Looks like he pulled piece "A" to a point to meet piece "c" at a point. Just tried that on a cube. Nope... didn't get 2 corner pieces. I now have no corner piece.



Still thinking...
I'm not sure myself... Oskar is so used to breaking the universe I think he can get geometry to jump at his whim. It's like a trick he's pulling off right in front of our faces and I still can't see how its done.

Carl

Got IT!!!



I first tried a conical cut which would produce this same surface pattern but the corner is still just one piece and turns with a, b, and c faces. However it allowed me to see the solution. Use a much shallower conical cut on top of a cylindrical cut and you get this same surface pattern. The c face is now also fixed to the top little "c" pieces. X turns with a and c faces and X' turns with b and c faces. The little "c" piece just turns with the "c" face. This is a uniaxial cube so note the c face turn includes the A piece.

Carl

For a while I was going to say "no you didn't" but then I realized the cuts were somewhere else. Here's a pic for other people who are going to be just as confused as I was.

And I now think you were right the first time...

Aren't the "new corners" just extensions of the x-centers next to them? The pieces connected by the lines in this pic.



Carl

No because they get separated when you turn c.
*EDIT* nvmd, was thinking of getting bandaged to the center, not that other piece. You're right, this extra cut doesn't do it.

I was reading through David Pitcher's Multi-Origin Puzzles Topic when this idea came to me.

The Rex version of 112


And the Rex 122


I'm sure there is some way to expand this into a G123 but I'm not sure how it would be done. Getting to the G123 required a hole new piece, and I'm assuming that this wouldn't be any different. However I just can't invision where it would be.

Am I beeting this Idea to death yet?

I need to make it over there too. Great ideas or coming to the surface faster then I can keep up... keep them coming though. That's not a request to slow down.

Keep at it... if you ask me you are just giving this idea more life. I LOVE both of these ideas. And do you see what I see? Look at GuiltyBystander post here...

You made the "corners" he was after and again I don't think any fudging was needed. But look where your "corners" are. And not only have you made pieces that can be moved on 3 axes you have some faces centers on the ends of your Rex version 112 that even move on 4 axes in their start position.

Great Great work...
Carl

I wanted to try to go just a little deeper then a rex cube today.

First I tried a Skewb, ironically it look exactly like the original dino G112 with flat cuts. Unfortunately I wasn't able to find a way to avoid pieces that would float above nothingness. I think that it would require curved cuts, but I don't think you can do that with a deepest cut. So swing and a miss.

So I brought the cut out slightly to a master skewb G112. This Seems much more feasible ... however. I Took the liberty of coloring in the orbits. These orbits only apply if someone can think of a way to make it possible to switch corner and edge pieces. If this can't be done, I think that the puzzle would still be doable. It would be very restricted in movement. Scrambling it would basically require algorithms.


enjoy

compydino and helicopter

For the sake of fluid conversation and clarity I have created a poll to vote on what to call this monster. I urge everyone who has contributed or followed this topic to vote. http://twistypuzzles.com/forum/viewtopic.php?f=9&t=21001 Poll closes April 28 2011!

Okay, I've vaguely read through this thread, and because a simpleton like myself finds this stuff really confusing, I don't know if this has been mentioned. I was just wondering whether it's possible to make a Boublezed Dodecahedron (or actually a truncated pentagonal trapezohedron):



I know the drawing is terrible, and I could have got the small edges to touch each other, but in principal it's the same and you should get the idea. I don't know the naming notation, although this is a simple Boublezed Dodecahedron. And yes, I got lazy and didn't draw the cuts on the back-right side, but oh well.

Welcome to the division Luke. This idea was briefly brought up before but quickly got brushed of to the side. I do like the idea however there seems to be an issue that I'm have a hard time raping my head around. I think we might need Oskar's help on this one. One of two things must happen for this to be possible.

1) The cuts would have to be at 2 different depth, meaning the angles would be symmetrical on each piece. This would require fudging, one of Oskar's very many specialties.

