Online since 2002. Over 3300 puzzles, 2600 worldwide members, and 270,000 messages.

TwistyPuzzles.com Forum

It is currently Tue Jul 22, 2014 12:47 pm

All times are UTC - 5 hours



Post new topic Reply to topic  [ 94 posts ]  Go to page Previous  1, 2
Author Message
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 3:49 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Noblesse oblige. Here is my attempt to provide some proper definitions related to "deeper than origin".

Definition 1: Deeper-than-origin mechanism
A twisty puzzle has a deeper-than-origin mechanism if at least one of its cut curves covers an area more than 180 degrees, measured from its single origin.
Note: this definition only applies to single-origin twisty puzzles.

Definition 2: Cut curve
A cut curve is the cross-section area between the moving sets of pieces while performing a turn.

Definition 3: Origin
An origin is a point where two or more rotation axes of the twisty puzzle coincide.

The sketch below illustrates the cut curves and origin of Mixup Cube. Note that Mixup Cube has only three bolts, namely the three bolts of the inner 2x2x2.
    -The magenta cut curve covers an area of around 40 degrees, measured from the origin.
    -The blue cut curve covers an area of around 170 degrees, measured from the origin.
    -The green cut curve covers an area of around 220 degrees, measured from the origin.
The green cut curve satisfy the requirement of Definition 1. Therefore Mixup Cube has a deeper-than-origin mechanism according to this definition.
Attachment:
Mixup Cube v5 - cut curve - view 2.jpg
Mixup Cube v5 - cut curve - view 2.jpg [ 59.16 KiB | Viewed 6133 times ]


TP definition debaters, please challenge these proposed definitions and identify possible ambiguities. And please propose alternative definitions, in order to debate which definition is the most elegant.

Thank you.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri Mar 29, 2013 6:21 pm 
Offline
User avatar

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Well I will say I'm about to give up my definition as I think I see a hole or two in it.

(1) It can't be applied to jumbling puzzles and as Bram points out a fair number of the puzzles considered deeper-then-deep-cut are jumbling puzzles.

(2) I'd have to make a distinction between a piece with a given signature and a location with a given signature. Otherwise you could take what I've written and claim the 2x2x2 is deeper-then-deep-cut. I think this is fixable but point (1) still bothers me.
Oskar wrote:
I continue my objection against the term "deeper-than-origin puzzles". The fundamental problem is that it requires people to hypothesize a mechanism. Either you look at the puzzle as it is played, but then your have only the outside looks and feels to work with.
I'd tend to disagree but I can actually argue this point both ways...

(1) With a Rubik's cube in addition to the surface you are also given very clearly defined axes of rotation. You are also given fairly obvious cut planes which cut 3D space up into 27 unique areas. Granted only 26 are seen from the surface of the cube.

(2) The other point of view here is who says a Rubik's cube needs planar cut surfaces. You could make a Rubik's cube using conical cut surfaces which meet at the origin and if you did then 3D space would only be cut up into 26 unique areas. As such Oskar does have a valid point here. Is it fair to talk about the core piece in an analysis of that puzzle?

Maybe the thing here is we should be defining deeper-then-deep-cut OPERATIONS. I'd propose an operation which rotates the entire axis-system of the puzzle back onto itself during its application is a deeper-then-deep-cut OPERATION.

This would mean face turns on a 3x3x3 are slallow operations. Slice turns on a 3x3x3 are deeper-then-deep-cut operations. Etc.
Oskar wrote:
Here is my attempt to provide some proper definitions related to "deeper than origin".
<SNIP>
Oskar wrote:
Therefore Mixup Cube has a deeper-than-origin mechanism according to this definition.
My biggest objection here is you are now appling this definition to say something about the puzzle. Using your definition I'd say Oskar's mechanism for the Mixup Cube is a deeper-than-origin mechanism. But your drawing doesn't necessarily say anything about a Mixup Cube which uses magnets for example.

Also let's look at VeryWetPaint's 4 part 1x2x2 seen here. I think we'd both agree this is a deep cut puzzle. Correct? Are we now going to say this particular deep cut puzzle has a deeper-than-origin mechanism which I believe it does have using your definition?

Carl

P.S. Just noticed that using the above definition of deeper-then-deep-cut operations one can claim that half the face turns on the Mixup Cube are deeper-then-deep-cut operations as they contain the corner attached to the core which drags the axis system of the puzzle with it. The slice turns on the Mixup Cube are NOT deeper-then-deep-cut operations. Hmmm... and I now also see this depends on the mechanism. Let's assume the corners of Maltese Gears was built up into a Mixup Cube. Does that make ALL face turns on such a Mixup Cube deeper-then-deep-cut operations? Does this give the Mixup Cube 9 axes of rotation? I guess we could define stored axes in much the same way we have defined stored cuts?

_________________
-
Image

Image


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Mar 30, 2013 3:35 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Oskar wrote:
Here is my attempt to provide some proper definitions related to "deeper than origin".
<SNIP>
Oskar wrote:
Therefore Mixup Cube has a deeper-than-origin mechanism according to this definition.
My biggest objection here is you are now applying this definition to say something about the puzzle. Using your definition I'd say Oskar's mechanism for the Mixup Cube is a deeper-than-origin mechanism. But your drawing doesn't necessarily say anything about a Mixup Cube which uses magnets for example.
Carl,

It does. The metal sphere inside a magnetic mechanism has a cut curve that covers 360 degrees. That would make all magnetic puzzles "having a deeper-than-origin mechanism". Such would be the consequence of accepting my Definition 1 in its current phrasing.
wwwmwww wrote:
Also let's look at VeryWetPaint's 4 part 1x2x2 seen here. I think we'd both agree this is a deep cut puzzle. Correct? Are we now going to say this particular deep cut puzzle has a deeper-than-origin mechanism which I believe it does have using your definition?
Assuming that it has at least one deeper-than-origin cut curve (I haven't checked this, but you are most likely correct), then yes, that would be the consequence of my Definition 1. If you wish, we could weaken my Definition 1 as follows.

Definition 1': Deeper-than-origin mechanism
A twisty puzzle has a deeper-than-origin mechanism if it requires at least one cut curves to cover an area more than 180 degrees, measured from its single origin.

According to Definition 1', Mixup Cube does not have a deeper-than-origin mechanism (it does not require the green cut, as the blue+magenta cuts suffice), nor do magnetic-sphere puzzles. The classification of Scott Elliott's 4-part 1x2x2 mechanism would probably be unaffected. I wonder how it would classify Dave Pitcher's RotoPrism 2, which has several hidden floating pieces and additional internal cut curves. Dave, could you provide sketches of your cut curves, please?
wwwmwww wrote:
... half the face turns on the Mixup Cube are deeper-then-deep-cut operations ... The slice turns on the Mixup Cube are NOT deeper-then-deep-cut operations ...
That is correct when using my Definition 1.
wwwmwww wrote:
I guess we could define stored axes in much the same way we have defined stored cuts?
My definition set is still lacking a definition for stored cut. Stored cuts are not part of "the cross-section area between the moving sets of pieces while performing a turn" by my Definition 2. Can you propose a proper definition for "stored cut", consistent with my definitions? (Let us stay away from the "stored axis" can of worms for the time being.)

