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 Post subject: Does the Fractured Cube really jumble?
PostPosted: Tue Nov 19, 2013 3:55 pm 
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http://www.shapeways.com/model/681644/f ... uzzle.html

It looks like it should be unbandageable to me, but perhaps I'm missing something.

BTW, David, if you're reading this, it would be great for your Fractured puzzles if you provided assembly diagrams for the intended initial configurations. :)


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Tue Nov 19, 2013 7:31 pm 
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Looks to me like it can be unbandaged.


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 3:05 pm 
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Jared wrote:
It looks like it should be unbandageable to me, but perhaps I'm missing something.
I posted this image (depicting a face of the Fractured Cube with some additional cut lines) in response to the same question in the original Fractured Cube post:
Attachment:
jumbling proof.JPG
jumbling proof.JPG [ 60.97 KiB | Viewed 1385 times ]
As you can see, additional cuts to "unbandage" the puzzle could be made, but they would result in missing material. I haven't drawn all the necessary cut lines here, only enough to show that the angles don't match up to create a regular polygon in the center. This is the equivalent of turning the Fracture-6 puzzle into Constellation Six. So while technically the puzzle could be unbandaged (ignoring whether or not enough material is left for the pieces to be held in properly), it is still a jumbling puzzle (at least, insofar as my understanding of jumbling goes).

Does that answer the question to your satisfaction?

Jared wrote:
BTW, David, if you're reading this, it would be great for your Fractured puzzles if you provided assembly diagrams for the intended initial configurations.
There are only a few different configurations that can be made with the Fractured Cube, Fractured Tetrahedron, and QuadStar puzzles. (Fracture-6, 10, and 12 are all very straightforward). I thought I'd let anyone purchasing the puzzles explore those possibilities on their own. Of course, if you want more information on the configuration in which I chose to arrange the parts in a particular puzzle, just send me a message.

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 4:19 pm 
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David Pitcher wrote:
I posted this image (depicting a face of the Fractured Cube with some additional cut lines) in response to the same question in the original Fractured Cube post:

I can't seem to find that post; link?

Why is this not sufficient?

Image

Oh... NM, I see now why that is not sufficient. OK, I understand your figure and most of your comments.

Re jumbling / bandaging, no, it only jumbles (by definition) if it is bandaged and cannot be finitely unbandaged. Which on second thought does seem likely here. I need to update my circle puzzle unbandaging program to work on Jaap spheres.

BTW I really like this puzzle concept. To me the Fractured Tetrahedron seems the most elegant in this line. I may just have to get one!


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 5:51 pm 
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bhearn wrote:
I can't seem to find that post; link?
Here's a link to the original Fractured Cube post.
bhearn wrote:
Re jumbling / bandaging, no, it only jumbles (by definition) if it is bandaged and cannot be finitely unbandaged. Which on second thought does seem likely here.
That is exactly what's going on. To unbandage the puzzle will result in portions of pieces being reduced to "dust", resulting in gaps on the exterior of the puzzle.

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 7:09 pm 
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David Pitcher wrote:
That is exactly what's going on. To unbandage the puzzle will result in portions of pieces being reduced to "dust", resulting in gaps on the exterior of the puzzle.
I haven't done the math to check... if some of these angles are irrational that may be enough. But your statement in the first thread...
David Pitcher wrote:
As you can see, the new cuts do not line up to create equal size and shaped pieces. In order for the puzzle to be unbandaged, the cut lines shown would need to form a five-pointed star, but the proportions of the triangular faces do not allow this.
...doesn't necessarily mean that the puzzle jumbles. As Bob has seen with the circle puzzles there can be cases where you must go through many many iterations of unbandaging but the process can still stop with a finite (but very very large) number of pieces. If this happens then technically the puzzle doesn't jumble per the current definition. From a practical point of view, as seen with some of the circle puzzles, you could still end up cutting pieces off that are so small the puzzle would have to be impractically large to give the smallest pieces any significant volume.

Now having said the above this behavior has really only been seen in the 2D circle puzzles. In the case of the few 3D puzzles which have been looked at in detail, I believe irrational numbers have been found early and thus pretty well proven they must jumble. Which is likely the case with Fractured Cube. I'm just not sure its been proven technically... though from a practical point of view I believe you have proven one can't make a fully unbandaged Fractured Cube.

I would LOVE to see a puzzle (any 3D puzzle) which appears to jumble but could actually be unbandaged with a large but finite number of unbandaging iterations. Large in this case could be anything more then 10 as I don't believe outside of the 2D circle puzzles any 3D puzzles have been found to fall into this catagory. I don't think we've really had to tools to look for them till Bob made his program.

Carl

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 7:42 pm 
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wwwmwww wrote:
I would LOVE to see a puzzle (any 3D puzzle) which appears to jumble but could actually be unbandaged with a large but finite number of unbandaging iterations. Large in this case could be anything more then 10 as I don't believe outside of the 2D circle puzzles any 3D puzzles have been found to fall into this catagory. I don't think we've really had to tools to look for them till Bob made his program.

I'm not sure I see any essential difference here between 2d and 3d puzzles. Except for multicore puzzles, I believe, can't you just look at a 3d twisty in terms of its Jaap sphere, which is a 2d surface? E.g., helicopter jumbling is apparent on the sphere. I expect to see the same kinds of behavior we saw for the two-circle puzzles, in general.

In this case (Fractured Cube), it should just be a matter of plugging in the right rotation centers, R, set N = 6, and see what happens.

