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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Thu Jan 20, 2011 11:22 pm 

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

PuzzleMaster6262 wrote: The fused cube fits the definition yet isn't isomorphic. This definition also would say the Pyraminx (minus the trivial tips) is deep cut.... yet... Allagem wrote: I think everyone agrees that the pyraminx (let's just ignore the tips) is not deepcut (the planes do not meet at a single point, there is a realvolumed core, etc.). Not sure what I think about that. Notice you could shape mod a fused cube into a sphere but you CAN'T place the intersection of the 3 cut planes at the center. As such these cuts DON'T cut the puzzle in half and I think that explains why its not isomorphic. That isn't the case with Shim's Constellation Six so what keeps it from being isomorpic? I do notice both puzzles have "inactive cuts" so maybe that is what allows a deepcut puzzle to be nonisomorphic. Then again my desire to call Oskar's Slice Kilominx a deep cut puzzle may have caused me to jump on this new definition a bit too soon. The fused cube which isn't (and can't be) cut in half being called deep cut seems very odd to me. I'm also not sure I want to accept the Pyraminx (minus the trivial tips) as deep cut. Heck... it's cut planes don't even meet at a single point. From a geometry stand point both just seem wrong... however from a puzzle standpoint it does seem to fit. It's odd how Shim's Constellation Six is now making me question the way I've look at the Pyraminx, a puzzle I've probably owned and known how to solve for 30 years. I'm back on the fence again. PuzzleMaster6262 wrote: Oskar a while ago I thought said something about how his mixup cube wasn't a "pure" puzzle. Also I made a thread about mine in the puzzle building and modding section. Oskar's Mixup Cube makes use of what I called slidey pieces. So it wasn't considered a "pure" twisty puzzle. See Matt's post here and some of my pictures posted later in that thread to see what you get when you try to make a "pure" twisty Mixup Cube. Just found yours. Need to look at that a bit more but it appears you used slidey pieces too. Not sure why you need the extra 24 slidey pieces at the moment. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Thu Jan 20, 2011 11:54 pm 

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

Andreas Nortmann wrote: Argh! What a discussion! Agreed... I'm loving it. I'm not sure how resiticted vs bandaged came into this... I'll have to think a bit more about that. To be honest, I'm not sure I understand the line you are trying to draw. The brain is fried and its too late in the day. But thanks for the links. Those help alot. If (big if) we accept bmenrigh definition of deep cut: On a deepcut puzzle, the effect of any possible twist can be achieved through a different twist + a puzzle reorientation.And these definitions from Bram: A 'doctrinaire' puzzle is one where if you were to remove all the coloration then every single position would look exactly the same.
A shape mod is a nondoctrinaire puzzle which can be shape modded to a doctrinaire puzzle.
A bandage puzzle is a nondoctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.
A jumble puzzle is one which is nondoctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle.Then we can finally say this about Shim's Constellation Six. It is a doctrinaire puzzle. So by definition it is NOT bandaged and it doesn't jumble. And it is deep cut. However I doubt that is the end of the story. Bram states here that the slice kilominx isn't a deep cut puzzle. And based on Bram's definition of deep cut here: In rubiktype puzzles, the axes of rotation generally all go through a single point. A slice is 'deep cut' if it goes through that point.I'm inclined to think he won't consider the fused cube as deep cut either. However I believe Shim's Constellation Six fits his definition. Carl
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Brandon Enright

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 3:14 am 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

wwwmwww wrote: It's odd how Shim's Constellation Six is now making me question the way I've look at the Pyraminx, a puzzle I've probably owned and known how to solve for 30 years. I'm back on the fence again. I think the hangup we are having over the Pyraminx (specifically the HM pyramid) being deepcut comes down to the lack of isomorphism between the halves. On tetrahedrons, a vertex is opposite a face (the polytop is selfdual) so if you pick vertex or face twisting there is no way to cut the puzzle so that it has isomorphic halves. If you make it edgeturning such as the Pyramorphix then you can make isomorphic halves (albeit orthogonal ones). My gut tells me all of the platonic solids should have deepcut instances for the face, vertex, and edge turning variants and indeed, ignoring the tetrahedron they do: Icosahedron: GB 2.1.5; GB 2.2.6; GB 2.3.1 Dodecahedron: Pentultimate; GB 1.2.9; Big Chop Octahedron: Skewb Diamond; <not made or named but very simple puzzle>; GB 4.3.3 Cube: Rubik's 2x2x2; Skewb; Little Chop Tetrahedron: ???; ???; Pyramorphix I think the fact that the tetrahedron is selfdual is what is giving us the trouble. All of the other puzzles have face opposite face, vertex opposite vertex, and edge opposite edge. This prevents the isomorphism of the halves for the tetrahedron. I just can't accept that the face/vertex turning tetrahedron can't be deepcut. Especially when the HM pyramid is just a shape mod of a deepcut puzzle. But my proposed definition is flexible enough to handle irregular shapes and shape mods without worrying about any isomorphism. So Shim's Constellation Six is vertex opposite edge  shrug, seems deepcut to me. EDIT: I just realized that I was using isomorphic in the geometric sense and you guys are using it in the grouptheory sense.
_________________ Prior to using my real name I posted under the account named bmenrigh.
Last edited by Brandon Enright on Fri Jan 21, 2011 7:00 pm, edited 1 time in total.


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 8:18 am 

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

bmenrigh wrote: I think the hangup we are having over the Pyraminx (specifically the HM pyramid) being deepcut comes down to the lack of isomorphism between the halves. On tetrahedrons, a vertex is opposite a face (the polytop is selfdual) so if you pick vertex or face twisting there is no way to cut the puzzle so that it has isomorphic halves.
Tetrahedron: ???; ???; Pyramorphix
I think the fact that the tetrahedron is selfdual is what is giving us the trouble. All of the other puzzles have face opposite face, vertex opposite vertex, and edge opposite edge. This prevents the isomorphism of the halves for the tetrahedron.
I just can't accept that the face/vertex turning tetrahedron can't be deepcut. Especially when the HM pyramid is just a shape mod of a deepcut puzzle. The HM pyramid IS isomophic. For the exact same reson the skewb is isomorphic. There exists a 1to1 mapping between the two halves. Sure not all the pieces mapped to each other are the same shape... but they can be made that way... again look at the Skewb. And with the tetrahedron being selfdual the HM pyramid fills in both the faceturning and vertex turning gaps in your table. All deep cut puzzles that I'm aware of that fill Barm's definition of deep cut are also isomorphic except Shim's Constellation Six. By they way... take a 3x3x3 and bandaged all the corners together. Is it now deep cut? Carl
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stardust4ever

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 9:43 am 

Joined: Sat Mar 24, 2007 6:58 pm Location: Louisiana, US

GuiltyBystander wrote: wwwmwww wrote: Is this puzzle deep cut? One definition says if all the cut planes meet at single point in the center of the puzzle it's deep cut. Here the 3 planes meet in a line.... not a point. Still based on that definition I'd be inclined to say yes, its deep cut. Think of a 1x2x2. I'd say that has 2 deep cut planes and they meet along a line.
However the other definition of deep cut states that each cut plane divides the puzzle into two isomorphic groups of pieces. Is that the case here? If so it certainly isn't obvious to me. The part that rotates has 25 pieces: I would definitely prefer the first definition. The second one seems to have a clear bias against puzzles with an odd number of axis. Under that system, you could never have a deepcut triangular/pentagonal floppy but I think we can all imagine what they would look like. So we are now defining a deep cut puzzle as a puzzle in which the cutting plane splits into two isomorphic groups. Why should oddordered geometry be a problem? Let us start with Rubik's Cheese: It has three axes, 120 degrees apart, on a single plane. I'm sure no one would argue, that while quite a trivially simple puzzle, it is obviously deep cut so that any move will undoubtedly divide the puzzle into two equal isomorphic regions. Next, shape mod the Cheese into a Trigonal Dipyramid with the points along the axes. It is still remains deep cut because it is a shape mod of a deep cut puzzle and therefore functionally the same. Additionally, it now has the same exact shape and rotational axes as Shim's Constellation Six. Now comes the tricky part: Rotate the corner 90 degrees. This blocks rotation of the other axes. The solution? Make new cuts in the mechanism to allow the other axes to rotate, fudging the angles so that the puzzle is not cut to dust. Which presets us with a unique paradox: One you begin slicing the pieces (whether it results in a jumbling puzzle with perfect geometry or a nonjumbling doctrinaire puzzle with invalid geometry), the halves on either side of the primary rotational plane cease to be isomorphic. By slicing up one part into fragments, you also slice the other two parts as well. This results in two identical divisions being performed on one side of each rotational plane, with only one corresponding division of the other side of the rotational plane. As a result of this fact, whenever a move is performed, there are an unequal number of fragments on either side of the rotational plane. This attribute means that Shim's Constellation Six is a deep cut puzzle with distinctly nonisomophic moves. wwwmwww wrote: Also speaking of stickers, you reminded me of something. Lets assume we add a sticker to the circular face center that gives it orientation. If I understand things correctly (not sure I do) then this piece has a continuum of orientation states. All other twisty puzzles that I'm aware of only allow discrete orietation states. Also, unrelated question, are the circles truly circular, or is it some sort of fudged Pentagon? If ithey are circular, then any kind of arbitrary orientation is meaningless, as the star centers (assuming low friction) could be made to rotate in place. In fact, the forces of friction against the rotation plane might even cause it to turn in place by itself. The Triskaidecagon may or may not have prevented this, but either way, it would've made a cool fashionstatement
_________________ My Creepy 3D Rubik's Cube Videocisco wrote: Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 11:18 am 