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Mar 30, 2013 11:47 pm 
Offline

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
As I see it, the obvious definition of a deeper than origin puzzle only applies to puzzles which only have one origin, and for those puzzles a slice is deeper than origin if the parts on the surface in the same plane as the origin don't move in tandem with the origin.

Which is a perfectly good definition, although it raises the question: Has anybody explored deeper cut versions of the corner turning box?


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 2:32 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Bram wrote:
As I see it, the obvious definition of a deeper than origin puzzle only applies to puzzles which only have one origin, and for those puzzles a slice is deeper than origin if the parts on the surface in the same plane as the origin don't move in tandem with the origin.
Bram,

That looks like a proper definition to me. It would classify HandiCube, Mixup Cube, Slice-only 3x3x3 and Slice Kilominx as deeper-than-origin. It would classify Enabler Cube as not deeper-than-origin. Am I correct?

If my interpretation is correct, then I would say that your's is a proper definition too, albeit different from my two definitions.

Now let us collect arguments in favor and against the three current definitions: Bram's definition above, Oskar's definition 1 and Oskar's definition 1'.

My argument in favor of Bram's definition (only if my interpretation above is correct), is that it is simple and unambiguous. An argument against it is that it does not distinguish RotoPrism2 from a whole zoo of puzzles that have slice moves. My two definitions do.

So I can accept Bram's as definition for "has a slice mechanisme", but I prefer something more strict for "has a deeper-than-origin mechanism".

Oskar

P.S. Carl, I checked the cut curve of Scott Elliott 4-part 1x2x2. It is not deeper-than-origin according any of our definitions. The cut curves does not extend beyond 180 degrees.

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 10:03 am 
Offline

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Oskar wrote:
Bram wrote:
As I see it, the obvious definition of a deeper than origin puzzle only applies to puzzles which only have one origin, and for those puzzles a slice is deeper than origin if the parts on the surface in the same plane as the origin don't move in tandem with the origin.
Bram,

That looks like a proper definition to me. It would classify HandiCube, Mixup Cube, Slice-only 3x3x3 and Slice Kilominx as deeper-than-origin. It would classify Enabler Cube as not deeper-than-origin. Am I correct?


Yes. To differentiate, we could define a slice slice as being one where the two points on the axis of rotation on the surface both move in tandem, and modify my definition of deeper than origin to specify 'except for slice slices'.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 10:10 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Bram wrote:
Oskar wrote:
That looks like a proper definition to me. It would classify ... Am I correct?
Yes. To differentiate, we could define a slice slice as being one where the two points on the axis of rotation on the surface both move in tandem, and modify my definition of deeper than origin to specify 'except for slice slices'.
Is you amended definition effectively identical to my definition 1'? If yes, then I would like to debate the wording, as I do not understand "the two points on the axis of rotation on the surface". If no, then please explain in what cases they would differ.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 10:59 am 
Offline

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Oskar wrote:
Is you amended definition effectively identical to my definition 1'? If yes, then I would like to debate the wording, as I do not understand "the two points on the axis of rotation on the surface". If no, then please explain in what cases they would differ.


I believe that in yours a sufficiently large slice - that is, one which is so... anti-deep? that its angle is greater than 180 degrees would qualify as deeper than origin, while in mine the opposite centers moving in tandem is sufficient to say that it's slice.

Shells can be applied to slice mechanisms as well, pulling them outwards. Unfortunately even single shell slice mechanisms tend to fail due to the non-orthogonal angles, except for the slice cube. But you could shell that into an impossible object where three rods seems to all go through each other in the middle of a ball, when they're actually the centers of an internal slice cube.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 4:36 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Bram wrote:
I believe that in yours a sufficiently large slice - that is, one which is so... anti-deep? that its angle is greater than 180 degrees would qualify as deeper than origin, while in mine the opposite centers moving in tandem is sufficient to say that.
I do not understand. Could you sketch some cut curves and evaluate those, please?

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Last edited by Oskar on Sun Mar 31, 2013 10:05 pm, edited 1 time in total.

Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 7:52 pm 
Offline

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
Oskar wrote:
Bram wrote:
Oskar wrote:
I believe that in yours a sufficiently large slice - that is, one which is so... anti-deep? that its angle is greater than 180 degrees would qualify as deeper than origin, while in mine the opposite centers moving in tandem is sufficient to say that.
I do not understand. Could you sketch some cut curves and evaluate those, please?


For example, you could have a megaminx where opposite sides always moved in tandem, because it was a slice mechanism which had been built outwards.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 10:01 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Bram wrote:
Oskar wrote:
Bram wrote:
I believe that in yours a sufficiently large slice - that is, one which is so... anti-deep? that its angle is greater than 180 degrees would qualify as deeper than origin, while in mine the opposite centers moving in tandem is sufficient to say that.
I do not understand. Could you sketch some cut curves and evaluate those, please?
For example, you could have a megaminx where opposite sides always moved in tandem, because it was a slice mechanism which had been built outwards.
I know Slice Kilominx. I guess that it could be modded into a Slice Megaminx. However, I still do not understand why that requires a slice cut larger than 180 degrees. Nor do I even understand what a slice cut larger than 180 degrees means. Sketch please!

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun Mar 31, 2013 11:26 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
I believe that the best way to look at this is to view the axis system as fixed in space, and the origin as unmovable. The puzzle then moves around this system, and it does not matter what the mechanism is, or if there is one at all. All that is required to understand the puzzle is the axis system, the exterior shape of the puzzle, and the slice pattern. It is easiest to simply think of the puzzle as a virtual model, with no mechanism. For the moment, let's consider puzzles with flat slice planes only.

Given those conditions, one can quickly see some interesting things about several of the puzzles in question. For example, a normal Rubik's Cube is shallow cut since the turns all fall short of a plane going through the origin. Slice turns are actually performed by turning two opposite sides of the puzzle.

The next example is a Mixup Cube. This gets interesting because now a 45 degree slice turn can be performed and then subsequent legitimate turns can be made on any other axis. Therefore the slice turn did not violate the rule of not allowing the axes to move. This being the case, we can see that the axes are not affixed to the face centers of the Mixup Cube as they are with the Rubik's Cube. We can then deduce that any physical mechanism for the Mixup Cube must have a separate inner layer that allows the outer rings of centers and edges to glide over them, in a manner similar to the Equator puzzle. I would say that this type of cut falls into a different category than any of the three groups (shallow, deep, deeper-than-origin) we've been using so far.