2) The dodecahedron would have to be stretch out so that all the cuts angles can remain the same. Now I don't know the proper way to stretch a dodecahedron. When thinking of stretching a dodecahedron the first name that comes to mind is once again Oskar.

I would really love to see this concept be taken all the way to a dodecahedron. I think the logical progression would be to first take it to a octahedron and see what we learn from that.

I realised those two things. The image I've drawn is an elongated dodecahedron, which is the same as a truncated pentagonal trapezohedron. I think the elongated dodecahedron would probably be the best way.

Remember, the elongated dodecahedron is the same thing as a dodecahedron cuboid; you would have the same issues if you tried to make all the Boublezed cuboids cubic.

The first G112 I drew was mad by taking 5 sides of a cube putting 4 corner cuts into it then flipping it over onto itself so that new edge pieces would be exactly half the size of the rest of them. Do these all fit in with our current classification system?

From left to right an over lap of; 100%, 75%, 66%, 50%.


Would this be a G335? Notice that it has one more type of piece the a G112.

I think it's possible to make a dino-dodecahedron "cuboid", but I don't think it's quite so simple as making the regular dino-cuboids. With a cube, when you stretch in in one direction (while maintaining all angles), you only add one new type of edge-length. The problem with stretching a dodecahedron in the way you described is that it creates two new edge-lengths at the same time (the extended edges become longer, and the pentagon polar-caps become smaller). I think this is kind of what Boublez was saying.

The significance of different edge lengths is it defines how much the corners overlap, and each new different edge length forces you to add new sets of pieces. I think what this means is that to make a dino-dodecahedron "cuboid" (stretched the way you have shown), you will need to jump to the case of having 3 different amounts of overlap, like the case of a 3x4x5 dino-cuboid for example. The cuts the way you sketched I don't think could work.

Someone correct me if I'm utterly wrong here. xD

EDIT: Hmmm, I just realized that the "extra" sets of pieces at the polar-cap-edges may get stuck in an orbit up there if you tried to do a 2x3x4, so you wouldn't necessarily have to split them up in that case... I guess it depends how much overlap you choose to use. I think a 3x4x5 would fully scramble.

Not what I was trying to say but you are very right. The ratio of edge lengths would have to be just right in order for this to work. Working with Sketchup I'm not going to be able to figure it out. I tried before DKwan offered this bit of knowledge so I have a picture to explain. I stopped mid process because I realized that this is more then I can handle.



The red yellow and blue edge pieces consequently become three different sizes. There should be a perfect ratio of edge lengths where the red and blue pieces would be the same size. The other option would be to have all three pieces be in proportion to each other. There might be more then one ratio where this could happen, but I'm not completely sure.

Actually, assuming all the angles are still 108 and the cut-depth is the same for all corners, it should be impossible for the red and blue pieces to be the same size (except for the case of a regular dodecahedron).

As you said though, you have to find a very specific set of edge lengths. For working with 3 edge lengths, if you label the lengths A, B, and C where A < B < C, you should need (B-A)/(C-B) to be rational. (I think)

You are right none of the pieces could be the same size. Im not sure why I thought that, long day I guess, my bad. As for this ABC stuff thats too many variable for me to understand but if some one could figure this out it would be great.

More nice ideas but I'd call that first one a F-Skewboid based on the F-Skewb. And the second one I'd call a... hmmm... not sure I should say this... a Sex-X based on Drew's Quad-X.
Quad = Quadruple so Sex = Sextuple.

I must confess I have a hard time seeing Mefferts selling a puzzle called the Sex-X. But I really really like that name.

Carl

So apparently I discovered that (B-A)/(C-B) will always equal the golden ratio, if the angles remain unchanged... It makes me a little happy on the inside to find the golden ratio, but sad at the same time because it means this won't work out so easily... =X

I suppose this means you have to find some other way of stretching the dodecahedron, or use fudging...

Yes... if I assume the top face is square and we are looking at the front face then from left to right we have:
G111(B111), G344(B544), G233(B433), G122(B322)

That is the G344(B544).