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

stardust4ever wrote: So we are now defining a deep cut puzzle as a puzzle in which the cutting plane splits into two isomorphic groups. That is one definition that I've seen. I'm still on the fence as to which is best. stardust4ever wrote: Why should oddordered geometry be a problem? I don't believe it is. stardust4ever wrote: Let us start with Rubik's Cheese... it is obviously deep cut so that any move will undoubtedly divide the puzzle into two equal isomorphic regions. Agreed. stardust4ever wrote: Next, shape mod the Cheese into a Trigonal Dipyramid with the points along the axes. It is still remains deep cut because it is a shape mod of a deep cut puzzle and therefore functionally the same. Additionally, it now has the same exact shape and rotational axes as Shim's Constellation Six. Also agreed. stardust4ever wrote: Now comes the tricky part: Rotate the corner 90 degrees. This blocks rotation of the other axes. The solution? Make new cuts in the mechanism to allow the other axes to rotate, fudging the angles so that the puzzle is not cut to dust. Which presets us with a unique paradox: One you begin slicing the pieces (whether it results in a jumbling puzzle with perfect geometry or a nonjumbling doctrinaire puzzle with invalid geometry), the halves on either side of the primary rotational plane cease to be isomorphic. By slicing up one part into fragments, you also slice the other two parts as well. This results in two identical divisions being performed on one side of each rotational plane, with only one corresponding division of the other side of the rotational plane. As a result of this fact, whenever a move is performed, there are an unequal number of fragments on either side of the rotational plane. Too tricky... let's remove one step. If we can do this to your shape modded Rubik's Cheese, surely we can also do it to the unmodded Rubik's Cheese. Correct? Attachment:
Cuts2.png [ 13.74 KiB  Viewed 6549 times ]
Ok the puzzle on the left is the Rubik's Cheese. The Middle and Right picture are the same puzzle now with one half turned 90 degrees. Consider one view and top view and the other a side view. Are the cuts needed to allow rotation in this position the 2 green cuts on each half? Doesn't this retain the isomorphic property. I'm lost when you say "only one corresponding division of the other side". I see two on both sides? Obviously I'm doing something wrong as I agree that Shim's Constellation Six is NOT isomorphic. And I think I almost see what it is... I see if I make these green cuts on BOTH sides that I now have a puzzle with 5 planes that allow rotation in this position. Shim's Constellation Six only has 3 in any given position. Where are you putting the 1 cut on the other side? stardust4ever wrote: If they are circular, then any kind of arbitrary orientation is meaningless, as the star centers (assuming low friction) could be made to rotate in place. In fact, the forces of friction against the rotation plane might even cause it to turn in place by itself. I believe they are circular and I agree in a real world / physical puzzle you are certainly correct. I was thinking about the doctrinaire Constellation Six from a mathematical point. Assume you had a model of it in a PC that you could play with that kept track of the face center orientation and didn't allow friction to come into play and didn't allow it to rotate in place. You'd have a puzzle with an infinite number of states. I just find that to be a very interesting property not that I expect it to have any real world consequences. Considering how close a Triskaidecagon is to a circle and the amount of fudging in this puzzle I doubt that would keep the face centers from being able to be rotated in place. Carl
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Last edited by wwwmwww on Fri Jan 21, 2011 5:33 pm, edited 2 times in total.


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Allagem

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 5:15 pm 

Joined: Sun Oct 08, 2006 1:47 pm Location: Houston/San Antonio, Texas

I JUST WROTE A LONG RESPONSE TO THIS AND BY THE TIME I WAS DONE AND I TRIED TO HIT SUBMIT I HAD TO RELOGIN, THEREBY LOSING MY ENTIRE POST..... Sadly, I don't have time now to retype it all... Short version: All the properties that you guys are observing about these socalled "deepcut" puzzles are a result of the puzzles either being one with a core that has opposite faces, or being one that shares a geometric duality with another shape that has opposite faces. These are tetrahedraloctahedral (in the case of Skewb), and trianglehexagon (in the case of Rubik's Cheese/Ufo as well as the simplified version of the Constellation Six [without the inactive cuts]). These are the EXCEPTIONS! In general, puzzles with cores that do not have opposite faces do not have any of these properties UNLESS they can be expressed as a different puzzle that does have a core with opposite faces. The Constellation Six is based on a puzzle that CAN be expressed as a puzzle with a core that has opposite faces (hexagonthink Rubik's Cheese) but abandons that duality when it applies these extra rotations to only the triangle symmetries of the puzzle. As an additional example, The Meteor Madness/More Madness puzzles have a core that do not have opposite faces and does not share this duality with any core that does have opposite faces. I also still think the term deepcut should not be associated with this particular aspect of puzzles because it originated from the progression of Megaminx to Brilic to Starminx to Master Pentultimate to Pentultimate, the last being the deepcut incarnation. Another argument I made here in my original post includes how the two "isomorphic groups" of a Skewb can't actually replace eachother due to orbital restrictions. This fact in itself also allows the HM Pyramid to retain what you identify as "deepcut" while losing the symmetry along cuts and causing.. not exactly issues, but at least confusion. Puzzles like the Skewb and the Rubik's Cheese have both the symmetries of a shape with opposite faces and a shape without opposite faces inherent within them and so have some properties of both simultaneously. They are EXCEPTIONS. Carl: I HAD a much longer and in depth discussion , but as far as the slice kilominx is concerned, there are similarities between it and traditional deepcut puzzles, specifically having essentially only one move per axis, but the isomorphic groups I believe are coincidence (the slice kilominx is basically the dodecahedral version of this and I don't think it can really be considered deepcut in any way, ESPECIALLY not the traditional way). Call it what you will but I still would like to reserve "deepcut" to specifically mean when the cuts from OPPOSITE faces of a puzzle meet in the middle. I had more but I can't recall it now... wish my original posting didn't get lost. Oh well Start with that, and if you see something I missed, perhaps I will recall more of my original post Peace, Matt Galla PS In case it wasn't clear enough, the main point of my 1st paragraph is simply that these deepcut properties go hand in hand with opposite faces, and I still personally believe we shouldn't use the term deepcut unless it deals with the meeting of cutting planes from opposite faces! Granted, some properties are also present on so called "deepcut" puzzles that don't have opposite faces but all of these are exceptions and still have some sort of connection to a puzzle with opposite faces  unless this geometric ambiguity is removed somehow (as in the case with the Constellation Six), the puzzle's attributes must satisfy all implications of looking at the puzzle from either point of view. It all goes back to opposing faces.


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 5:50 pm 

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

Allagem wrote: I JUST WROTE A LONG RESPONSE TO THIS AND BY THE TIME I WAS DONE AND I TRIED TO HIT SUBMIT I HAD TO RELOGIN, THEREBY LOSING MY ENTIRE POST..... I've had that happen to me more times that I want to admite. If I'm going to write a long post I usually try to write it in word or notepad and then copy and past it into a post here. For shorter replies I still usually try to highlight and copy my entire post before I hit submit just in case. Even doing that I still forget at times or hit the wrong button and have to start over. I only have that issue with twistypuzzles... not sure why but it gets under my skin as well. As for the rest... I don't disagree. Its certainly one way to look at deep cut puzzles. I just feel (not very scientific granted) that term deep cut should be able to be applied to any twisty puzzle... or at least any twisty puzzle that can be shape modded into a sphere. Granted more interesting things happen when you have opposite cuts joining in the center but I think we all agree that a Rubik's Cheese is deep cut too and that isn't happening there. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 6:07 pm 

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

Allagem wrote: Carl: I HAD a much longer and in depth discussion , but as far as the slice kilominx is concerned, there are similarities between it and traditional deepcut puzzles, specifically having essentially only one move per axis, but the isomorphic groups I believe are coincidence (the slice kilominx is basically the dodecahedral version of this and I don't think it can really be considered deepcut in any way, ESPECIALLY not the traditional way). Call it what you will but I still would like to reserve "deepcut" to specifically mean when the cuts from OPPOSITE faces of a puzzle meet in the middle. Think of a slice turn only rubik's cube. It has "essentially only one move per axis" but its NOT isomorphic and its cut surfaces do NOT meet at the point all the axes of rotation go through. It is only deep cut per bmenrigh's definition. If you look at a slice kilominx like this: You can see one way to make it is to use 6 conical cuts that do all meet at the point all the axes of rotation go through. So this puzzle meets Bram's definition of deep cut even though Bram himself doesn't view this as a deep cut puzzle. It's also isomorphic and I do NOT believe that is a coincidence. I believe it is directly related to the fact this puzzle DOES meet Bram's definition of deep cut. Up till Shim's Constellation Six I didn't believe it was possible to have a puzzle meet Bram's definition of deep cut and NOT be isomorphic so I felt those two definitions were redundant. Obviously that isn't the case and I still don't quite see how Shim's Constellation Six pulls that off. But I think stardust4ever has his finger on it.... I'm just not seeing it yet. By the way, I really enjoy these conversations. I love seeing other's point of view. I'm not sure its necessary we all agree or come to the same set of "best" definitions so I hope no one is taking this personally. Carl
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PuzzleMaster6262