Next we can look at a puzzle like RotoPrism 2 or Trapentrix, and it can readily be seen (more easily with Trapentrix) that in order for a turn to not violate the principle of maintaining the axis system fixed in space, the portion of the puzzle that turns must extend over the origin point of the axes. Thus the puzzles are deeper-than-origin turning.

In sum, I believe my originally proposed definition of deeper-than-origin turning of "for the puzzle to maintain the orientation of its axis system, a slice that includes the origin point must be turned" is the best way to classify these (and other) puzzles. Perhaps extra wording is needed to clarify that the origin and axis system should be viewed as fixed in space, with the pieces of the puzzle moving around them. This definition has the added benefit of teaching us something about how any physical model of the puzzle must behave, even if a virtual model is the only current representation of the puzzle.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 12:26 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
David Pitcher wrote:
I believe that the best way to look at this is to view the axis system as fixed in space, and the origin as unmovable.
That is one way of looking at the RotoPrism 2. But the solver could also imagine it as a shallow-cut magnetic puzzle, and some hidden bandaging. As you explained to me privately, the Prismatic RotoPrism 2 has all the bandaging visible, which make the visualization of the bandaging even easier. Whereas we can debate which of the two views is the most effective in solving, both views are equally valid. You know, Einstein, relativity, ...
David Pitcher wrote:
"for the puzzle to maintain the orientation of its axis system, a slice that includes the origin point must be turned" is the best way to classify these (and other) puzzles.
I object against this definition as it uses "its axis system". RotoPrism 2 is a jumbling puzzle, which means that one can remove some bandaging, which results in additional axes. If we unbandage it to Bram's proverbial jumble dust, then the result is a fractal axes system with an infinite number of axes, all of which are equivalent. Actually, that is the axes system of the shallow magnetic version.

Also, your definition is ambiguous for Enabler Cube.

We agree that RotaPrism 2 has a deeper-than-origin mechanism. We also agree that for some puzzles (RotoPrism 2, Enabler Cube) imagining a puzzle having a deeper-than-origin mechanism is an effective way of analyzing the puzzle. But imaginary mechanisms cannot be part of a definition. What if Carl starts playing Rubik's Cube in a deeper-than-origin way? Does it then suddenly become a deeper-than-origin puzzle?

I propose to keep things simple. A deeper-than-origin puzzle is a puzzle with a deeper-than-origin mechanism.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 12:08 pm 
Offline
User avatar

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
I'm moving into a new house and should get internet access again later this week. Things have also been picking up at work so I'll try to stay rather quiet until I can get access from home again. So sorry for dropping out there for a bit.
Oskar wrote:
P.S. Carl, I checked the cut curve of Scott Elliott 4-part 1x2x2. It is not deeper-than-origin according any of our definitions. The cut curves does not extend beyond 180 degrees.
Could you draw your understanding of its cut curve? The two halves each take up 180 degrees and one MUST cut into the other half in order to hold the puzzle together. So I don't see how you can claim they don't extend beyond 180 degrees.

Carl

_________________
-
Image

Image


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 12:16 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Oskar wrote:
I propose to keep things simple. A deeper-than-origin puzzle is a puzzle with a deeper-than-origin mechanism.
I think that I am actually keeping things as simple and as consistent as they can be with my fixed origin and fixed axis system model.

Let's look at RotoPrism 2 and Trapentrix (I use Trapentrix because it's a bit easier to visualize) for a moment. Using my model (fixed origin and axes), these puzzles have three and two axes respectively. Given the shapes and slice patterns of the puzzles as constrained by the fixed origin/axis systems, both puzzles can be readily understood, the only caveat being that we must break our previously held assumption that no puzzle can be deeper-than-origin turning (I know, we have already parted with this assumption, but bear with me for the rest).

Another way to view these puzzles is to say that deeper-than-origin turning is heresy, and we must view them as shallow-turning puzzles. If we do this, then we can see that phantom axes begin to appear as the shallow side of the puzzle turns, and the axis systems become reoriented in space with each move. As we know, RotoPrism 2 is a jumbling puzzle (Trapentrix is as well, as evidenced by the internal voids in the puzzle), meaning that we now have an infinite proliferation of axes being created and destroyed as the puzzle moves. Clearly this is not what is happening with the physical manifestation of either puzzle, so the fixed axis system with deeper-than-origin turning is the most consistent and readily understood model used to analyze the nature of the puzzles. Yes, you can view them in other ways, but they require difficult to grasp and impossible to physically implement concepts such as infinite numbers of axes.

Oskar wrote:
Also, your definition is ambiguous for Enabler Cube.
I see the proposed model as being a very clear and unambiguous way of understanding Enabler Cube. Let's actually look at the Bingo Cube rather than the Enabler Cube for the sake of simplicity (two axes rather than three). The same principles apply directly to the Enabler Cube. First we fix our origin in space, then add the two axes (working with the rule that axes do not pass through the origin (okay, I've added that one just now, but it's important). Placing the sliced cube around this origin/axis system, it becomes apparent that to turn the cube on a shallow slice requires moving the axis system, or creating new axes and vanishing others. This is the same situation as with RotoPrism 2 or Trapentrix. The difference (and the confusion) lies in the fact that Bingo Cube (and Enabler Cube) is not a jumbling puzzle. So the number of axes being created is limited to six, and they are all in orthogonal alignment with the original two axes. So if we are consistent with the model I'm proposing, no new axes are needed, and the puzzle(s) can readily be seen as deeper-than-origin turning.

Oskar wrote:
But imaginary mechanisms cannot be part of a definition. What if Carl starts playing Rubik's Cube in a deeper-than-origin way? Does it then suddenly become a deeper-than-origin puzzle?
Contrarily, I believe that in defining the nature of the puzzle, we must ignore any potential physical manifestation. After all, every puzzle made in physical form can also be made and used in exactly the same way as a computer model, without ever having to design a mechanism. Don't misunderstand, I find mechanisms to be the most fascinating part of twisty puzzles, and only use real (as opposed to virtual) puzzles myself. I simply see the particular mechanical manifestation of any given puzzle as separable from the geometry that defines the puzzle.

Looking at Carl's example, I see this as again proving my point. The Rubik's Cube cannot be viewed as deeper than origin because any turn that includes 2/3 of the puzzle will violate the rules of the proposed model by rotating the axis system. While this is certainly possible to do in reality, it is the equivalent of making a shallow turn on RotoPrism 2. The difference being that with RotoPrism 2 you pay the price of quickly losing track of where your axis system is, whereas with Rubik's Cube your axis system remains plainly visible at all times. At the risk of speaking for Carl, I think this is precisely what he was getting at when he said that his example fails because the puzzle defined in the way he did does not have a "holding point". The difference between the model Carl was proposing with holding points, and my fixed origin/axis system model is that in my model the origin/axis system is always the holding point. No other place on the puzzle may be used as a holding point.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Last edited by David Pitcher on Mon Apr 01, 2013 3:09 pm, edited 1 time in total.

Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 12:42 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
The way in which I am proposing to define deeper-than-origin turning puzzles also requires us to agree on a new model for understanding all single-origin planar-sliced puzzles. I suspect that my proposed model will work just as well for puzzles with non-planar slices, and perhaps even multi-origin puzzles, but I will leave it to others to test the model against multiple other puzzles and puzzle types to determine its' effectiveness.

That said, here is my attempt to define the model in a clear and precise manner:

Fix a point in space as the origin, then define the axis system with all axes originating at that point. The origin point and the axis system once defined are considered immovable. The puzzle geometry will be located on this axis system, and the "pieces" (whether real or virtual) of the puzzle will orbit around the origin as allowed by the axis system.

If the slice planes of the puzzle as defined do not touch the origin point, and none of the moving parts of the puzzle cross over the origin, the puzzle is shallow-turning.

If the slice planes of the puzzle as defined touch the origin point, the puzzle is deep-turning.

If the slice planes of the puzzle as defined do not touch the origin point, and some of the moving parts of the puzzle cross over the origin, the puzzle is deeper-than-origin turning.

If multiple slice planes must be engaged to move pieces of the puzzle, whether or not those pieces move over the origin or adjacent to the origin, the puzzle is slice-turning.

Puzzles may be hybrid. In other words, some axes may be deep-turning, and some axes may be shallow-turning within the same puzzle. Similarly, some axes may be slice turning, and some axes may be shallow-turning within the same axis of a puzzle. Any combination of these qualities may be valid within the same puzzle.

I believe that by using this method, we can categorize most, if not all twisty puzzles in regard to their fundamental geometries.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 3:15 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
Scott Elliott 4-part 1x2x2 ...Could you draw your understanding of its cut curve? The two halves each take up 180 degrees and one MUST cut into the other half in order to hold the puzzle together. So I don't see how you can claim they don't extend beyond 180 degrees.
Here you are:
Attachment:
1x2x2 Scott Elliott - view 1.jpg
1x2x2 Scott Elliott - view 1.jpg [ 19.92 KiB | Viewed 5840 times ]
Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 4:18 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
David Pitcher wrote:
Fix a point in space as the origin, then define the axis system with all axes originating at that point. The origin point and the axis system once defined are considered immovable. The puzzle geometry will be located on this axis system, and the "pieces" (whether real or virtual) of the puzzle will orbit around the origin as allowed by the axis system. ...
Dave,

Thank you for your precise formulation. I understand where you want to go. However, I keep having questions.

You are talking about THE axis system. However, not all axes are active all the time. Potential moves are blocked by mid-turn states, visible bandaging and hidden bandaging. So how can we map the axis system onto itself between consecutive turns? The axes have no name or label, and they disappear and reappear all the time. The opening move for scrambling RotaPrism 2 is a 180-degrees turn. The 3-axes system maps onto itself if I model that turn as a shallow turn. So is that first move a shallow turn by your definition?

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 4:56 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Oskar wrote:
However, not all axes are active all the time. Potential moves are blocked by mid-turn states, visible bandaging and hidden bandaging. So how can we map the axis system onto itself between consecutive turns? The axes have no name or label, and they disappear and reappear all the time. The opening move for scrambling RotaPrism 2 is a 180-degrees turn. The 3-axes system maps onto itself if I model that turn as a shallow turn. So is that first move a shallow turn by your definition?
True, in any given puzzle geometry, some axes may be blocked at any given time. This will be particularly true in a mid-turn state, which is not an allowable finish position for a turn of a puzzle. Bandaging may also block some axes at times, but this allowed by the geometry sometimes (as in many jumbling puzzles), and not at other times (as in fracture-cut puzzles). Either way, the origin point and axis system that was used to define the puzzle remains fixed in space.

For RotoPrism 2, the opening move is always a 180 degree turn, but that turn should not be a shallow turn. Yes, you can get away with an initial shallow turn for much the same reason you can get away with a slice turn on a Rubik's Cube, because the axis system maps back onto itself and it is easy enough to follow. With RotoPrism 2 though, if shallow turns are used you quickly run into trouble. As you point out, phantom axes seem to disappear and reappear constantly. This is why it is best to play the puzzle as a deeper-than-origin puzzle. The three axis system that the puzzle is actually built around remains fixed in space, allowing the user to always know where the next turns can be made. This is precisely why RotoPrism 2 is a deeper-than-origin puzzle by the definition I've proposed.

Oskar wrote:
Thank you for your precise formulation. I understand where you want to go. However, I keep having questions.
I'm hopeful that my wording was precise enough to be a good starting point. I'm sure it can be simplified though. I tend to use ten words where two might suffice. Keep asking questions! I do believe that this proposal will hold up to scrutiny, but the more it is tested the better.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 01, 2013 4:57 pm 
Offline
User avatar

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Oskar wrote:
Here you are:
So doesn't this mean the top half of that drawing is shallow or deep cut and the bottom half of the drawing is deeper-then-origin?

Carl

_________________
-
Image

Image


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Apr 02, 2013 7:00 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
wwwmwww wrote:
So doesn't this mean the top half of that drawing is shallow or deep cut and the bottom half of the drawing is deeper-then-origin?
I have no clue what you are suggesting. The drawing shows a single cut that covers exactly 180 degrees from the origin (the dot at the bottom center). This is exactly the boundary case between "shallow cut" and "deeper-than-origin cut". The current term for such a cut is "deep cut".

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Apr 02, 2013 12:16 pm 
Offline

Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I generally like David Pitcher's definition, but it raises an interesting question: Is there any jumbling puzzle for which it's impossible to interpret all slices as being produced by slice planes which only point in a finite number of directions? I suspect that this property is downright common for deep cut jumbling puzzles. In particular, Matt Shepit commented on the shell mechanism deep cut little chop that it's possible to get into positions which look like the solved state but there are slices which can't be done. I suspect that for that puzzle there are sequences of moves which are possible from the start state, but only if it starts in some orientations, and blocked from others.

Obviously any puzzle which has this property can't be completely unbandaged as long as it uses a shells mechanism with a core. You could go coreless, but that presents a lot of problems with keeping the insides aligned, and even if you go with a geared internal mechanism the jumbling will present deep issues, unless you manage to fudge the gear meshings...