Carl

... Oskar!!! Or any one else that know how to fudge a puzzle!!!

It just occurred to me that another way of handling this dodecahedron dinoid would be to make a Lattice Dodecahedron and restrict it so that only one layer could turn. On some vertexes you could restrict the outer layer and on the others you could restrict the inner layer. It might be interesting to see what new kind of orbits this would bring to the table.

Hi Carl or Anthony,

Carl's coloring convention is not good for stickering a puzzle. Do you have any suggestion how this puzzle should be stickered?

Thanks.

Oskar

How about like this? I think it would make it quite hard this way

How about something like this? you would need a lot of colors though... 18 by my count... maybe you could use colors more than once?

Here is my idea...



Add raised or recessed numbers and letters to the pieces and then order them such that each wheel has the pieces in order in a clockwise/inward-outward fashion.

The piece F13 makes me think of Friday the 13th so you could call if the Friday13 puzzle.

You may need to add an underscore to the pieces names so you could tell piece "p" from piece "d" for example. Capital letters come before numbers, and numbers before lower case letters. This way you can easily tell a lower case L from a capital i.

Carl

Getting closer to seeing what Oskar is doing. Still trying to apply his Bobbles in a Plane idea to a cube and I've got this so far:



I'll call this the SUB-Cube. That's short for Shallow Uniaxial Boublezized Cube. If Oskar's trick can be pulled off "off the plane" I think this may be a lower order version of that trick. No jumbling and fudging not required yet.

Carl

I think the tone//Lsd///Gradian (different ways to say it) version would suit this puzzle best, making it extemely hard!

How about a simple 'color wheel' overlapping pattern? It would make a rather 'simple' solution and (in my opinion, at least) is quite pretty.
I'll post other ideas as I get them.

Hmmm.... someome PLEASE tell me I'm wrong... I think I just noticed what Oskar did... and its... he cheated.

This puzzle doesn't have blue pieces. Each pair of blue pieces should be combined with one of the yellow pieces to form a purple piece as shown here. If I'm wrong PLEASE tell me how you seperate this purple piece into the yellow and blue pieces that it was made from. I really hope there is some jumbling move that I'm over looking.

Carl

I'm afraid to say you're right...



I really like that one!

If that is the case then the cubic version of this puzzle is simply the DUB-Cube or Deep Uniaxial Boublezized Cube. Here is what it would look like. In this form it doesn't jumble and no fudging is required.



Carl

Here is an animation which shows how this turns:



This is actually a subset of the normal Deep Uniaxial 3x3x3.

Carl

Carl,

Thank you for spotting my mistake. It was an honest mistake, by the way. Would adding a fourth circle solve the problem that you identified?

Oskar

Yes, nice illustration as now it seems it does.
And this must be the minimal structure required to do so.




Pantazis

You are welcome.
Didn't mean to imply that it wasn't. I just thought it funny. I sort of looked at it as the universe getting even and tricking you for a change. Granted it took me a while to see it myself.
Yes.... but you've done FAR more then add a fourth circle. You've changed the symmetry of the 3 existing circles as well.

The two puzzles on the left are basically the same and both have the "problem". In both cases you have two types of circles. On a cube this maps to cuts of different depth. Circle type "A" has 2 orange triangles between each blue piece. Circle type "B" has only 1 orange triangle between each blue piece.

This new proposed puzzle with 4 circles has just one type of circle, called type "C". Each type "C" circle has 180 degree symmetry. There are two gaps between blue pieces that require 2 orange triangles that are opposite each other and the other gaps only require 1 orange triangle. I see what you have done but with these distorted haxagonal face centers I think you are stuck in a plane. If we can cut this from 6 edges per face to 3 edges per face I think we get something very similiar to boublez's Octahedron seen here. Can the cut depth on that puzzle be increased to create some edge overlap (i.e. the orange pieces seen here)? I'll have to think about that.