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 21, 2011 7:13 pm 

Joined: Sun Jun 13, 2010 1:00 am Location: Colorado

To me a deep cut is a great circle on a sphere transformation of the puzzle. This idea would allow any puzzle regardless of geometry to be classified quickly as deep cut or not. Also wwwmwww, could you post a image of the sphere transformation of this puzzle?(you seem to have amazing POV Raytrace powers) I think this image could greatly help figure out what this puzzle is.
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 22, 2011 11:18 am 

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

PuzzleMaster6262 wrote: To me a deep cut is a great circle on a sphere transformation of the puzzle. We sure don't have any lack of definitions for deep cut here. Two issues with that definition. One you need to specify that the sphere is centered on the point where the axis cross. And two, you limit yourself to planar cuts. Maybe only planar cuts should be allowed but I'm just not convinced. PuzzleMaster6262 wrote: This idea would allow any puzzle regardless of geometry to be classified quickly as deep cut or not. Also wwwmwww, could you post a image of the sphere transformation of this puzzle?(you seem to have amazing POV Raytrace powers) I think this image could greatly help figure out what this puzzle is. Working on that now. I made the model last night but its rendering very slowing due to the way I'm trying to deal with the fudging aspect of this puzzle. At least now I think I see why its not isomorphic and where I went wrong tring to apply these cuts to a Rubik's Cheese. You don't put the green cuts on both sides of the puzzle... just one. You rotate the puzzle 120 degrees and add 2 more cuts. And the you rotate the puzzle 240 degrees and add two more cuts. I thought you'd be done at this point... but NO! It's an iterative process that needs to be repeated an infinite number of times to get the doctrinaire puzzle. I tried 18 iterations last night and POVRay ran out of memory. 17 iterations is trying to render now. Carl
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Allagem

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 22, 2011 5:28 pm 

Joined: Sun Oct 08, 2006 1:47 pm Location: Houston/San Antonio, Texas

wwwmwww wrote: Granted more interesting things happen when you have opposite cuts joining in the center but I think we all agree that a Rubik's Cheese is deep cut too and that isn't happening there.
Not, but it COULD be happening there. I have seen things like this pop up all the time in geometry. If something can be viewed two different ways and in each instance there is some implicit result of looking at it in one of the ways, then whatever the something we are looking at is; it must retain the implicit results of both ways of looking at it. Rubik's Cheese and Skewb each can be constructed two different ways. One way uses cores that do not have opposite faces; the other DOES. Because it is POSSIBLE to come from the geometry of a shape with opposite faces, each of the Rubik's Cheese and Skewb must retain the properties of it being built from a shape with opposite faces. Specifically, they must behave as if they are deepcut in the traditional sense and this includes splitting into isomorphic groups, having the cut planes interset at a point, etc. Here is the cross section of a nondeepcut Rubik's Cheese built up from a hexagonal prism core: Attachment:
HexagonalCheeseShallowerCut.png [ 24.73 KiB  Viewed 6398 times ]
If we continue pushing the cut planes from opposite sides together, we get a puzzle that looks and functions exactly like a Rubik's Cheese > thus a Rubik's Cheese COULD be built from a shape with opposite faces. (If it's not immediately obvious how a Skewb can be constructed from an octahedral core via bandaged Master Skewb design, I can give more details on that) wwwmwww wrote: Think of a slice turn only rubik's cube. It has "essentially only one move per axis" but its NOT isomorphic and its cut surfaces do NOT meet at the point all the axes of rotation go through. It is only deep cut per bmenrigh's definition.
Agreed wwwmwww wrote: If you look at a slice kilominx like this: You can see one way to make it is to use 6 conical cuts that do all meet at the point all the axes of rotation go through. So this puzzle meets Bram's definition of deep cut even though Bram himself doesn't view this as a deep cut puzzle. It's also isomorphic and I do NOT believe that is a coincidence. I believe it is directly related to the fact this puzzle DOES meet Bram's definition of deep cut. Up till Shim's Constellation Six I didn't believe it was possible to have a puzzle meet Bram's definition of deep cut and NOT be isomorphic so I felt those two definitions were redundant. Obviously that isn't the case and I still don't quite see how Shim's Constellation Six pulls that off. But I think stardust4ever has his finger on it.... I'm just not seeing it yet. Very interesting idea... I have to admit I didn't quite understand what you were trying to point out with this picture, but I do now. I do have two concerns though. 1) We just agreed that the Slice Cube is not deepcut/does not form isomorphic groups. However, couldn't you form the topology of a slice cube using conical cuts exactly like you did here, except on a cube? Unfortunately, the only CAD program I have capable of doing this is currently not working, so I have to rely on you for a picture (if you want ). 2) As I am sure you are fully aware, there are "cracks" in this puzzle, empty space that can be mathematically assigned to pieces that are not drawn here. Doing this results in a puzzle that no longer forms isomorphic groups as there are many more piece within the slice than outside of it. Shouldn't that be an issue if we are looking for some sort of mathematical relationship? Just some thoughts to that idea. (I'm still leaning towards coincidence, although you have at least generated some doubts to that effect ) wwwmwww wrote: By the way, I really enjoy these conversations. I love seeing other's point of view. I'm not sure its necessary we all agree or come to the same set of "best" definitions so I hope no one is taking this personally.
Of course not! Any hostility noted in my previous post was directly caused by the loss of my post And yes it has happened to me before, but not for a good three years and I momentarily forgot I had a time limit Peace, Matt Galla


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 22, 2011 9:17 pm 

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

Allagem wrote: Not, but it COULD be happening there. Very nice point. Thank you. Allagem wrote: (If it's not immediately obvious how a Skewb can be constructed from an octahedral core via bandaged Master Skewb design, I can give more details on that) No I see it. Allagem wrote: Very interesting idea... I have to admit I didn't quite understand what you were trying to point out with this picture, but I do now. I do have two concerns though. 1) We just agreed that the Slice Cube is not deepcut/does not form isomorphic groups. However, couldn't you form the topology of a slice cube using conical cuts exactly like you did here, except on a cube? Unfortunately, the only CAD program I have capable of doing this is currently not working, so I have to rely on you for a picture (if you want ). 2) As I am sure you are fully aware, there are "cracks" in this puzzle, empty space that can be mathematically assigned to pieces that are not drawn here. Doing this results in a puzzle that no longer forms isomorphic groups as there are many more piece within the slice than outside of it. Shouldn't that be an issue if we are looking for some sort of mathematical relationship? Just some thoughts to that idea. (I'm still leaning towards coincidence, although you have at least generated some doubts to that effect ) Two of the best arguments I've seen yet against allowing nonplaner deep cuts. Nice!!! I'll try to counter with this: Lets start with your point 2, the conical cuts can be distorted a bit to fill in those cracks, at least on the surface. You'd still need some voids on this inside for the puzzle to be able to turn though. And your point 1 is very interesting. If I did EXACTLY what I did here except on a cube you do (well almost) get another deep cut puzzle. See all the pieces on the slice kilominx are defined as the intersection of 3 cones. That is why there are the gaps or you would have face centers and edges, the pieces you mention in your second point which are required to fill the cracks. You do this exact same thing to a cube and you are just left with the 3x3x3 corners. Give the cones an opening angle approaching 180 degrees to minimize those gaps and you are left with something that looks like a 2x2x2. However looking closer I don't really think this is a 2x2x2. The top half and the bottom half are in the same cone so nothing turns. Hmmm... I think your argument is better then mine. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 22, 2011 10:12 pm 

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

PuzzleMaster6262 wrote: Also wwwmwww, could you post a image of the sphere transformation of this puzzle?(you seem to have amazing POV Raytrace powers) I think this image could greatly help figure out what this puzzle is. Ok... Start with a spherical Rubik's Cheese with red cut planes. Rotate the left side 90 degrees and continue the red cuts from the right side over onto the left as blue cut planes and you have this puzzle. Attachment:
C6_2.png [ 59.87 KiB  Viewed 6343 times ]
This is 1 iteration. Now rotate the puzzle as a whole by 120 degrees to redefine what is left and right and repeat. Attachment:
C6_3.png [ 62.82 KiB  Viewed 6343 times ]
This is the puzzle after 2 iterations. 3 iterations gives this... Attachment:
C6_4.png [ 68.59 KiB  Viewed 6343 times ]
And after 6 iterations... Attachment:
C6_7.png [ 80.62 KiB  Viewed 6343 times ]
After 9 iterations... Attachment:
C6_10.png [ 82.12 KiB  Viewed 6343 times ]
You now can start to see the Constellation Six in this. The square pieces are the corners. The circle face centers are starting to form. And all the arms of the star are now there as well. The little extra bits will be cut out with future iterations. And I miscounted before, POVRay will only let me do 16 iterations. That one is still rendering... has been now for 21.5 hours. And I think it has atleast another day to go. It will still have a few extra bits that haven't been cut away yet. I think if I could run this to 100 or 1000 iterations you would get something that would show all the curved edges of the fudged pieces. I just don't think POVRay will get me there. Carl
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Kapusta

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 22, 2011 10:37 pm 

Joined: Tue Mar 10, 2009 7:06 pm Location: Nowhere in particular.

