vain attempt to unbandage this extremely simple jumbling puzzle with only two cut planes.

Oskar
Attachment:

Unbandaging.jpg [ 49.62 KiB | Viewed 8416 times ]

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 1:21 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
You had already been proven wrong when Bram and I published our paper "Puzzles that Jumble" in Cubism For Fun and at the Twisty Puzzles Forum last year. The drawing below illustrates the vain attempt to unbandage this extremely simple jumbling puzzle with only two cut planes.
I was just about to counter and say that's a 2D puzzle and not a 3D twisty puzzle but I guess those could just as easilly be planar cuts on a sphere. Well done.

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 2:00 pm

Joined: Thu Jul 23, 2009 5:06 pm
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Oskar wrote:
Oh, and here is FlatBoublez. With a bit of fantasy, one can wrap this around a sphere.
As for classification, how about F53342 (F=Flat)?

Oskar

Attachment:

FlatBoublez - view 1.jpg [ 114.92 KiB | Viewed 8403 times ]

I like the flat version very much. I think it's pretty neat. A quick analysis of the pieces is as follows:

There are 6 types of pieces: green, cyan, red, orange, blue, and yellow. They can be classified into 2 categories:

Category-1: Green, red, and blue. These pieces can be flipped in place. There is an odd-orbit and an even-orbit. That is, if the positions are numbered from left to right (gap is also numbered), then the positions with the odd numbers are in the odd-orbit and the same thing for the even-orbit.

Pieces with different colors have different range of movement: Blue pieces can run the farthest, from all the way to the left to all the way to the right. Red pieces are limited into two clusters. Green ones doesn't move at all, but only flips in place.

Category-2: Cyan, orange, and yellow. There is no parity constraint for the positions. They cannot be flipped in place, that is, once a piece is in the correct location, its orientation is automatically correct. Actually if an odd piece goes to any even position, the orientation is reversed and if it goes to any odd position, the orientation is correct.

Again, pieces with different colors have different range of movement. Yellow can go anywhere. Orange can go anywhere except for the circle at the right end (coz it doesn't have orange pieces). Cyan is constrained in the left cluster (well, you can find them only there).

It looks like a fun puzzle to solve. I can see a pure 3-cycle algo with 8 moves ([3,1]) for the yellow pieces.

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 2:10 pm

Joined: Thu Jul 23, 2009 5:06 pm
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GuiltyBystander wrote:
Ah, much better. Now we can take the dual of the dual and see it better. I've added 3 red lines to show some non-jumbling orbits.

What exactly do you mean "dual of dual" of the cuboctahedron? When I analyzed the orbits of it, I drew this kind of projection on a piece of paper and worked on it.

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 2:57 pm

Joined: Sun Dec 06, 2009 9:00 am
wwwmwww wrote:
boublez wrote:
So I was sitting in a conference when this thought crossed my mind. I drew up a quick sketch, then transferred the thought to google sketch up.
Wow!! So much has now grown from this idea through such a collaborative effort that I have to ask... what was this conference you were sitting in about? Just more curious then anything...

Carl

I was at a conference for my job, I work for a energy efficiency/reduction. It was a best practice for air sealing and insulating a building. I'm not sure how my mind wondered onto cubes but then again it does that often.

To be honest with you all this has kind of gotten way over my head. You are all very bright people with a much better understanding of designing and creating process then me. I have read every post to this thread since it's started and I will continue to but as things progress I am understanding less and less. I think this forum needs a dictionary, so that we can exact definitions for words like fudging, olzing, and jumbling, readily available for everyone to see.

I am very interested to see how much further this idea will go. I've toyed around with the idea of edge turning cuboids but that would be a really bandaged puzzle ... or a really fudged one. I'll leave that up to you to figure out.

Thank you everyone for there input.