This makes me wonder what the jumbling properties of a little chop based on a 2x2x1 as the core would be. It would certainly jumble a lot less, I wonder if it would have to jumble at all.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Apr 02, 2013 1:19 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
David Pitcher wrote:
... This is why it is best to play the [RotoPrism 2] puzzle as a deeper-than-origin puzzle ...
Actually, I play RotoPrism as a mix of shallow turns and deeper-than-origin turns. I make any 180-degrees turn as shallow turn and any 90-degrees turn as deeper-than-origin turn. This is for me the easiest way to play the puzzle. You use only deeper-than-origin turns, which works fine for you. At IPP, I saw a person making only shallow 90- and 180 degrees moves, and a constant reorientation of the puzzle. And if I understand our private conversations correctly, your friends back home play the puzzle making only 180-degrees shallow turns and keeping some cuts unused.

By your earlier definitions, a player can make various types of turns:
    -Shallow turn
    -Slice turn
    -Deep turn
    -Deeper than origin turn

I propose that we make "deeper than origin" a property of an algorithm, instead of the puzzle itself. Here are some examples of definitions that would work for me.
    -A twisty puzzle algorithm is partially deeper-than-origin if it uses at least one deeper-than-origin turn. Example: Oskar's algorithm for RotoPrism 2.
    -A twisty puzzle algorithm is fully deeper-than-origin if it uses only deeper-than-origin turns. Examples: Dave's algorithm for RotoPrism 2, Carl's approach to Rubik's Cube.
We could add further definitions to include the algorithm used by your shallow friends.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Apr 02, 2013 2:44 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Here are some sketches of different types of cut curves, as input to our TP-mechanism terminology discussion. I have noted the following types.
    -Shallow cut curve
    -Origin cut curve
    -Deeper-than-origin cut curve
    -Slice cut curve
    -Offset cut curve
    -Ball cut curve
    -Composite cut curve
Have I missed any type?

Mixup Cube has a composite cut curve, consisting of a deeper-than-origin -, a slice - and a shallow cut curve. Depending on whether we use Oskar's Definition 1, Oskar's Definition 1', or Bram's definition, Mixup Cube has a deeper-than-origin mechanism, or not.

The red dot (.) in the sketches indicates the origin. The green dashed line (---) indicates the turning axis.

Oskar
Attachment:
Classtifying TP cuts - Shallow cut.jpg
Classtifying TP cuts - Shallow cut.jpg [ 34.71 KiB | Viewed 5681 times ]

Attachment:
Classtifying TP cuts - Origin cut.jpg
Classtifying TP cuts - Origin cut.jpg [ 27.84 KiB | Viewed 5421 times ]

Attachment:
Classtifying TP cuts - Deeper-than-origin cut.jpg
Classtifying TP cuts - Deeper-than-origin cut.jpg [ 42.12 KiB | Viewed 5681 times ]

Attachment:
Classtifying TP cuts - Slice cut.jpg
Classtifying TP cuts - Slice cut.jpg [ 36.39 KiB | Viewed 5681 times ]

Attachment:
Classtifying TP cuts - Offset cut.jpg
Classtifying TP cuts - Offset cut.jpg [ 20.14 KiB | Viewed 5681 times ]

Attachment:
Classtifying TP cuts - Ball cut.jpg
Classtifying TP cuts - Ball cut.jpg [ 28.85 KiB | Viewed 5679 times ]

Attachment:
Classtifying TP cuts - Composite cut.jpg
Classtifying TP cuts - Composite cut.jpg [ 34.29 KiB | Viewed 5674 times ]

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Last edited by Oskar on Mon Apr 15, 2013 12:21 pm, edited 1 time in total.

Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 15, 2013 1:15 am 
Offline
User avatar

Joined: Wed Jan 07, 2009 6:46 pm
Location: San Francisco, CA
Wow. What a debate.

I thought I might chip in one little tidbit. A while back, Eric and I were talking about puzzles, and I proposed that we call puzzles whose cuts go perfectly through the origin "Origin Cut" rather than "Deep Cut." I feel it's more accurately descriptive. Some might think it's a little late for a nomenclature change, but perhaps the best time to introduce a better way of saying things is when the theory is in flux.

EDIT: I just watched the video again after reading the debate, and I realized that Oskar mentioned a way to see that it is deeper than origin-cut that I don't remember seeing in the debate. He says that in certain scrambled states, it may look like a cut goes all the way around, but there is some internal bandaging. I'm not sure I agree with this, since I recall there is internal bandaging that happens on More Madness, which it seems everyone agrees is not "deeper than origin."

Don't mind me. While you guys talk, I'm going to go design more puzzles. :P

-Eitan

_________________
Eitan = "EIGHT-ahn"
Buy a Radio Cube 3! Only $150 at Eitan's Shapeways Shop
Check out my video: Twisty Puzzles a la Vi



Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 15, 2013 12:38 pm 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
pirsquared wrote:
... I proposed that we call puzzles whose cuts go perfectly through the origin "Origin Cut" rather than "Deep Cut."
I agree, so I have updated my classification of cuts accordingly.
pirsquared wrote:
I just watched the video again after reading the debate, and I realized that Oskar mentioned a way to see that it is deeper than origin-cut that I don't remember seeing in the debate. He says that in certain scrambled states, it may look like a cut goes all the way around, but there is some internal bandaging. I'm not sure I agree with this, since I recall there is internal bandaging that happens on More Madness, which it seems everyone agrees is not "deeper than origin."
Given the same outside looks, what how can an outsider find out whether a puzzle is implemented with a deeper-than-origin mechanism or not? Say, sort-of a Turing test. One answer is taking the puzzle apart, of course. A more intellectual answer is to use turning and deduction. This way, one can discover that the turning and hidden bandaging of Deeper Madness and Rotoprism 2 is consistent with a deeper-than-origin mechanism, and inconsistent with a shallower-than-origin mechanism. I proposed that a solving-method-based-on-a-deeper-than-origin-mechanism-mental-model-of-the-puzzle is called a deeper-than-origin algorithm for short.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Apr 15, 2013 8:13 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
pirsquared wrote:
I proposed that we call puzzles whose cuts go perfectly through the origin "Origin Cut" rather than "Deep Cut."
I agree that this is perfectly acceptable, and is actually a more correct terminology. Of course, one could then argue that "deep-cut" should actually mean what is currently termed "deeper-than-origin-cut". The progression would be: shallow-cut, origin-cut, deep-cut (slice cuts would need to be defined as combinations of these types of cuts).

Further, I believe that it is most useful to explore and classify the geometries of the puzzles independently from their mechanisms. I also think that this is the best way to define the puzzles regarding their cut types. This does become more complex with puzzles such as Deeper Madness, where it is not clear (at least, not to me yet) which side should be seen as the moving side. If it does not make any difference (and Oskar says that it does although I have yet to wrap my head around the geometry), then it seems that the puzzle is ambiguous, and should default to shallow cut. In the case of Deeper Madness though, I'm inclined to believe that Oskar is correct, and we have another example of a deeper-than-origin-cut (or simply deep-cut) puzzle. But, it would be interesting and informative to know if it is possible to design a mechanism for the puzzle that does not place the core axis pieces (the ones screwed into the core) under the face centers, but rather under the edges opposite the face centers, in a manner similar to the TriTangle puzzle. Would such a puzzle operate in the same way as Deeper Madness?