Wow, that's brilliant. I love how as soon as I start to understand something, you guys come along and make it a thousand times more complicated.
_________________ ~Kapusta
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Current Goals: 7x7 sub 6:30 4x4 sub 1:10


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Brandon Enright

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 22, 2011 10:59 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

Carl, this is looking great. I'd love to see > 16 iterations. Can I lend you computing resources? My fastest box has 8 CPUs (cores) and 48 GB of RAM but if you need more RAM I have a box with 64 GB but 8 slower CPUs. I just compiled POVRay 3.7 Beta 40 but I can match whatever version you are running if you think that is necessary. With regard to your sphere shapemod, it looks like all of those planes meet at the center. Do they? Based on your described iterative construction I don't see how they couldn't all pass through the center point. As for feelings being hurt, no way, I'm enjoying the conversation. If my ideas are bogus you should tell me so and not sugarcoat it. I can't tell where you stand on my deepcut definition but maybe you don't know where you stand on it either . Per my proposed definition the Gear Cube is also deepcut and I don't see how any other definition could really tackle it. As for losing posts due to session timeout, I've had that happen to me before and it is NO FUN AT ALL. Depending on how technically inclined you guys are though, if you lose a post in the future, you can recover it by dumping your browser's process and extracting your post out of the memory footprint. This is easier than it sounds and I have used it several times to recover lost posts for both myself and others. It's basically just procdump + strings + grep. Posts get memcpy()'d around so much that there are literally hundreds of copies of it which means the post is recoverable even after further browser use.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 1:02 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

bmenrigh wrote: On the subject of what "deepcut" means, my definition is nearly the same as Matt's. That is:
On a deepcut puzzle, the effect of any possible twist can be achieved through a different twist + a puzzle reorientation. First reaction: I like it. Second reaction: Isn't every cut be deep under that definition? Here's doing an L turn vs. doing 2 layers of R + reorient. http://www.randelshofer.ch/cube/rubik/?Lhttp://www.randelshofer.ch/cube/rubik/?TRCR'Allagem wrote: 1) We just agreed that the Slice Cube is not deepcut/does not form isomorphic groups. However, couldn't you form the topology of a slice cube using conical cuts exactly like you did here, except on a cube? Unfortunately, the only CAD program I have capable of doing this is currently not working, so I have to rely on you for a picture (if you want ). 2) As I am sure you are fully aware, there are "cracks" in this puzzle, empty space that can be mathematically assigned to pieces that are not drawn here. Doing this results in a puzzle that no longer forms isomorphic groups as there are many more piece within the slice than outside of it. Shouldn't that be an issue if we are looking for some sort of mathematical relationship? Just some thoughts to that idea. (I'm still leaning towards coincidence, although you have at least generated some doubts to that effect ) I was thinking the exact same thing. It seems the isomorphic slice deep cut theory only works for a small fraction of piece types. A few examles are the dino cube and a corners only helicopter cube. I thought I'd try my hand at "unbandaging" this monster manually with Rhino and here are my results. I basically just repeatedly did (R' L'). I wasn't counting so I don't know how many iterations this was. Attachment:
whole.png [ 23.36 KiB  Viewed 6284 times ]
Here's a close up of 1 face Attachment:
face.png [ 6.6 KiB  Viewed 6284 times ]
It looks like a nice 13 sided polygon in the middle, but if we zoom in and look at the next cut (in green) you'll see that it clearly isn't. Attachment:
not13sides.png [ 4.07 KiB  Viewed 6284 times ]
Based on these results, I believe this is what the full unbandaging would look like. Attachment:
asymptote.png [ 8.79 KiB  Viewed 6284 times ]
The black region would contain an infinite number of pieces. I don't know if these infinite number of pieces would be infinitely small or if there would be some kind of fractal shrinking pattern. I'm leaning towards the latter. I'd like to write a program to slice it up a few hundred or thousand times to know for sure.
_________________ Real name: Landon Kryger


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Brandon Enright

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 1:46 am 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

GuiltyBystander wrote: bmenrigh wrote: On the subject of what "deepcut" means, my definition is nearly the same as Matt's. That is:
On a deepcut puzzle, the effect of any possible twist can be achieved through a different twist + a puzzle reorientation. First reaction: I like it. Second reaction: Isn't every cut be deep under that definition? Here's doing an L turn vs. doing 2 layers of R + reorient. I wanted the definition to be succinctly stated in one sentence without extraneous explanation. By "twist" I mean any minimum set of movable pieces. "2 layers of R" is two twists and violates the definition. As Carl has pointed out, with this definition puzzles like the SliceKilominx and Gelatinbrain's 1.1.50 are deepcut. I'm not sure if that is telling me the definition is wrong or that the puzzles are deepcut. I'm leaning towards the latter.
_________________ Prior to using my real name I posted under the account named bmenrigh.


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stardust4ever

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 5:05 am 

Joined: Sat Mar 24, 2007 6:58 pm Location: Louisiana, US

stardust4ever wrote: By slicing up one part into fragments, you also slice the other two parts as well. This results in two identical divisions being performed on one side of each rotational plane, with only one corresponding division of the other side of the rotational plane. wwwmwww wrote: I'm lost when you say "only one corresponding division of the other side". I see two on both sides? Obviously I'm doing something wrong as I agree that Shim's Constellation Six is NOT isomorphic. And I think I almost see what it is... I see if I make these green cuts on BOTH sides that I now have a puzzle with 5 planes that allow rotation in this position. Shim's Constellation Six only has 3 in any given position. Where are you putting the 1 cut on the other side? Let me rephrase that for clarity: The axis you are making the unbandaging cuts on, these cuts must also be replicated on the other axes. Suppose we label the cutting planes "A", "B", and "C", along with their perpendicular axes, "a", "b", and "c". Any cut to the part connected to the "a" axis will have to be duplicated on parts connected to the "b" and "c" axes. Therefore, a cut made to a part orbiting the "a" axis on one side of the "A" plane, must also be duplicated on the "b" and "c" axes, creating two cuts on the opposite side of the "A" plane. And because the "B" and "C" planes are both congruent to the "A" plane, they also exhibit the same inequality. This symmetry is wholly a threefold rotation about the centroid, never a twofold reflection across any cutting plane. There obviously exists certain twofold symmetry within the puzzle, for example the top half related to the bottom half, but these reflectional symmetries do not exist about any cutting plane. It is interesting that people bring up the Skewb geometry with regards to deepcut isomerism, as the two hemispheres are not reflectional like the Junior Cube, but instead involve a 180 degree inversion of sorts. So we now seem to have three catagories of deepcut puzzles: Even deepcut puzzles, with reflective isometric symmetry (2x2x2 Cube) Odd deepcut puzzles, with inverse isometric symmetry (Skewb, pentultimate) And NonIsometric deepcut puzzles , such as the Constellation Six. I am sure there are several others that fit this definition as well; I just cannot name them. I borrow my terminology from the mathematical definitions of even and odd functions: http://en.wikipedia.org/wiki/Even_and_odd_functionsFor example, in Trigonometry, the Sine function is odd while the Cosine function is even. Many functions also exist, such as Exponent (y=e^x), which are neither even nor odd.
_________________ My Creepy 3D Rubik's Cube Videocisco wrote: Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 10:42 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

bmenrigh wrote: I wanted the definition to be succinctly stated in one sentence without extraneous explanation. By "twist" I mean any minimum set of movable pieces. "2 layers of R" is two twists and violates the definition. Playing devil's advocate, if you don't think of the Fused Cube as a bandaged 3x3x3, it has 2 layers per axis and several inactive cuts.
_________________ Real name: Landon Kryger