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 Post subject: Re: Compy/Dino CubiodPosted: Sat Apr 09, 2011 6:52 pm

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
wwwmwww wrote:
GuiltyBystander wrote:
2. The huge orbit I drew shows 1 spot for each position. This means that if you're trying to draw a graph for all states (position + orientation), it'd be twice as big
Could you draw the graph for all states? Just curious... Not trying to ask for alot of work. It's just I don't have a clue how you are making these and I find them fasinating.
I'll try to show my process showing the entire state space for the blue/cyan pieces.
1. I start by drawing the positions. Here I drew lines to represent the 9 possible positions and I will be using each end to represent a sticker so that we can distinguish orientation.
2. Then I draw the connections between each position.
3. Next I turn on the control points of the lines so I can drag them around.
4. I keep tugging and pulling on the points until it looks like something reasonable. I don't really have a defined method, I just keep moving them. It's like trying to untangle a ball of string.
Attachment:

Jumble-Blue+Orient.png [ 5.53 KiB | Viewed 8355 times ]

Trying to untangle the yelow/red one seems like a nightmare. I have a hunch on an easy way to do it but I'm not positive yet.

schuma wrote:
GuiltyBystander wrote:
Ah, much better. Now we can take the dual of the dual and see it better. I've added 3 red lines to show some non-jumbling orbits.
What exactly do you mean "dual of dual" of the cuboctahedron? When I analyzed the orbits of it, I drew this kind of projection on a piece of paper and worked on it.
I don't mean the "dual of dual" of the cuboctahedron, I mean the "dual of dual" of the position graph of pieces on the cuboctahedron. I'm not sure where you're getting lost, so I'll just start at the beginning so it's clear for everyone else too.
First when I say dual, I mean like a dual polyhedron. When you find a dual of a polyhedron, faces become vertices and vertices become faces. The dual of the cube is a octahedron and the dual of a octahedron is a cube. The dual of a dual of X should be X.
But our connected graph of the orbits isn't a polyhedron so the definition of "dual" has to change a bit. I hope I'm not stepping on any Math ppls toes here. In a normal orbit graph we've had so far, vertices represent positions a piece can be and edges mean that a piece can go from one position to another. Example
In my dual graph (left side of example), I'm almost reversing these roles. First imagine what a jumbling piece's orbit looks like when you're only using 1 axis. It's a simple quadrilateral. In my dual graph, I take ever one of these and turn it into a vertex. Edges in my dual graph now represent positions that are shared by two vertices.
This dual basically represents the same thing but in a different form. The one fault with this version is that it can't represent positions that are only on one turning axis. Actually, you could just have a line leading out of a vertex that connects to nothing else, but I've choosen to leave it out. The advantage of this dual representation is that it has much fewer vertices so it's easier to play with. Once I get it sorted out (right side of example), I can just undo the dual process and get back to the graph of positions (example).

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 2:36 am

Joined: Thu Jul 23, 2009 5:06 pm
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GuiltyBystander wrote:
I don't mean the "dual of dual" of the cuboctahedron, I mean the "dual of dual" of the position graph of pieces on the cuboctahedron. I'm not sure where you're getting lost, so I'll just start at the beginning so it's clear for everyone else too.
First when I say dual, I mean like a dual polyhedron. When you find a dual of a polyhedron, faces become vertices and vertices become faces. The dual of the cube is a octahedron and the dual of a octahedron is a cube. The dual of a dual of X should be X.
But our connected graph of the orbits isn't a polyhedron so the definition of "dual" has to change a bit. I hope I'm not stepping on any Math ppls toes here. In a normal orbit graph we've had so far, vertices represent positions a piece can be and edges mean that a piece can go from one position to another. Example
In my dual graph (left side of example), I'm almost reversing these roles. First imagine what a jumbling piece's orbit looks like when you're only using 1 axis. It's a simple quadrilateral. In my dual graph, I take ever one of these and turn it into a vertex. Edges in my dual graph now represent positions that are shared by two vertices.
This dual basically represents the same thing but in a different form. The one fault with this version is that it can't represent positions that are only on one turning axis. Actually, you could just have a line leading out of a vertex that connects to nothing else, but I've choosen to leave it out. The advantage of this dual representation is that it has much fewer vertices so it's easier to play with. Once I get it sorted out (right side of example), I can just undo the dual process and get back to the graph of positions (example).

Thank you for explaining it. Now I understand it. The analysis of position graph looks interesting. But I believe it can be improved:

One turn on the puzzle moves several pieces SIMULTANEOUSLY. The simultaneity is usually why a puzzle is hard to solve. This property is never reflected in the current version of the position graph. For example, on your "dual of dual" graph, there are many quadrilaterals. Each turn rotates not only one quadrilaterals, but two quadrilaterals. Therefore all the quadrilaterals can be grouped into pairs, and the two quadrilaterals in a pair are always "synchronized". Once we identify the pairs (and label the relative directions of rotation) on the position graph, it is sufficient to simulate all the red pieces in the puzzle (let's pretend the pieces are orientation-insensitive). At that time we can throw away the puzzle and focus on the position graph. I think the graph should be more clear than the original puzzle, because the graph is untangled.