Looking at the geometry of the puzzles independently from their mechanisms though does not mean I think the mechanisms aren't worthy of their own classifications (as Oskar has provided). One might certainly argue that we could talk about the puzzles' geometries, mechanisms, and playing operations (algorithms) as independent entities. Most often these will probably align with one another, but there may be cases where they don't. For example, one could imagine a mechanism that is (unnecessarily) complex, and actually employs deep cuts for a shallow-cut puzzle. More realistically, one can certainly use a shallow-cut puzzle in a deep-cut (or slice-cut) manner. This happens with great regularity with the Rubik's Cube.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Tue Apr 16, 2013 3:39 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
David Pitcher wrote:
... Deeper Madness, where it is not clear (at least, not to me yet) which side should be seen as the moving side. If it does not make any difference (and Oskar says that it does although I have yet to wrap my head around the geometry), then it seems that the puzzle is ambiguous, and should default to shallow cut.
Dave,

What we are discussing here, is a scientific method to classify twisty puzzles without looking at their mechanism. A good null hypothesis is that a puzzle has a non-bandaged magnetic-core-equivalent type of mechanism, which is shallow-cut by definition. Once the TP scientist starts turning, and finds a turned (blocked or enabled) that is inconsistent with the null hypothesis, then he has to reject it and find a better null hypothesis about the mechanism. The new null hypothesis could be about hidden bandaging, deeper-than-origin cuts and other, using Ockham's razor to select the simplest alternative hypothisis first.

So when the TP scientist starts turning RotoPrism 2, he might first hypothesize it as a simple three-axes shallow-cut puzzle. He would need to revise that hypothis as some point, as there are turns inconsistent with that hypohesis. His next hypothesis might be a nine-axes puzzle, which he would need to reject as well. Ultimately, he would find that RotoPrism 2 moves consistently with a deeper-than-origin mechanism. But he would need to continue trying. Perhaps he finds yet another type of move that would be inconsistent with that hypothesis as well.

For Deeper Madness, shallow-cut is a good null hypothesis. I have managed to find blocked turns inconsistent with that hypothesis, so I believe that the movement of this puzzle is inconsistent with that of a shallow-cut puzzle. However, as my findings have not yet be independently confirmed by another TP scientist (peer review) like Carl, Bram, Eitan or you, its status remains ambiguous.

Oskar

P.S.
David Pitcher wrote:
Of course, one could then argue that "deep-cut" should actually mean what is currently termed "deeper-than-origin-cut". The progression would be: shallow-cut, origin-cut, deep-cut (slice cuts would need to be defined as combinations of these types of cuts).
I see no reason to revise the semantics of "deep-cut". It is the combination of a mechanism that looks origin-cut from the outside, and that has some additional symmetries. For example, whereas the mechanism of Vladimir's Cheese is origin-cut (just check its mechanism), it does not qualifies as fully deep-cut as the puzzle is lacking some of the required symmetries for that.

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Apr 18, 2013 8:16 am 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Oskar wrote:
I see no reason to revise the semantics of "deep-cut".
I agree with this statement, if only to avoid confusion since we have been using "deep-cut" to mean "origin-cut" for so long that it is deeply ingrained.

Oskar wrote:
What we are discussing here, is a scientific method to classify twisty puzzles without looking at their mechanism. A good null hypothesis is that a puzzle has a non-bandaged magnetic-core-equivalent type of mechanism, which is shallow-cut by definition.
My starting point is actually based on a virtual model of the puzzle, where no mechanism (not even a magnetic core) is required to hold the pieces together. I just posted a lengthy explanation in the Deeper Madness thread, so I've just linked to that here rather than repeating the whole thing. I come to the conclusion that Deeper Madness is a shallow-cut geometry, although the mechanism may be deeper-than-origin-cut.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Apr 18, 2013 8:36 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
David Pitcher wrote:
... Deeper Madness thread, ... I come to the conclusion that Deeper Madness is a shallow-cut geometry, although the mechanism may be deeper-than-origin-cut.
Please repeat this analysis for Rotoprism 2, which would help me understand.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Apr 18, 2013 9:56 am 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Oskar wrote:
Please repeat this analysis for Rotoprism 2, which would help me understand.
I've made a preliminary attempt at this with RotoPrism 2, but due to the stored cuts (and my antiquated version of Solidworks) this will be a much more difficult task than it was for the "simpler" puzzles already analyzed. (Not to disparage those puzzles in any way, just that no stored cuts makes them simpler to carve up in the computer). I'll try to do something that comes close, but don't have the time to fully work it through right now.

I will be publishing a new version of RotoPrism 2 in the next couple of days, once I can find time to shoot video and photos. This new version is intended to resolve the "phantom axis" issue by making the blocking that prevents turns on those non-axes visible. Hopefully that will help to make the geometry of the puzzle clearer.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Last edited by David Pitcher on Thu May 30, 2013 10:49 am, edited 1 time in total.

Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu Apr 18, 2013 11:11 am 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Deleted, post was obsolete.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Last edited by David Pitcher on Mon Jun 03, 2013 2:31 pm, edited 1 time in total.

Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Thu May 09, 2013 10:04 pm 
Offline
User avatar

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
Let me start off by saying that I'm no expert on cut geometries, and that I may be totally wrong, but I think there have been a group of puzzles that behave strangely and need to be addressed: what about slice-turn-only puzzles?
The reason I bring this up is because there seem to be several slice-turn-only puzzles which have a deeper-than-origin mechanism but do not have cut geometries of normal deeper-than-origin puzzles; we will use a dodecahedron as an example in this case.
Now, for slice-turn-only puzzles, the cut depth naturally starts out very close to the center of the puzzle as shallow cut and gradually moves away from it when it gets deeper. Why do we know this? Because when we do this, the close cut depths result in the core of the puzzle and as they move away, the core is hidden. Look at a slice-turn-only master pentultimate:
Attachment:
File comment: STO MP.
slice pentultimate 1.jpg
slice pentultimate 1.jpg [ 206.67 KiB | Viewed 4779 times ]

This is a 'shallow cut' puzzle because the core is actually visible; here's a look at the insides. (NOTE: these images are not intended to present a working mechanism-they simply demonstrate the underlying geometry):
Attachment:
File comment: THIS is the core. Notice the extended sections which hold the other pieces.
slice pentultimate 2.jpg
slice pentultimate 2.jpg [ 252.25 KiB | Viewed 4779 times ]

Attachment:
File comment: A look at the floating parts.
slice pentultimate 3.jpg
slice pentultimate 3.jpg [ 273.19 KiB | Viewed 4779 times ]