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 10:10 pm 

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

bmenrigh wrote: Carl, this is looking great. I'd love to see > 16 iterations. Can I lend you computing resources? My fastest box has 8 CPUs (cores) and 48 GB of RAM but if you need more RAM I have a box with 64 GB but 8 slower CPUs. I just compiled POVRay 3.7 Beta 40 but I can match whatever version you are running if you think that is necessary. Here is my POVRay code: http://wwwmwww.com/Puzzle/C6.povThis is the file I have running now for 16 iterations. Its very basic so it should be easy to add more iterations if you want. However I fear my code is probably horribly inefficent. I'm sure there are some experts in the POVRay forums that could get this image in seconds instead of the week it looks like it may take my PC to finish it. And a 3D raytracer is probably way over kill if you are after the 2D fractal patern GuiltyBystander expects to find. My guess he may have it before my 16 iteration image is finished rendering. Then again you may be able to get my image before I do with that PC of yours and I've got a 1 day 21 hour head start. I'm currently rendering at a rate of 25 pixils per minute and the image is 600x600... if I did my math correctly I think that is coming out to 10 days. bmenrigh wrote: With regard to your sphere shapemod, it looks like all of those planes meet at the center. Do they? Based on your described iterative construction I don't see how they couldn't all pass through the center point. Yes, all cuts are arcs of great circles and cut to the center point. Only the 3 cut planes that allow rotation go all the way through the puzzle. bmenrigh wrote: As for feelings being hurt, no way, I'm enjoying the conversation. If my ideas are bogus you should tell me so and not sugarcoat it. I can't tell where you stand on my deepcut definition but maybe you don't know where you stand on it either . Per my proposed definition the Gear Cube is also deepcut and I don't see how any other definition could really tackle it. As you say, I'm not sure where I stand. I think there are merits to several. I sort of like stardust4ever's idea of having several catagories of deepcut puzzles. Up till Matt's post I was convinced a conical cut could be deepcut and now that has pretty will been shot down in my mind. At the moment I think I'm leaning toward something like this: (I think this is basically Bram's definition) A deep cut puzzle = a puzzle which has all its axis of rotation meet at a point and all its cut planes which allow rotation pass through this point. A 4x4x4 does have deep cuts but the 2x2x2 is the pure deep cut puzzle of that geometry. Just wait till the next puzzle comes along and turns that definition on its ear... if not sooner... I may feel differently tomorrow. bmenrigh wrote: As for losing posts due to session timeout, I've had that happen to me before and it is NO FUN AT ALL. Depending on how technically inclined you guys are though, if you lose a post in the future, you can recover it by dumping your browser's process and extracting your post out of the memory footprint. This is easier than it sounds and I have used it several times to recover lost posts for both myself and others. It's basically just procdump + strings + grep. Posts get memcpy()'d around so much that there are literally hundreds of copies of it which means the post is recoverable even after further browser use. My technically inclined rating must be zero. procdump + strings + grep might as well be batmobile + string theory + sexy nurse to me. Boy do I feel old... and to think 20 years ago I was building my own PCs from the motherboard on up. I paid over a $1K for my first 800MB hard drive. Could you make a post... let it get lost... and walk us through the recovery with say a YouTube video? Knowing me even then I'll get lost but I don't have a clue how to do what you are talking about. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 10:32 pm 

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

GuiltyBystander wrote: The black region would contain an infinite number of pieces. I don't know if these infinite number of pieces would be infinitely small or if there would be some kind of fractal shrinking pattern. I'm leaning towards the latter.
I'd like to write a program to slice it up a few hundred or thousand times to know for sure. Please do... I'd love to see that as well. In POVRay I have to give the cut some width and I was expecting all these pieces to get cut away. But do all the cutting with 2D planes and I too think you might find an interesting fractal pattern. I'm curious what the surface area of the largest piece might be... if there is a largest. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 10:43 pm 

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

stardust4ever wrote: Let me rephrase that for clarity: The axis you are making the unbandaging cuts on, these cuts must also be replicated on the other axes. Suppose we label the cutting planes "A", "B", and "C", along with their perpendicular axes, "a", "b", and "c". Any cut to the part connected to the "a" axis will have to be duplicated on parts connected to the "b" and "c" axes. Therefore, a cut made to a part orbiting the "a" axis on one side of the "A" plane, must also be duplicated on the "b" and "c" axes, creating two cuts on the opposite side of the "A" plane. And because the "B" and "C" planes are both congruent to the "A" plane, they also exhibit the same inequality. This symmetry is wholly a threefold rotation about the centroid, never a twofold reflection across any cutting plane. There obviously exists certain twofold symmetry within the puzzle, for example the top half related to the bottom half, but these reflectional symmetries do not exist about any cutting plane. Thanks, that helps. stardust4ever wrote: It is interesting that people bring up the Skewb geometry with regards to deepcut isomerism, as the two hemispheres are not reflectional like the Junior Cube, but instead involve a 180 degree inversion of sorts.
So we now seem to have three catagories of deepcut puzzles: Even deepcut puzzles, with reflective isometric symmetry (2x2x2 Cube) Odd deepcut puzzles, with inverse isometric symmetry (Skewb, pentultimate) And NonIsometric deepcut puzzles , such as the Constellation Six. I am sure there are several others that fit this definition as well; I just cannot name them. Nice! I like this idea. I believe the Constellation Six is the only puzzle in the last catagory at the moment. Others are probably in the works as we speak. stardust4ever wrote: Many functions also exist, such as Exponent (y=e^x), which are neither even nor odd. That's the understatement of the year. Not only are there infinitely many such functions but I believe there are an uncountably infinite amount. I hung out with too many math types when I was in grad school. LOL!!! Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 23, 2011 10:47 pm 

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

GuiltyBystander wrote: Playing devil's advocate, if you don't think of the Fused Cube as a bandaged 3x3x3, it has 2 layers per axis and several inactive cuts. Yes, that puzzle has been mentioned already above. It does share some interesting properties with deep cut puzzles... the Constellation Six in particular in having inactive cuts. But is it deep cut? All depends on the definition you like most... Carl
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Timur

Post subject: Re: Shim's Constellation Six Posted: Mon Jan 24, 2011 3:05 am 

Joined: Mon Mar 22, 2010 7:00 am Location: Germany, Siegerland

Quote: After 9 iterations... Very nice picture! I've been wondering how the projection on a sphere would look like, but could not imagine that. stardust4ever wrote: And NonIsometric deepcut puzzles , such as the Constellation Six. I am sure there are several others that fit this definition as well; I just cannot name them. I can propose at least one more  take a pentagonal dipyramid and try making similar cuts. I didn't try myself, but I think we'll see similar "stars" with a circle in the center. This is a picture that I got before the pieces were rounded: Attachment:
cs.jpg [ 42.89 KiB  Viewed 6117 times ]
On the picture you can see that the 13gon's position is unstable and loose  it's fixed only at 5 corners.
_________________ Timur aka Shim Signed Vulcanos ($48), Pillow Pyraminxes ($22) Come visit my Shapeways shop


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Mon Jan 24, 2011 2:27 pm 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

GuiltyBystander wrote: I'd like to write a program to slice it up a few hundred or thousand times to know for sure. Bleh, I kind of did this sloppy so there are quite a few graphical glitches from numerical errors. I'll probably fix it later, but was kind of excited to show some results. So here's what the center region looks like after (R' L') x 100. I wanted to do (random(R,R') random(L,L'))x100 but I think numerical errors/bugs causes that thing to explode. Click to see full size Attachment:
100xRL.png [ 108.95 KiB  Viewed 6075 times ]
Close up of some corners. Attachment:
100xRLzoom.png [ 91.86 KiB  Viewed 6075 times ]
I tried sketching this up in Rhino and saw a startling pattern. Attachment:
fib1.png [ 20.97 KiB  Viewed 6075 times ]
1  purple 2  red 3  orange 5  yellow 8  green 13  blue Infinite  black region Can anyone tell me why the Fibonacci sequence is popping up here? Based on my program though, I don't see how we'll get 21 of anything so maybe it's just a coincidence. I can't tell for sure, but I think there's 18 of the next piece type.
_________________ Real name: Landon Kryger


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Jeffery Mewtamer

Post subject: Re: Shim's Constellation Six Posted: Mon Jan 24, 2011 9:08 pm 

Joined: Sun Nov 23, 2008 2:18 am

GuiltyBystander wrote: Can anyone tell me why the Fibonacci sequence is popping up here? Based on my program though, I don't see how we'll get 21 of anything so maybe it's just a coincidence. I can't tell for sure, but I think there's 18 of the next piece type. The overall shape of the array is a distorted pentagram, and a regular pentagram has four powers of the golden ratio among its segment lengths. Also, the golden ratio is the limiting ratio between consecutive terms of the Fibonacci sequence. Whether this connection is mathematically significant or merely coincidence, I cannot say.
_________________ Just so you know, I am blind.
I pledge allegiance to the whole of humanity, and to the world in which we live: one people under the heavens, indivisible, with Liberty and Equality for all.
My Shapeways Shop


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Mon Jan 24, 2011 11:59 pm 

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

GuiltyBystander wrote: 1  purple 2  red 3  orange 5  yellow 8  green 13  blue Infinite  black region
Can anyone tell me why the Fibonacci sequence is popping up here? Based on my program though, I don't see how we'll get 21 of anything so maybe it's just a coincidence. I can't tell for sure, but I think there's 18 of the next piece type. Very very interesting. The Fibonacci sequence shows up in alot of odd places. Here are a few examples: http://www.branta.connectfree.co.uk/fibonacci.htmBut I see what you mean as the next number appears to be 18. I think I'm more surprised that this deviates from the Fibonacci sequence. Could the real sequence be one of these? http://oeis.org/A120761 or http://oeis.org/A081612Is the next number 31 or 24? I can't say I'm that familiar with either sequence or why either might be showing up here if in fact it is one of these. Looks like we have a new way to make a "higher order" puzzle. You could make a "higher order" Constellation Six using the exact same cuts but with the 8 green pieces added. And who is going to be the first to post a picture of the a pentagonal dipyramid with similar cuts that Timur proposed. I have POVRay tied up at the moment and would love to see it. Carl
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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 1:49 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