And the bottom right puzzle in this picture still has me confused. The blue pieces here don't occur in pairs and even the orange pieces I'm not sure are related to the orange pieces in the other puzzles. I'm wondering if you have fudged some pieces out of existance that should be there or... what!? You again clearly have two types or circles present. The top and bottom circles have just 1 orange piece present between the inner row yellow pieces. The left and right circles have 2 orange pieces present between the inner row yellow pieces. To me this again implies different cut depths on a non-planar puzzle to me and this appears that it should be mapable to a cube. I'm just not seeing how to do that. I think (assuming there isn't some cheating going on here that neither of us has spotted yet) that this puzzle is a "deeper" cut puzzle then the other 3 in this image and that is creating some new piece types. I say this because there are orange pieces in this puzzle that are inside 3 of the turnable circles. Let's call these corner pieces. None of the other 3 puzzles in this image have pieces like that. All of there pieces are in at most 2 turnable circles. So you can think of them as just having edge pieces.

Carl

Carl,

How do bandaging, jumbling and boublezizing relate? The figure below is my attempt. However, given the introduction of boublezizing, we may need to revise the definitions of bandaging and jumbling. Not to mention fudging ...

Oskar

I don't think that grouping is right... If there exist non-jumbling boublezized puzzles then there must exist non-bandaged boublezized puzzles.

boublez's first picture is a perfect example.

This is a discussion about definition. I would consider boublez' first picture bandaged, as I consider the picture below bandaged. Obviously, we need a sharper definition of "bandaged".

There has been requests made by others to get a lock thread in place about definitions and I agree with that idea as even those of us that use them get confused at times. Part of the problem though is I know some of the definitions used here differ from person to person, take the "order" of a puzzle for example. In this case I think we are actually in pretty good shape.

For "doctrinaire", "jumbling", and "bandaged" Bram's post here is a good place to start.
By this definition the FusedCube is doctrinaire. As noted here by Bram. This means the presence of "stored cuts" by itself doesn't make a puzzle bandaged.

Stored Cuts = cuts which seperate pieces which can't be seperated with a single rotation/twist of the puzzle.

Boublezized so far to me means puzzles which have been inspired by this thread and that is a bit vague. There are two new basic concepts here. Puzzles with more then one core, i.e. a puzzle in which all the axes of rotation don't cross at a single point. And the other being a mix of my uniaxial concept of having different cut depths along different axes.

So I'd say Boublez's first puzzle in this thread IS doctrinaire. The one you just posted would also be doctrinaire if it weren't for the horizontal slice. With the horizontal slice I'm not sure if it jumbles or is bandaged. I'm guessing it jumbles as I don't see an easy way to unbandage it.

Carl

P.S. Looking closer at your puzzle I'd say without that horizontal slice it would be a shape mod. It does change shape with the vertical turns but that is just due to pieces of the same "type" having different shapes. Just as with your Mixup Cube. That horizontal slice still complicates the picture.

P.S.S. Just saw this quote on ShapeWays:

I'll take your word for it and as such I'd call this a Shape Mod without the horizontal slice, and bandaged with the horizontal slice. It does sort of run counter to the idea that you bandage a puzzle by taking away a cut here or there. Here you've added a cut and bandaged a puzzle that wasn't bandaged before.

Whenever Carl or I have a question about doctrine/bandage/jumble, we usually go to a post made by Bram. We both think Bram's definitions should be used as the standard. Bram defines doctrine asBy that definition, I think boublez's first idea and your Hexagonish Blob are doctrinal puzzles.

*edit* took to long writing it and Carl beat me.

I completely agree that Boublez's first several puzzles are doctrinaire. I like the concept of stored cut. But in the recent days, I was thinking in another direction, to describe boublezized puzzles regarding symmetry. I haven't got a very clear thought. But here's what I was thinking:

When we make puzzles based on platonic solids, there's always an underlying group of symmetry. Let's take the example of the Rubik's cube. There are several ways to talk about symmetry.