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 10:51 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
schuma wrote:
Thank you for explaining it. Now I understand it. The analysis of position graph looks interesting. But I believe it can be improved:

One turn on the puzzle moves several pieces SIMULTANEOUSLY. The simultaneity is usually why a puzzle is hard to solve. This property is never reflected in the current version of the position graph. For example, on your "dual of dual" graph, there are many quadrilaterals. Each turn rotates not only one quadrilaterals, but two quadrilaterals. Therefore all the quadrilaterals can be grouped into pairs, and the two quadrilaterals in a pair are always "synchronized". Once we identify the pairs (and label the relative directions of rotation) on the position graph, it is sufficient to simulate all the red pieces in the puzzle (let's pretend the pieces are orientation-insensitive). At that time we can throw away the puzzle and focus on the position graph. I think the graph should be more clear than the original puzzle, because the graph is untangled.
This is kind of what I did on my first orbit diagrams except I forgot about rotation directions then.
While attempting to do this for the the cubeoctahedron, I realized that I majorly screwed up the position order. When converting from the dual back to the normal mode, (on the dual) the edged going into a vertex don't have and order. When I was doing triangles, it didn't matter, but on quadrilaterals, it does. Consequently, half of the quadrilaterals in my last picture should look like bowties instead of squares/retangles. I hope to have a picture this afternoon, but I'm kind of sad that it isn't going to be as pretty. And I'm kind of doubting that I can untangle all of the bowties.

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 11:05 am

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
... ALL fudged puzzles are doctrinaire. Jumbling puzzles are not. So you can't really have a jumbling fudged puzzle if my definition is valid... not 100% sure that it is.
Carl,

Here is your counter example, see picture below and the 3D PDF.
This geometry jumbles.
The puzzle is fudged.
Look at the orange triangles, which I fudged into equilateral ones. As a result, there are some cracks between some pieces. All fudged puzzles have such cracks.

Oskar

Note that you can click and drag the 3D PDF to get views from all sides.
Attachment:

Tetraboublez v4 - view 1.jpg [ 82.08 KiB | Viewed 8299 times ]

Attachment:

Tetraboublez v4 - view 2.jpg [ 82.95 KiB | Viewed 8299 times ]

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 11:51 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Here is your counter example, see picture below and the 3D PDF.
This geometry jumbles.
The puzzle is fudged.
Did you edit this? I was sure it said "This geometry jumbles as any boublezized geometry..." when I clicked the quote button. Which isn't ture. Look at the cuboids. Though if this said that I guess you already saw that and thus why it was edited. Either that or I was seeing things.
Oskar wrote:
Look at the orange triangles, which I fudged into equilateral ones.
Hmmm... You've done something very weird here and I'm still not sure what to make of it. The orange pieces in all the above puzzles have two surface areas. They run between two blue pieces and under a yellow piece. That doesn't appear to be the case here.
Attachment:

JumbleFudge.png [ 68.32 KiB | Viewed 8275 times ]

Is the orange piece at the black arrow tied to the one at the red arrow or the purple arrow. I'm now guessing neither. Actually in the 3D PDF the purple arrow points to a white space. Is that a void on the surface of this puzzle?
Do those pieces need to be equilaterial triangles for this puzzle to function? If not but they do partially unbandage it then I'd say this is something along the lines I proposed in my definition above. You've started to take a few steps towards turning this into a doctrinaire puzzle but have stopped short. If that is the case, I still can't quite see what is actually going on with this puzzle, then I'd be tempted to call this "partially fudged". It would be a third class which falls between fudged puzzles and jumbling puzzles. For clarity maybe I need to change my definition of "fudged" to that of "fully fudged".

Do you have a mech for this puzzle? Or can you even cut up the sphere into this puzzles basic pieces? I don't need a mech to hold them together... just assume its held together by a clear sphere on the outside. I'm not even sure if I'd call this boublezized or something else... because at first glance it certainly appears something else is going on here.