Jeffery Mewtamer wrote: The overall shape of the array is a distorted pentagram, and a regular pentagram has four powers of the golden ratio among its segment lengths. Also, the golden ratio is the limiting ratio between consecutive terms of the Fibonacci sequence. Whether this connection is mathematically significant or merely coincidence, I cannot say. Ah, good catch. I had forgotten about the golden ratio in the pentagram. However, if I had to guess, the sequence seems to be coming from the distortedness of the pentagram, rather than from the pentagram itself. Can't say for sure either way, so you could still be right. Wikipedia link for those curious: http://en.wikipedia.org/wiki/Pentagram#Golden_ratiowwwmwww wrote: But I see what you mean as the next number appears to be 18. I think I'm more surprised that this deviates from the Fibonacci sequence. Could the real sequence be one of these? http://oeis.org/A120761 or http://oeis.org/A081612Is the next number 31 or 24? I can't say I'm that familiar with either sequence or why either might be showing up here if in fact it is one of these. It seems great minds think alike. I already tried checking the integer sequence database. As far as I can tell, the next pieces are significantly smaller such that their count increase is possibly linear for a while. Namely (5 + 8 *n) which would make the sequence 1,2,3,5,8,13,18,23,28,33,38... i.e. none of the above. I'm not sure when this pattern ends and when a new pattern takes over. The pieces are incredibly slender at this point so it's hard to tell what's happening. I'm thinking about trying to render them by taking their polar coordinates and plotting them as if they were XY coordinates so that it would unroll it. I'd obviously make the "r" axis log so that it would stretch them out more as they got tinier. But seeing as I have several bugs already, it won't be soon. wwwmwww wrote: Looks like we have a new way to make a "higher order" puzzle. You could make a "higher order" Constellation Six using the exact same cuts but with the 8 green pieces added. I was thinking this too would be an interesting puzzle. The added pieces are at an increased "depth" but it's a different kind of depth than the megaminx>brilic>starminx>pentultimate progression. Someone has already posted a solving thread on this puzzle. It looks like they go from the outside > in, so adding more pieces in this fashion wouldn't screw up your solving method. Of course, making a physical version with these extra parts would be tricky I'm sure. The pieces are quite small and you can't use any curved cuts to make them bigger wwwmwww wrote: And who is going to be the first to post a picture of the a pentagonal dipyramid with similar cuts that Timur proposed. I have POVRay tied up at the moment and would love to see it. I could probably get a rough sketch tomorrow. One thing that initially bugged me about the pentagonal dipyramid is that it's sort of been done before but it didn't have this infinite number of pieces. This confused me because you could have a square prism (Rubik's cube) and pentagonal prism (Oskar's Illegal Cube) with 90 degree turns, but the triangular prism doesn't allow them (without jumbling). I'm guessing it has something to do with the fact that those pieces are from a shallow cut and have finite volume or something whereas the pieces on the Constellation 6 have infinite volume.
_________________ Real name: Landon Kryger


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stardust4ever

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 3:12 am 

Joined: Sat Mar 24, 2007 6:58 pm Location: Louisiana, US

GuiltyBystander wrote: Close up of some corners. Attachment: The attachment 100xRLzoom.png is no longer available I tried sketching this up in Rhino and saw a startling pattern. Attachment: The attachment fib1.png is no longer available 1  purple 2  red 3  orange 5  yellow 8  green 13  blue Infinite  black region Can anyone tell me why the Fibonacci sequence is popping up here? Based on my program though, I don't see how we'll get 21 of anything so maybe it's just a coincidence. I can't tell for sure, but I think there's 18 of the next piece type. I don't think that this is necessarily the start of an infinite progression of the Fibonacci sequence. You will likely not find any larger numbers in the sequence for a while. The shape Timur has created is an excellent nearmiss of a triskaidecagram, or 13gram (13 point star). The number 13 (despite the superstitious stigma associated with it) happens to be prime as well as a member of the Fibonacci sequence. A near miss, when refering to mathematics, simply means that the exact value of something is extremely close, but not exactly, equal to something else. The rational approximation of 355/113 for Pi is an excellent example of a mathematical nearmiss. The nearmiss Triskaidecagram has been bandaged as follows, using GuiltyBystander's coloring scheme: Attachment:
triskaidecagram.png [ 43.02 KiB  Viewed 5971 times ]
Attachment:
triskaidecagram Constellation Six.png [ 56.25 KiB  Viewed 5971 times ]
_________________ My Creepy 3D Rubik's Cube Videocisco wrote: Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 8:57 am 

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

GuiltyBystander wrote: Someone has already posted a solving thread on this puzzle. Nice!!! GuiltyBystander wrote: I'm guessing it has something to do with the fact that those pieces are from a shallow cut and have finite volume or something whereas the pieces on the Constellation 6 have infinite volume. You lost me with infinite volume. You mean Infinitesimal volume talking about the many small pieces not present on the Constellation 6? And speaking of Oskar's Illegal Cube, would it be possible to apply an iterativie process to io's jumbling version seen here. To turn it into the fudged version. If so what does that say about fudging and jumbling. Are they really one in the same? Something like... a fudged puzzle = a jumbling puzzle where an infinitely many tiny pieces have been removed. And does that make the Mixup Cube a fudged puzzle? The fudging just being done on the inside of the puzzle and not on the surface. This could remove the term "slidey pieces" which hasn't been very well defined yet from the vocabulary. Carl
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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 11:57 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

stardust4ever wrote: The nearmiss Triskaidecagram has been bandaged as follows, using GuiltyBystander's coloring scheme: Yeah, that makes perfect sense now. I didn't think it would fully replicate the Fibonacci sequence because the next number was 18. I just thought it was kind of odd that it partially had it. wwwmwww wrote: You lost me with infinite volume. You mean Infinitesimal volume talking about the many small pieces not present on the Constellation 6? Sorry, yeah, should have clarified that. If you take an infinitely size volume and apply the Constellation 6 cuts to it, the pieces would have infinite volume. This would not be true for example with megaminx edges or corners. wwwmwww wrote: Something like... a fudged puzzle = a jumbling puzzle where an infinitely many tiny pieces have been removed. I was thinking the exact same thing. However, io's picture shows that the Illegal Cube doesn't jumble so fudge puzzle = a puzzle with a finite or infinite number of pieces removed. wwwmwww wrote: And does that make the Mixup Cube a fudged puzzle? The fudging just being done on the inside of the puzzle and not on the surface. I assume you're referring to your work in this thread (and previous page a bit). And in that case, I guess so.
_________________ Real name: Landon Kryger


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Brandon Enright

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 12:01 pm 

Joined: Thu Dec 31, 2009 8:54 pm Location: Bay Area, California

wwwmwww wrote: GuiltyBystander wrote: Someone has already posted a solving thread on this puzzle. Nice!!! The easiest way to solve puzzles like this with a ton of pieces is to start from the most shallowcut pieces and work your way deeper. Think my solution to the multidodecahedron or Julian's to the multiedgecube. This isn't exactly how pytlivyj_1 tackles the Constellation 6 but all of his solving methods I have seen seem pretty unique. Solving shallow > deep should allow one to continually adapt their solution to new versions of the Constellation 6 with more and more small pieces. wwwmwww wrote: GuiltyBystander wrote: I'm guessing it has something to do with the fact that those pieces are from a shallow cut and have finite volume or something whereas the pieces on the Constellation 6 have infinite volume. You lost me with infinite volume. You mean Infinitesimal volume talking about the many small pieces not present on the Constellation 6? I was a bit lost too but I thought he meant infinite surface area. A fractal like the Mandelbrot set has an infinite perimeter but a finite area. The sum total of the pieces of the Constellation should have a finite volume but an infinite surface area. wwwmwww wrote: And speaking of Oskar's Illegal Cube, would it be possible to apply an iterativie process to io's jumbling version seen here. To turn it into the fudged version. If so what does that say about fudging and jumbling. Are they really one in the same? Something like... a fudged puzzle = a jumbling puzzle where an infinitely many tiny pieces have been removed. And does that make the Mixup Cube a fudged puzzle? The fudging just being done on the inside of the puzzle and not on the surface. This could remove the term "slidey pieces" which hasn't been very well defined yet from the vocabulary. I'm skeptical. If you tried to unjumble / unfudge a Tutt's Minx do you get an infinite number cuts (and tiny pieces)? If so then is the nonfudging version of Tutt's Minx a jumbling puzzle just because a hexagonal face can block a pentagonal one?
_________________ Prior to using my real name I posted under the account named bmenrigh.