-- The shape of the Rubik's Cube is a cube. The symmetry of a cube is the octahedral group. This symmetry is not really important because we can always do shape-mod to change it. Let's forget about this symmetry.

-- Symmetry of axes: There are six axes. The symmetry of axes happens to be the same: the symmetry of a cube (octahedral group). Because axes determine behavior, this symmetry is really what matters.

-- Equivalency of pieces based on the symmetry of axes: Take any corner piece, transform it according to the symmetry of axes (the octahedral group in this case), we can get other corners. Therefore all corners are equivalent with respect to the symmetry of axes. Similarly, all edges are equivalent.

-- Orbits of pieces based on reachability of valid twists: Take any piece, the places it can reach by valid twisting form an orbit. All corner pieces are in one orbit, all edges another.

For the Rubik's Cube, the equivalence classes coincide with the orbits

For the Skewb, all corners are equivalent w.r.t. the symmetry of a cube. But a certain corner can only reach four out of eight corners. So an orbit is smaller than the equivalence class.

For the face turning 3x3x4, the symmetry of axes is the symmetry of the cuboid (certain dihedral group). And just like the Rubik's Cube, the orbits coincide with the equivalence classes.

Let's look at boublez's Dino cuboid. The symmetry of axes is the symmetry of the cuboid (certain dihedral group).



All the red pieces in the figure are equivalent, but not together with the blue or green ones. But an orbit includes some green ones, some blue ones and some red ones (not all of them are in the same orbit) So the orbits are not contained in the equivalence classes. I find this property interesting and maybe this is the characterization of all boublezized puzzles.

This is only my tentative thought. Maybe this criterion is too narrow so that it excludes jumbling boublzed puzzle. Maybe it's too broad so that it includes the mixup cube and Shim's constellation Six (seems so). I'm not sure if they are just the same kind or we should find another criterion to distinguish them.

Anyone else spot this yet?

http://www.shapeways.com/model/247067/



Oscar, mind if I ask how much time you put into designing and thinking up puzzles in a typical week? I believe you've said this isn't your full time job but HOW do you have time for one of those AND are able to pump out this stuff at the rate you do?

This makes me think of the The Time, Money, and Energy Conundrum...

When you’re young, you’ve got all the time and all the energy to enjoy life, but no money. When you’re in your middle years, you’ve got all the money and all the energy, but no time. And when you’re retired, you’ve got all the money and all the time, but no energy.

People like you and clauswe seem to make me think that a very high percentage of those of you able to solve this conundrum end up here at TwistyPuzzles.

It makes me feel like I'm short of time, money, and energy.

Very nice puzzle by the way, I'm quite eager to see this one printed.

Carl

Sorry for the Bump but I think these needs to be done, and all the information I'm going to use is already in this thread. There are far to many terms in this community with no definition, therefore a definition of boubling/boublized is in order.

I'm going to be looking to 5 puzzles that everyone can agree are boublized in order to try to set up some criteria. They are:
The Mosaic Block, Bubbles in a Plane, The Hexagonish Block, The Bubble Block, and Carl's DUB-Cube.

I believe there are 4 plausible criteria for defining something as being boublized. They are:
Multiple Origins/ Offset Axes
Varying Depth of Cuts
Varying Angels Between Axes
Stored Cuts.
I think the definition should go something along the lines of in order to be boublized the puzzle should fit one of the first 3 criteria and must fit the 4th.

With this definition the list of boublized puzzles would grow to include; David Pitcher's Sunrise Sunset, Eric Vergo's Compound Crystal, Carl's Deep Uniaxial 3x3x3 Cube, and probably a few others I haven't thought of.

I was thinking for a while that since they all have stored cuts we could just use that as our only criteria. Doing this would add a lot of puzzles to the list that I don't think really "feel" boublized such as; Shim's Constellation Six, All/Most off David Pitcher's "Compound" and "Roto" Puzzles.

Sorry for the long post I've had this on my mind all day . Can anyone else add to this or at least nod and smile.