Notice that the core is essentially the opposite of that of the normal puzzle, since the deep-cut pieces of a normal master pentultimate make up the core here.
What happens if we make the cut 'deeper?' If we do this, we get a slice-turn-only Pyraminx Crystal:
Attachment:
File comment: STO PC.
slice pentultimate 4.jpg
slice pentultimate 4.jpg [ 220.52 KiB | Viewed 4779 times ]

The edge pieces are extended out, which hold the deep-cut corner pieces from the slice kilominx:

There really isn't anything strange or unexpected so far. Let's make the cut even 'deeper' and see what happens. The result is a slice-turn-only megaminx:
Attachment:
File comment: STO Megaminx.
slice pentultimate 5.jpg
slice pentultimate 5.jpg [ 204.14 KiB | Viewed 4779 times ]

Here's a view of the mechanism. (NOTE: the core is highlighted orange):
Attachment:
File comment: A look at the internals.
slice pentultimate 6.jpg
slice pentultimate 6.jpg [ 244.83 KiB | Viewed 4779 times ]

But, what if we make the cut even deeper? In order to do this to make a change, we will also make the cuts conical. The result is a slice-turn-only shallower-cut edges-only Megaminx:
Attachment:
File comment: STO shallow edges-only Megaminx. Notice the 'anti-core' which is represented by the 'corners.'
slice pentultimate 7.jpg
slice pentultimate 7.jpg [ 189.98 KiB | Viewed 4779 times ]

This is where something very unusual appears. Notice the new type of "corner" pieces. An attentive viewer will notice that these pieces make up the 'anti-core' of the puzzle. Here's a view of both the core and 'anti-core' of the puzzle. (NOTE: the 'anti core' is not one solid piece, it is made up of 20 individual parts. The original core is highlighted in orange):
Attachment:
File comment: A view of both the core and 'anti-core.'
slice pentultimate 8.jpg
slice pentultimate 8.jpg [ 268.32 KiB | Viewed 4779 times ]

The reason we can tell that there are two cores is that when one slice turn is made on the puzzle, the orientation of the 'anti-core' changes relative to the normal core. Similarly, if we turn 2 opposite faces, the normal core will change its orientation relative to the 'anti-core.'
So since there is both a core and opposing 'anti-core' in the mechanism at the same time, it appears as though this resulting puzzle has a deeper-than-origin mechanism.
But look at the cut sketch:
Attachment:
File comment: A look at the cut sketch. Notice the total angle is LESS than 180 degrees.
slice pentultimate sketch.jpg
slice pentultimate sketch.jpg [ 213.58 KiB | Viewed 4779 times ]

As you can see, the depth of the cut does NOT exceed 180 degrees relative to the origin (look at the vertical and horizontal line to guide your observations).
So what is going on here? Do different rules apply to slice-turn-only puzzles for determining if they have a deeper-than-origin mechanism? Do slice-turn-only puzzles even have a definable cut depth where it is deeper-than-origin? Is it obvious that I don't understand deeper-than-origin puzzles? Or is this post misleading and entirely false?
I'm really intrigued by this. Let me know your thoughts.

_________________
My Youtube channel
My Shapeways Shop


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri May 10, 2013 1:16 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
Hi Ben,

Fascinating analysis!

Please look again at the classification of cuts.
viewtopic.php?p=299395#p299395
You cut would be classified as a big symmetrical slice-cut.

The reason why Bram suggested any slice cut puzzle to be considered as deeper-than-origin is that the cut crosses the equator. To better understand Bram's point, make a mirror-image copy of the cut, mirrored over the rotation axis.

I would be interested in the views of Bram, Carl, David P and others how they would classify your new puzzle design.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri May 10, 2013 9:47 am 
Offline

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
I don't think slice-turn only puzzles can be classified that way - I would classify them as bandaged instead. But that's just me...


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Fri May 10, 2013 3:43 pm 
Offline
User avatar

Joined: Fri Dec 28, 2012 1:50 pm
Location: Near Las Vegas, NV
Jared wrote:
I don't think slice-turn only puzzles can be classified that way - I would classify them as bandaged instead. But that's just me...

If they are bandaged, then what would an un-bandaged version of a slice-turn-only puzzle look like and how would it function? If you're saying that they are bandaged versions of normal functioning puzzles, you're right-but only in the sense of their behavior.
In the slice Kilominx for example, it is a 'bandaged' puzzle in its behavior (opposite corners behave as though they are connected) but there is no easy way to physically and solidly connect opposite corners in the mechanism, and the mechanism itself is un-bandaged. So are you classifying them by their behavior, or by their mechanism?

_________________
My Youtube channel
My Shapeways Shop


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat May 25, 2013 4:52 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Very interesting series Ben, and apologies for not replying sooner.

Regarding the question of whether slice only puzzles are deeper-than-origin, I think they are a fundamentally different type of puzzle. Certainly many (most?) of them have turning slices that pass over the origin of the puzzle, but those slices are not connected to a face slice. The quality of the deeper-than-origin puzzles that makes them different is that more than half of the puzzle (inclusive of the origin point) must be turned in order to keep the axis system stationary. Most (if not all) slice turn puzzles' slices are not attached to the origin, and so any turn of the puzzle does not affect the orientation of the axis system.

The analysis that you've done is very interesting, although I do think that it should be viewed as an analysis of the mechanisms of the puzzles, rather than of the geometry of the puzzles themselves. I have found it most helpful when examining the geometry of a puzzle to do so without the complications of a mechanism in the way. I'm not certain if this would change much in the way you see this set of puzzles, but it might shed some additional light. For example, if you look at a Mixup Cube in this way (as a set of 27 cubies and an axis system), it readily becomes apparent that the visible face centers cannot be attached to the core of the puzzle as with a normal Rubik's Cube.
benpuzzles wrote:
Is it obvious that I don't understand deeper-than-origin puzzles?
Not at all, and in fact the same might be said of my reply above. I think we're all learning as we go here.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sun May 26, 2013 9:33 am 
Offline
User avatar

Joined: Mon Nov 30, 2009 1:03 pm
David Pitcher wrote:
Regarding the question of whether slice only puzzles are deeper-than-origin, I think they are a fundamentally different type of puzzle.
A twisty puzzle mechanism can have any combination of deeper-than-origin cuts, slice cuts and other types of cuts, as defined in my previous post. The question whether slice-only puzzles are deeper-than-origin, this is a semantic discussion: a discussion about meaning of terminology. The only way to resolve semantic discussions is by providing sharp definitions. I am still missing some good mechanism-agnostic definitions.