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 12:45 pm 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

GuiltyBystander wrote: I could probably get a rough sketch tomorrow. Well a deepcut pentagonal dipyramid will shape shift because it doesn't have rotational symmetry about the vertices like the triangular dipyramid does. Attachment:
pentdipyramind.png [ 7.97 KiB  Viewed 5888 times ]
So after (R' L') x40, my program ends up with this. I'm only cutting the stickers at the moment so that's why it looks like something threw up. Attachment:
const540xRL.png [ 175.29 KiB  Viewed 5888 times ]
You can still see some interesting developments however. There appear to be several circles with varying size. Reminds me of the circle packing problem. I wonder how far this pattern goes. So, I thought I'd try it with a sphere. I'm approximate one with a 16x16 array of rectangles. It's not a perfect sphere so there's still a few gaps in it. Again, here's (R' L') x40. Attachment:
const5sphere40xRL.png [ 264.72 KiB  Viewed 5888 times ]
You can see a giant circle, but that is a product of me restricting myself to (R' L') which will keep that 1 piece in the center and it is constantly being "polished." So here it is again with (R' L) x40 which should hopefully chew that big piece up. Attachment:
const5spherechew40xRL.png [ 307.43 KiB  Viewed 5888 times ]
That seemed to break up the big piece. I don't see any overlying fractal pattern taking over though. Perhaps that's just due to my crude rendering, but I thinking that it probably isn't. For my last trick, I decided to try turning all vertices in a staggered fashion: (a1' a2 a3' a4 a5')x20. Attachment:
const5sphere20x12345.png [ 441.14 KiB  Viewed 5888 times ]
It appears that all but the large circles have been killed. There are 10 of them that I assume correspond to the 10 faces of the pentagonal dipyramid. Other than that, I don't see any pattern. There are a few small circles scattered about, but because they don't appear at regular intervals, I believe they are a side effect of the limited scramble I'm doing.
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Timur

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 1:11 pm 

Joined: Mon Mar 22, 2010 7:00 am Location: Germany, Siegerland

GuiltyBystander wrote: Well a deepcut pentagonal dipyramid will shape shift because it doesn't have rotational symmetry about the vertices like the triangular dipyramid does. I meant this kind of a model: Attachment:
5DP.jpg [ 22.95 KiB  Viewed 5876 times ]
Because edges going to the tops are now longer (as well as for any other kind of a dypiramid), it's not deep cut (the cutting layers do not meet in the central axis), but its rotational properties are similar to those of Constellation 6.
_________________ Timur aka Shim Signed Vulcanos ($48), Pillow Pyraminxes ($22) Come visit my Shapeways shop


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 1:18 pm 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

Timur wrote: I meant this kind of a model:(image) Yeah, after finishing my previous renderings, I started to work on that. Here's what 1 face looks like after (R' L) x40 Attachment:
const5shallow40xRL.png [ 115.1 KiB  Viewed 5867 times ]
Here's the start of the piece count progression: 1,3,4,7,10,13,23...
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 11:38 pm 

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

GuiltyBystander wrote: Here's the start of the piece count progression: 1,3,4,7,10,13,23... There are two 1's. The circle center and the very top piece. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Tue Jan 25, 2011 11:53 pm 

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

GuiltyBystander wrote: Sorry, yeah, should have clarified that. If you take an infinitely size volume and apply the Constellation 6 cuts to it, the pieces would have infinite volume. This would not be true for example with megaminx edges or corners. Ok... but that isn't true of all shallow cuts. Look at the edges and corners on a 3x3x3. So I don't really think the volume of the piece comes into play here... but you never know. GuiltyBystander wrote: I was thinking the exact same thing. However, io's picture shows that the Illegal Cube doesn't jumble so fudge puzzle = a puzzle with a finite or infinite number of pieces removed. Lets take a look back at these definitions by Bram: A 'doctrinaire' puzzle is one where if you were to remove all the coloration then every single position would look exactly the same.
A shape mod is a nondoctrinaire puzzle which can be shape modded to a doctrinaire puzzle.
A bandage puzzle is a nondoctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.
A jumble puzzle is one which is nondoctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle.I believe Oskar's Illegal Cube doesn't jumble. It's doctrinaire so it doesn't jumble by definition. But the puzzle io shows in that thread obviously isn't doctrinaire. After one 90 degree turn stored cuts are in different places. So let's try this... Fudged Puzzle = a jumbling puzzle which has been made doctrinaire via the removal of an infinite number of pieces. Is a 2x2x2 fudged? It can be viewed as a 3x3x3 with the edges and face centers removed. Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Wed Jan 26, 2011 12:18 am 

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

bmenrigh wrote: I'm skeptical. If you tried to unjumble / unfudge a Tutt's Minx do you get an infinite number cuts (and tiny pieces)? I see the TuttMinx and the FuttMinx as two different puzzles. The TuttMinx simply doesn't have 60 degree turns on its hexagonal faces. It is a doctrinaire puzzle so its NOT jumbling and nothing needs to be unfudged. See Brams comment about the 24Cube here. If it were limited to 180 degree turns it would be doctrinaire too. To turn a TuttMinx into a FuttMinx one approach could be to turn a hexagonal face 60 degrees and add the cuts in that allow the neighboring pentagonal face to turn. You could then turn this into an iterative process just as we did above to make the "fudged" Constellation 6. And yes I do believe you end up cutting off an infinite number of tiny pieces in the process of making a fudged TuttMinx, or FuttMinx. You would still need to adjust the exterior angels on the edges to make them all the same and give the corners rotational symmetry for it to be truely doctrinaire but that would simply be an external shape mod of the puzzle created via the iterative process. At least that's how I think it would work out... POVRay's still tied up at the moment. bmenrigh wrote: If so then is the nonfudging version of Tutt's Minx a jumbling puzzle just because a hexagonal face can block a pentagonal one? Nope... its not jumbling as it IS doctrinaire. A 45 degree face turn on a 3x3x3 blocks the rotation of neighboring faces too. Does that make the 3x3x3 a jumbling puzzle? Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Wed Jan 26, 2011 12:23 am 

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

GuiltyBystander wrote: It appears that all but the large circles have been killed. ... and turned into art. Those are some seriously cool looking tile paterns you have going there. Carl
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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Wed Jan 26, 2011 2:29 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

It seems that the regular polygon dipyramids all seem to jumble (square dipyramid excluded). So I wondered what angle between the axises would it take for it to not jumble or require fudging. I start with a trapezoid dipyramid Attachment:
quaddipyramid.png [ 21.75 KiB  Viewed 5779 times ]
and I vary the angle theta from 90>0. Cuts are applied to the green axis. Here's an animation showing what happens to the inactive cuts as you tweak theta. On the legend, theta is the value in the diagram above. N is how many sides it would have if it was a Npolygon dipyramid. I am doing (R' L)x84 Attachment:
consttrap.gif [ 886.38 KiB  Viewed 5779 times ]
You can see a few states where there are indeed a finite number of pieces. Only problem is that you can't make a regular polygon dipyramid out of them without hyperspace. Would nonplanar cuts somehow magically fix this or are we stuck with fudging?
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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Thu Jan 27, 2011 1:23 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

wwwmwww wrote: GuiltyBystander wrote: Here's the start of the piece count progression: 1,3,4,7,10,13,23... There are two 1's. The circle center and the very top piece. Yeah, I'm not really sure how to count the circle piece. I almost want to say that it should be "1,3,4,7,10,13,23...1" because it's in the middle of everything. I kind of felt like I cheated when I put it at the front the first time. wwwmwww wrote: But the puzzle io shows in that thread obviously isn't doctrinaire. After one 90 degree turn stored cuts are in different places. Ugh. I feel like such a fool. For months I had looked at that image and thought that it didn't jumble and have been posting under that faulty assumption. I don't know why I didn't see that earlier. wwwmwww wrote: Fudged Puzzle = a jumbling puzzle which has been made doctrinaire via the removal of an infinite number of pieces. Either of your Fugded definitions is fine. I was wrong.
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stardust4ever

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 28, 2011 3:22 am 

Joined: Sat Mar 24, 2007 6:58 pm Location: Louisiana, US

GuiltyBystander wrote: It seems that the regular polygon dipyramids all seem to jumble (square dipyramid excluded). So I wondered what angle between the axises would it take for it to not jumble or require fudging. I start with a trapezoid dipyramid Attachment: quaddipyramid.png and I vary the angle theta from 90>0. Cuts are applied to the green axis. Here's an animation showing what happens to the inactive cuts as you tweak theta. On the legend, theta is the value in the diagram above. N is how many sides it would have if it was a Npolygon dipyramid. I am doing (R' L)x84 Attachment: consttrap.gif You can see a few states where there are indeed a finite number of pieces. Only problem is that you can't make a regular polygon dipyramid out of them without hyperspace. Would nonplanar cuts somehow magically fix this or are we stuck with fudging? Why are all of the stars seemingly fourpointed? The Constellation Six contains an clear but highly distorted five point star pattern in it's faces. I have viewed all of the 90 GIF frames but did not see this pattern appear. The Constellation Six star, as I pointed out in an earlier post, is much closer to 13/5 rather than 5/2, hence the illusion of finding a Fibonacci sequence.
_________________ My Creepy 3D Rubik's Cube Videocisco wrote: Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 28, 2011 12:10 pm 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

stardust4ever wrote: Why are all of the stars seemingly fourpointed? The Constellation Six contains an clear but highly distorted five point star pattern in it's faces. I have viewed all of the 90 GIF frames but did not see this pattern appear. The Constellation Six star, as I pointed out in an earlier post, is much closer to 13/5 rather than 5/2, hence the illusion of finding a Fibonacci sequence. I don't know for sure, but here's my guess: On the Constellation 6, the cut is deep so it goes through the center of the puzzle. On the frames of my animation, the cut only goes as deep as the adjacent vertex. Another possible answer is that the square dipyramid is some kind of cross over point. It is one of the few doctrinal puzzles in this sequence and there is only 1 piece type. I'm sure there's a real math reason for it that I just haven't tried to work out yet. Here's another animation for the Constellation 6 > square dipyramid progression where everything is deepcut. Attachment:
tritoquad.gif [ 247.82 KiB  Viewed 5658 times ]
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 28, 2011 9:16 pm 