Carl

P.S. Studying the 3D PDF a bit more it appears your orange piece doesn't relate to the orange pieces above at all. It looks like you are using cuts a bit deeper then typically allowed in the boublezized geometry and you have blue pieces intersection other blue pieces. This intersection is then cut off both pieces to make a new piece. It's still not clear to me if these new pieces can be rotated by 120 degrees or why they need to be fudged.

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 12:00 pm

Joined: Mon Nov 30, 2009 1:03 pm
Carlc,
wwwmwww wrote:
Did you edit this?
Yes, I did. Not every Boubez geometry jumbles. A counter example is a flat Boubez geometry wrapped around a sphere.
wwwmwww wrote:
Is that a void on the surface of this puzzle?
No, that is just an error in the automated 3D-PDF generation. Imagine orange triangles in those voids.
wwwmwww wrote:
Do those pieces need to be equilaterial triangles for this puzzle to function?
Yes they do. They move around and rotate similar to the corners of a Rubik's Cube.

Oskar

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 12:27 pm

Joined: Thu Dec 02, 2004 12:09 pm
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Oskar wrote:
A counter example is a flat Boubez geometry wrapped around a sphere.

Your F123 jumbles... at least I think it does.
Oskar wrote:
Yes they do. They move around and rotate similar to the corners of a Rubik's Cube.

Pieces like this don't exist in the other Boublezized geometries we've looked at so you are doing something "more" here. I think its that you are taking the cuts a bit deeper and pulling out new interactions. Though the corners of a Rubik's Cube turn with 3 faces/cut planes. Here I don't see a single orange piece that turns with 3 cut planes at once. I think I'd almost need to be able to hold this thing in my hands to fully appreciate what is going on here

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 1:02 pm

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Your F123 jumbles... at least I think it does.
No, it doesn't. You can completely unbandage it, like you can fully unbandage the flat F53342.

Oskar

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 1:28 pm

Joined: Mon Nov 30, 2009 1:03 pm
Carl,

It jumbles, as the angles are like 55.0337 degrees.
The little orange equilateral triangles are obviously fudged

Oskar
Attachment:

FlatBoublez v3 - view 1.jpg [ 92.35 KiB | Viewed 8228 times ]

Attachment:

FlatBoublez v2 - view 3.jpg [ 108.88 KiB | Viewed 8228 times ]

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 2:44 pm

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
Oskar wrote:
That pic looks strangely familiar...

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 4:57 pm

Joined: Thu Jul 23, 2009 5:06 pm
Location: Berkeley, CA, USA
GuiltyBystander wrote:
This is kind of what I did on my first orbit diagrams except I forgot about rotation directions then.
While attempting to do this for the the cubeoctahedron, I realized that I majorly screwed up the position order. When converting from the dual back to the normal mode, (on the dual) the edged going into a vertex don't have and order. When I was doing triangles, it didn't matter, but on quadrilaterals, it does. Consequently, half of the quadrilaterals in my last picture should look like bowties instead of squares/retangles. I hope to have a picture this afternoon, but I'm kind of sad that it isn't going to be as pretty. And I'm kind of doubting that I can untangle all of the bowties.

Sorry I missed that part of conversation about G123.

About the cuboctahedron, I realized that even if you can untangle the bowties, the graph cannot represent the puzzle in the way I originally thought, because of jumbling. In my original thought, a quadrilateral represents a 4-cycle like a->b->c->d->a. But 4-cycle never occurs on the puzzle because of jumbling. If you turn it by around 70 deg, a->b, c->d, but b and d end up in the middle of nowhere. So one has to take this into account. Therefore the graph for cuboctahedron is not as useful as the graph for G123.

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 5:38 pm

Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Great post guys!

I also really like the "dinoids" idea, and indeed starting from the ("simplest")
tetrahedron case will answer many questions before expanding it more.

Nice to see many points of view (graphs, renders, 2D representations), as the
images and the comments will surely be of huge value in (near) future designs!

As for the topology, no hardcore stuff seem to be needed. Instead, more graph-
theoretical and computational approaches are needed, but those have either been
done already (orbits, nods etc), or they will require some more tedious research.
(It took me an hour just to read this topic!).