Oskar

_________________
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Image.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Jun 03, 2013 1:13 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Oskar wrote:
Please repeat this analysis for Rotoprism 2, which would help me understand.
I have finally managed to get my antiquated version of Solidworks to cooperate enough to fulfill this request, at least as far as I can. The first step in the process was to make the images for the analysis of the Rototribute puzzle. These can be seen in the Rototribute thread. Next I needed to add the cuts that allow all of the face pieces to exchange places with one another. But when I do that the interior of the puzzle gets cut to shreds, and nothing recognizable is left. So for this set of images, I've cleared out the remaining "shards" from the inside of the puzzle, and replaced them with a miniature Fracture-6. As you can see the Fracture-6 actually fills the void exactly, with the vertices of the form touching the internal vertices of the RotoPrism 2 pieces. The first image is the exterior geometry of RotoPrism 2:
Attachment:
rotoprism 2 geometry solid.jpg
rotoprism 2 geometry solid.jpg [ 51.07 KiB | Viewed 4172 times ]

Next, a turning slice is removed, revealing the hollow interior:
Attachment:
rotoprism 2 geometry solid turning slice removed.jpg
rotoprism 2 geometry solid turning slice removed.jpg [ 56.98 KiB | Viewed 4172 times ]

Then I've taken away the top face to reveal more of the interior:
Attachment:
rotoprism 2 geometry solid turning slice and top face removed.jpg
rotoprism 2 geometry solid turning slice and top face removed.jpg [ 60.17 KiB | Viewed 4172 times ]

Now let's take a look at the Fracture-6 that becomes the core of the puzzle. The three turning axes are yellow, and the rest of the pieces are clear. Because this is an origin-turning (or deep-turning) puzzle, there is no core:
Attachment:
fracture-6 geometry.jpg
fracture-6 geometry.jpg [ 10.24 KiB | Viewed 4172 times ]

Now let's place the Fracture-6 into the RotoPrism 2, and you can see how it aligns with the outer parts:
Attachment:
rotoprism 2 geometry solid turning slice and top face removed and fracture-6 core.jpg
rotoprism 2 geometry solid turning slice and top face removed and fracture-6 core.jpg [ 69.52 KiB | Viewed 4172 times ]

And now a couple of different fully transparent views:
Attachment:
rotoprism 2 geometry with fracture-6 core.jpg
rotoprism 2 geometry with fracture-6 core.jpg [ 15.91 KiB | Viewed 4172 times ]
Attachment:
rotoprism 2 geometry solid turning slice and top face removed and fracture-6 core top view.jpg
rotoprism 2 geometry solid turning slice and top face removed and fracture-6 core top view.jpg [ 66.42 KiB | Viewed 4172 times ]

The last top view most clearly shows the alignment of the turning planes between the outer pieces and the inner deep-cut core. Since the core of the RotoPrism 2 is actually a miniature Fracture-6 (although it is highly modified to hold the outer pieces on), this is a true representation of the geometry of the RotoPrism 2 puzzle, sans mechanism.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Jun 03, 2013 2:41 pm 
Offline
User avatar

Joined: Wed Dec 10, 2008 6:26 pm
Location: Boston area
Oskar wrote:
The question whether slice-only puzzles are deeper-than-origin, this is a semantic discussion: a discussion about meaning of terminology. The only way to resolve semantic discussions is by providing sharp definitions. I am still missing some good mechanism-agnostic definitions.
I agree that it is a semantic discussion, but I also believe that the definitions I provided earlier are mechanism-agnostic, and provide the necessary clarity to answer this question:
David Pitcher wrote:
Fix a point in space as the origin, then define the axis system with all axes originating at that point. The origin point and the axis system once defined are considered immovable. The puzzle geometry will be located on this axis system, and the "pieces" (whether real or virtual) of the puzzle will orbit around the origin as allowed by the axis system.

If the slice planes of the puzzle as defined do not touch the origin point, and none of the moving parts of the puzzle cross over the origin, the puzzle is shallow-turning.

If the slice planes of the puzzle as defined touch the origin point, the puzzle is deep-turning.

If the slice planes of the puzzle as defined do not touch the origin point, and some of the moving parts of the puzzle cross over the origin, the puzzle is deeper-than-origin turning.

If multiple slice planes must be engaged to move pieces of the puzzle, whether or not those pieces move over the origin or adjacent to the origin, the puzzle is slice-turning.

Puzzles may be hybrid. In other words, some axes may be deep-turning, and some axes may be shallow-turning within the same puzzle. Similarly, some axes may be slice turning, and some axes may be shallow-turning within the same axis of a puzzle. Any combination of these qualities may be valid within the same puzzle.
The answer is that slice-only puzzles are not deeper-than-origin cut because multiple slice planes must be engaged in order to make a turn.

_________________
Visit Pitcher Puzzles where you can buy the IPP award-winning RotoPrism 2, Fracture-10, and many, many more.


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jun 15, 2013 1:50 pm 
Offline
User avatar

Joined: Sat Feb 03, 2007 10:03 pm
Location: Mississippi
I'm a little late to the party here, but I first wanted to say thanks to Oskar for posting the diagram of the mixup cube cuts. For whatever reason, this is the first time I really got a grasp on how the shells mechanism works!

Perhaps someday I will catch up to all you puzzle building geniuses and make something else new!

_________________
Space for rent


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Sat Jun 15, 2013 3:23 pm 
Offline

Joined: Sun Aug 26, 2012 10:01 am
I'm sorry this post isn't productive to the debate, but this is getting out of hand, it's affecting me in the world outside twisty puzzles! I was walking down the street to go pick my brother up from his football match and I was holding a bottle of Pepsi in my hand; after I had finished taking a sip I put the lid back on the bottle though instead of holding the bottle and twisting the lid on the top, I held the lid and twisted the bottle underneath and without meaning to I asked myself if the bottle was deeper than origin... And then carried on the debate in my head... I have issues :lol:

_________________
My YouTube, including a FF Siamese 2x2x2 Walkthrough


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Jun 17, 2013 9:26 am 
Offline
User avatar

Joined: Fri Nov 05, 2010 2:20 am
Location: Wherever
I had a thought that I would like to add.

We all agree that skewbs are origin cut. (Right?)

Well, if it is modded into a Jing's Pyraminx, would that change the status?

Because an origin cut face turning tetrahedron is trivial, but the skewb isn't trivial.
Would it make any difference if the Jing's pyraminx's mechanism has the spindles connected to the face centers or the vertices? Is the Jing's pyraminx deeper than origin?

_________________
A budding puzzle designer!

Check out my Shapeways shop!


Top
 Profile  
 
 Post subject: Re: Deeper-than-Origin Puzzles
PostPosted: Mon Jun 17, 2013 9:59 am 
Offline
User avatar

Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland
A Jing's Pyraminx is origin-cut, too.

_________________
My pen-and-paper puzzles


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 94 posts ]  Go to page Previous  1, 2

All times are UTC - 5 hours


Who is online

Users browsing this forum: No registered users and 13 guests


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  

Forum powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group