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

bmenrigh wrote: Carl, this is looking great. I'd love to see > 16 iterations. Can I lend you computing resources? Thanks to bmenrigh here in my POVRay coded verson after 16 iterations... Despite having a 2 day head start by PC is still only 65% complete at the moment on this same image. Thanks, Carl
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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Fri Jan 28, 2011 9:31 pm 

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

GuiltyBystander wrote: Here's another animation for the Constellation 6 > square dipyramid progression where everything is deepcut. Nice images and animations and certainly much much faster the POVRay. Can I put in a request? Take a look at the puzzle io presents here: http://twistypuzzles.com/forum/viewtopic.php?p=219040#p219040I can see how one face is turned 90 degrees and the cuts of the neighboring faces are continued onto the turned face to make the smaller pieces. Can you continue this iterative process to cut this piece up into smaller and smaller pieces and in effect turn it into Oskar's Illegal Cube shown in that same thread if the small pieces are removed? Can you show the pattern of the cuts in this case? If so can you also do the same thing starting with a Tuttminx and show how this process can turn it into a Futtminx? Thanks... Carl
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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 29, 2011 1:33 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

wwwmwww wrote: Can I put in a request? Take a look at the puzzle io presents here: http://twistypuzzles.com/forum/viewtopic.php?p=219040#p219040I can see how one face is turned 90 degrees and the cuts of the neighboring faces are continued onto the turned face to make the smaller pieces. Can you continue this iterative process to cut this piece up into smaller and smaller pieces and in effect turn it into Oskar's Illegal Cube shown in that same thread if the small pieces are removed? Can you show the pattern of the cuts in this case? (R' F)x63 Attachment:
pentprism.png [ 111.86 KiB  Viewed 5584 times ]
Again, I'd like to note that I'm only cutting the stickers and rotating them. I'm also using backface culling so that I can quickly easily render a closed convex shape. The pentagonal prism doesn't have rotational symmetry so it ends up not being closed anymore hence the fractured look. This result shouldn't be too surprising considering it's basically a shape mod of the pentagonal dipyramid. I know it's kind of hard to tell with just a single static image, but to me it looks like the renderings I did for the pentagonal dipyramid. wwwmwww wrote: If so can you also do the same thing starting with a Tuttminx and show how this process can turn it into a Futtminx? Gonna be a little while longer on that one since I kind of have to manually construct these. I have all of the platonic solids so I'll try doing a partial truncation to get the soccer ball shape then try to figure out how to make a tuttminx out of that. I think we may need an addendum to the fudge definition. The corners of the Illegal Skewb have 3 rotational states. The true version of the puzzle that has them fully unbandaged has infinite number of rotational states and any bandaging of this corner will block moves. To not block moves, you're also kind of adding another axis of rotation just for that corner so it can get realigned. You kind of did a similar thing on your Mixup Master Skewb. There are some pieces that should be circles, but you turned them into octagons to limit their rotations. These tiny micro twist aren't allowed on doctrinal puzzles are they? I'm not entirely sure what to call that property because the Constellation 6 doesn't do that with it's circle pieces so you can't say it's a fudged thing. The Constellation 6 doesn't look like it tries to restrict the rotation of the circles like the Illegal cube does on the corners. Hmm, can those circles spin in place? I guess maybe it does have this property. I feel like I'm rambling here a bit so if it's confusing, it's not you it's me.
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Timur

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 29, 2011 2:50 pm 

Joined: Mon Mar 22, 2010 7:00 am Location: Germany, Siegerland

GuiltyBystander wrote: The Constellation 6 doesn't look like it tries to restrict the rotation of the circles like the Illegal cube does on the corners. Hmm, can those circles spin in place? I guess maybe it does have this property. Yes, it's pretty easy to spin them by a fingertip. Too bad, because you can't draw an arrow on it and utilize the property of infinite number of states.
_________________ Timur aka Shim Signed Vulcanos ($48), Pillow Pyraminxes ($22) Come visit my Shapeways shop


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wwwmwww

Post subject: Re: Shim's Constellation Six Posted: Sat Jan 29, 2011 5:13 pm 

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

GuiltyBystander wrote: Again, I'd like to note that I'm only cutting the stickers and rotating them. I'm also using backface culling so that I can quickly easily render a closed convex shape. The pentagonal prism doesn't have rotational symmetry so it ends up not being closed anymore hence the fractured look. NICE!!! From context and the picture I take backface culling to simply mean you aren't drawing the back faces... correct? GuiltyBystander wrote: This result shouldn't be too surprising considering it's basically a shape mod of the pentagonal dipyramid. I know it's kind of hard to tell with just a single static image, but to me it looks like the renderings I did for the pentagonal dipyramid. Very interesting... and the corners are now the circle pieces. I see that. GuiltyBystander wrote: I think we may need an addendum to the fudge definition. The corners of the Illegal Skewb have 3 rotational states. The true version of the puzzle that has them fully unbandaged has infinite number of rotational states and any bandaging of this corner will block moves. To not block moves, you're also kind of adding another axis of rotation just for that corner so it can get realigned. You kind of did a similar thing on your Mixup Master Skewb. There are some pieces that should be circles, but you turned them into octagons to limit their rotations. These tiny micro twist aren't allowed on doctrinal puzzles are they? Here is Bram's definition: A 'doctrinaire' puzzle is one where if you were to remove all the coloration then every single position would look exactly the same. The Illegal Cube is doctrinaire. It looks the same after a 90 degree turn as it did before. So yes... it must be allowed. And yes the definition of fudged does need to include this. To turn a normal Pentagonal Prism into an Illegal Cube we have done MORE then just remove an infinite number of tiny pieces. The shape of the corners (aka circular pieces) has been altered to limit them to just 3 rotational states. This property of fudging is made possible by the empty space created by the removed pieces and this allows the corners to "micro twist" into the position they need to be in to remain doctrinaire. Nice catch... I never noticed that the corners were doing MORE then a simple 90 rotation in the Illegal Cube. What would the dual of the Constellation Six look like? Shouldn't you end up with a Triangular Prism with the circle pieces again being the corners? In the dual shape would it be easier to "fudge" the corners to limit them to (say three) rotational states? GuiltyBystander wrote: I'm not entirely sure what to call that property because the Constellation 6 doesn't do that with it's circle pieces so you can't say it's a fudged thing. The Constellation 6 doesn't look like it tries to restrict the rotation of the circles like the Illegal cube does on the corners. Hmm, can those circles spin in place? I guess maybe it does have this property. I feel like I'm rambling here a bit so if it's confusing, it's not you it's me. I'd say, yes it is a property of fudging and the Constellation 6 doesn't need to take advantage of it as the circular face centers have no apparent orientation. The first part of the "fudging" process cuts these pieces into circles... the second part of "fudging" is the altering of shapes to limit rotational states. So... I think we have: Fudged Puzzle = a jumbling puzzle which has been made doctrinaire via the removal of an infinite number of tiny pieces created through an iterative planar cutting process. Pieces which remain that have gained an infinite number of orientation states must either remain circular thus having NO orientation to remain doctrinaire OR they may be shape altered using the empty space left by the removed pieces to give them a finite number of orientation states. The consequence of the later is that a turn of the puzzle that contains one of these such pieces doesn't move as one unified whole. The pieces are loosely held together allowing the pieces within the layer to change their relative position/orientation slightly during a turn. And yes, my Mixup Master Skewb would be another great example of this. There all the pieces in the slice layer are these circulartype pieces and all change their relative orientation during a slice turn. Granted the nice definition of fudging went from a one liner to a paragraph so I suspect its open to further improvement/correction. Again thanks for the pic and thanks in advance for the effort on the tuttminx example, Carl
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Last edited by wwwmwww on Sun Jan 30, 2011 10:15 am, edited 1 time in total.


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GuiltyBystander

Post subject: Re: Shim's Constellation Six Posted: Sun Jan 30, 2011 3:21 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

wwwmwww wrote: NICE!!! From context and the picture I take backface culling to simply mean you aren't drawing the back faces... correct? Exactly. If the shape was closed, you'd never see the backside of a face so there's no point in rendering it. I assume it's convex so that I can render the faces in any order. wwwmwww wrote: Here is Bram's definition: A 'doctrinaire' puzzle is one where if you were to remove all the coloration then every single position would look exactly the same. The Illegal Cube is doctrinaire. It looks the same after a 90 degree turn as it did before. So yes... it must be allowed. Silly me. I need to go back to the basics more often. Doctrinal says nothing about the cuts, how pieces move, or even that it has to be a twisty Rubik's like puzzle. wwwmwww wrote: What would the dual of the Constellation Six look like? Shouldn't you end up with a Triangular Prism with the circle pieces again being the corners? In the dual shape would it be easier to "fudge" the corners to limit them to (say three) rotational states? Yeah, it's be a Triangular Prism. I don't know if you could build the corners to limit to 3 rotational states. The difference in angles (60 vs 90) is way more than it is on the on the Illegal Cube (90 vs 108). The edges are clipped quite a bit too. Attachment:
triprism.png [ 102.35 KiB  Viewed 5481 times ]
wwwmwww wrote: Granted the nice definition of jumbling went from a one liner to a paragraph so I suspect its open to further improvement/correction. lol
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