Orbits of types of pieces have also been my favourite way to solving and (somehow)
representing puzzles, but I must admit, I had never considered their important role

It has to be said, well done to everyone contributing to this topic, it is one of the

Pantazis

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 8:49 pm

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
GuiltyBystander wrote:
While attempting to do this for the the cubeoctahedron, I realized that I majorly screwed up the position order. When converting from the dual back to the normal mode, (on the dual) the edged going into a vertex don't have and order. When I was doing triangles, it didn't matter, but on quadrilaterals, it does. Consequently, half of the quadrilaterals in my last picture should look like bowties instead of squares/retangles. I hope to have a picture this afternoon, but I'm kind of sad that it isn't going to be as pretty. And I'm kind of doubting that I can untangle all of the bowties.
Leaving the bowties in but colored them so you can tell which ones are linked. Had to add arrows so you could tell the direction a bowtie rotates. All rotation arrow are for 1 CCW turn.
Attachment:

Jumble-Yellow-Flat+Orient.png [ 11.15 KiB | Viewed 8157 times ]

schuma wrote:
About the cuboctahedron, I realized that even if you can untangle the bowties, the graph cannot represent the puzzle in the way I originally thought, because of jumbling....
I thought about this for a bit too. I'm still on the fence on if it is useful, but I'm leaning towards yes. I think that because you never need more than 2 layers lined up at a time for a move, you shouldn't be restricted in your moves really so it should be an okay method for solving with. You'd have to put an arrow on it for orientation, but that's no biggie. If you build this puzzle on a sphere, you won't have any overhang bandaging to worry about either.

kastellorizo wrote:
(It took me an hour just to read this topic!).
lol. I think it would take me much longer. Some of this stuff is pretty dense. I had to re-read some of Oskars and Carls stuff before I realized that Oskar's cubeoctahedron wasn't a icosahedron

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 Post subject: Re: Compy/Dino CubiodPosted: Sun Apr 10, 2011 8:53 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:

Hmmm...
GuiltyBystander wrote:
That pic looks strangely familiar...

I think you need to look at a pic a few before that. Look at the the same one that sparked my idea for the Uniaxial 3x3x3's. In fact the more I "ponder" this topic the more I think the boublezization process is even more closely related to the process I used making the Uniaxial 3x3x3's then I initially thought. Boublez's process was initially applied to corner turns on a cuboid. My Uniaxial process was applied to face turns on a cube. However on a tetrahedron face turns ARE corner turns and vice versa. So do these processes become one in the same on a tetrahedron? What would the dual of one of these Dino Cuboids look like? Take a look at this:
Attachment:

Ponder.png [ 404.51 KiB | Viewed 8155 times ]

The only real difference I see is that Oskar has managed to split the corner into two. I'm wondering if that could be done on a cube...

Carl

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 Post subject: Re: Compy/Dino CubiodPosted: Mon Apr 11, 2011 1:50 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
wwwmwww wrote:
What would the dual of one of these Dino Cuboids look like?
I think this would be similar to creating deeper cut cuboids. When turning a Dino Cuboid into an octahedron, it's going to make the axes overlap more and create new pieces. But yeah, I would still like to see a pic of one also to know for sure.

wwwmwww wrote:
Take a look at this:
When you label the pieces showing the similarities, that's just creepy. I literally had chills run down my spine. I almost want to make it my desktop wallpaper.
For some strange reason, this pic changed my view of the Uniaxial. I use to think it has 2 cuts that were deeper than the other. Now I see it as having 4 cuts being shallower than the rest.

wwwmwww wrote:
The only real difference I see is that Oskar has managed to split the corner into two. I'm wondering if that could be done on a cube...
I stared at the differences between his and mine for a while trying to figure out why he has 2 and I didn't. I think the difference between his and mine is the ratio of overlaps amounts.
Is there anyway that using non-planar cuts on the Uniaxial would split it? Initially I thought it would, but the more that I think of it, the less I think it will.

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 Post subject: Re: Compy/Dino CubiodPosted: Tue Apr 12, 2011 8:27 am

Joined: Sun Dec 06, 2009 9:00 am
GuiltyBystander wrote:
wwwmwww wrote:
What would the dual of one of these Dino Cuboids look like?

I think this would be similar to creating deeper cut cuboids. When turning a Dino Cuboid into an octahedron, it's going to make the axes overlap more and create new pieces. But yeah, I would still like to see a pic of one also to know for sure.

Not sure if this is what you were talking about but either way it seems to be another interesting concept. It would be very shape shifting, I'm half temped to compare it with a golden cube and say that it would prove a harder challenge unstickered.

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