Among the things you list, having not all axes go through the same point is by far the most novel, although that isn't a property shared by all the things you list as boubleized.

Here's a question for you - do you consider TomZ's cube which looks like a 5x5x5 but only allows cuts at depth 2 on one axis but depth 1 on the other axes to be boubleized? It has stored cuts and varying depth of cuts, but no varying angles. What about puzzles with icosahedral symmetry and fairly deep cuts? Those have varying depth and angles, but no offset axes or stored cuts.

Talking with Oskar at g4g, it seems like the rule for offset axes is that adjacent faces must have axes going through the same point, but nonadjacent ones don't, which makes it difficult and interesting to find cases where that condition is met.

I really need to re-read all of this thread. Who was the first to coin the term "boublized"? I fear the definition could grow too big to be useful if it includes too many things. As it all grew out of the discussion about the Compy Dino Cuboid, I certainly feel it should include the "Multiple Origins/Offset Axes" puzzles but even there you need to be careful. What about all the Siamese cubes? I think what differentiates the Siamese puzzles from the boublized puzzles is that some (not necessarily all) pieces can be moved from one core (intersection of two or more turning axes) to another. However that now makes me wonder about the Fully Functional Fused 2x3x3 which I was about to say doesn't make use of stored cuts. But looking closer I see that it does. There is a 2x2x1 block of cubies fuzed together in the center of the white and yellow faces. The cuts between cubies outside this block which appear to go through this block are stored cuts. So I guess the Fully Functional Fused 2x3x3 should be considered a boublized puzzle.

Maybe the most unique puzzle which grew out of this discussion is Oskar's Bubble Block. And I'm glad I can say I got to see this puzzle first hand at G4G10. However its geometry still confuses me. It has 4 axes of rotation but I don't know if they all meet at the same point or not. Oskar could you make an image which shows how the axis intersect inside the Bobble Block? I'd really hate to propose a definition of "boublized" which didn't include this puzzle.
That is the Deep Uniaxial 3x3x3 and I hadn't thought of that as boublized before but based on the definition we come up with I see how it could be.I'm confused by this statement. When you say with "fairly deep cuts" I assume the cuts are all the same depth and if it has icosahedral symmetry then all the angles between adjacent axes should be the same. What am I missing? Could you give an example?
See Oskar's post here. This is his first rule but you didn't cover his second rule.

And now having read those rules maybe its puzzles which follow those rules which should be considered "boublized". Not if you did that I think the Fully Functional Fused 2x3x3 would fall out again as just a Siamese puzzle. The new axis which was added to make it "Fully Functional" doesn't have any adjacent axes which it crosses.

Still not sure this covers the Bubble Block.

Carl

OK, let's check for the four Oskar puzzles from your list.

Multiple Origins/ Offset Axes --> Not true for Bubbles in a Plane, Hexagonish Block, and Bubble Block.

Varying Depth of Cuts --> Not true for Mosaic Block and Hexagonish Block.

Varying Angels Between Axes --> Not true for Mosaic Block, doubtful for Hexagonish Block.

Stored Cuts --> True for all examples.

Apparently, the stored cuts is the only thing that these four have in common. But also doctrinaire, bandaged and jumbling puzzles may have stored cuts. So I agree that, unlike jumbling, "boublizing" has no clear-cut definition. Nor am I sure that "boublizing" is a single phenomenon.

Oskar

P.S.On 12 April, I am the last speaker at the 2012 Dutch Mathematical Congress. My primary goal is to introduce jumbling and boublizing to professional mathematicians and get at least one student or PhD to start studying these phenomena.

I am not convinced that there is any one thing that we can identify as "boublized". The term has actually been used to describe a set of concepts (and combinations thereof) which already have their own self-explanatory names (multiple origins, varying angles between axes, stored cuts, etc.). Given this, I'm inclined to say that despite the incredible creative outpouring that this thread inspired, we don't really have a new phenomenon that needs a new name. Fortunately we have many great new puzzles!

Which reminds me, is anyone going to produce this beautiful puzzle idea?