(Actually, and this gets at an issue I wanted to raise after doing some more remedial reading on jumbling... the helicopter cube seems problematic to me. It's considered bandaged because you can get to states where some moves are blocked. But only by making "partial" turns. Of course, turn a Rubik's cube face by 30°, and you've just blocked four faces from turning, but we don't consider that bandaged. It seems to me that to properly describe the helicopter cube as bandaged (and as a consequence jumbling), you need to define moves in terms of transitions of a given edge from one of its six "permissible" rotational states to another. Then, the current rotational states must become part of the configuration. This is not necessary where a move is always defined as rotation by a fixed angle. Here, with the Fractured Cube, we don't have this problem. Has this issue been discussed already?)


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 8:32 pm 
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bhearn wrote:
I'm not sure I see any essential difference here between 2d and 3d puzzles. Except for multicore puzzles, I believe, can't you just look at a 3d twisty in terms of its Jaap sphere, which is a 2d surface? E.g., helicopter jumbling is apparent on the sphere. I expect to see the same kinds of behavior we saw for the two-circle puzzles, in general.

In this case (Fractured Cube), it should just be a matter of plugging in the right rotation centers, R, set N = 6, and see what happens.
Oh I certainly agree. That is a valid way to view and approach this problem. I'm just wondering if the fact you are dealing with a curved (spherical) surface and not a flat surface will affect the onset of jumbling. The Helicopter Cube I believe was the first mass produced jumbling puzzle and one of the simplier ones yet I'm pretty sure that its been proven to jumble. Some of your planar 2D puzzles appear to go through 1000's of iterations and then stop to produce a finite non-jumbling puzzle. I simply haven't seen this yet in ANY 3D puzzle. I'm curious if there is something that is keeping that kind of behavior from happening on a sphere or that its just a case we haven't looked close enough before. I honesly have no idea at the moment.
bhearn wrote:
...Then, the current rotational states must become part of the configuration. This is not necessary where a move is always defined as rotation by a fixed angle. Here, with the Fractured Cube, we don't have this problem. Has this issue been discussed already?)
Not 100% sure I understand the "problem". A Helicopter Cube which only allowed 180 degree turns would be doctrinaire. The Fractured Cube does contain configuration information in each "state" for lack of a better word. By that I simply mean you can tell which moves are allowed and which aren't based on the locations of the pieces. There isn't simply one doctrinare state. So I'm confused about the exact "issue" you are wanting to discuss. However, and not sure this is relevant or not, not all jumbling puzzles can be turned into doctrinare puzzles like the Helicopter Cube by only considering one type of turn (say the 180 degree turn). There are puzzles where all turns (aside from 360 degrees) are jumbling turns and a puzzle which only allows 360 degree turns isn't much of a puzzle.

Carl

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Wed Nov 20, 2013 10:02 pm 
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And I see it was me who asked this question in the original topic. I sure feel dumb now. :oops:


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 10:42 am 
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Sorry for the double post but... it alllmost looks like with a little more unbandaging you'd get a regular octagon. Could the shape and/or mechanism be fudged to make it happen, or am I pipe dreaming?


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 10:51 am 
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wwwmwww wrote:
bhearn wrote:
...Then, the current rotational states must become part of the configuration. This is not necessary where a move is always defined as rotation by a fixed angle. Here, with the Fractured Cube, we don't have this problem. Has this issue been discussed already?)
Not 100% sure I understand the "problem". A Helicopter Cube which only allowed 180 degree turns would be doctrinaire. The Fractured Cube does contain configuration information in each "state" for lack of a better word. By that I simply mean you can tell which moves are allowed and which aren't based on the locations of the pieces. There isn't simply one doctrinare state. So I'm confused about the exact "issue" you are wanting to discuss.

No, there is a difference. The Fractured Cube, as I understand it, essentially allows 60° turns at each vertex, when not blocked by bandaging. The Helicopter Cube has these additional, irrational-angle turns. When you have made one, that edge is in an inherently different state. You can't turn by that angle again. So, you cannot simply characterize a Helicopter Cube as a particular instance of a circle puzzle on a sphere, with given center points, R, and N. But you *can* (I believe) so characterize a Fractured Cube. Thus, in order to speak of the Helicopter Cube as jumbling, it's necessary to define exactly what it is in a somewhat more complicated way. And the same seems to be true for most jumbling puzzles.

Basically, it comes down to what configurations should naturally be considered as bandaged. You want to consider any position you can physically reach as "valid", thus any moves made to get there should count as moves. Except that partial moves that don't enable other moves are not really moves (like turning a Rubik's Cube face 30°)... except when they are. I can imagine a bandaged position in which only one turn axis is unblocked, when one would "expect" others to be unblocked. (I think More Madness has states like this.) So then, it seems to me a little fuzzy, still, defining a position as bandaged, without explicitly specifying the desired unbandaged states (which in the case of the Helicopter Cube means adding state, a kind of parity, describing the allowed and current angles of each edge). Therefore, I'm not 100% sure it's even meaningful to describe a puzzle as bandaged without reference to some subjectively defined notion of what one should be able to do with it, above and beyond what is directly implied by the physical object itself. Perhaps one could come up with a definition. Maybe it's already out there. But if this issue doesn't seem familiar to you (I haven't been on the forums much lately), maybe not.

Sorry to derail somewhat from the Fractured Cube -- which I now believe is unusual among physically made twisties in that it seems to be jumbling without any extra specification of allowable moves or additional internal state (though we do see this with the 2d circle puzzles, and I expect to see it generically in 3d as well).


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 11:17 am 
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bhearn wrote:
No, there is a difference. The Fractured Cube, as I understand it, essentially allows 60° turns at each vertex, when not blocked by bandaging. The Helicopter Cube has these additional, irrational-angle turns. When you have made one, that edge is in an inherently different state. You can't turn by that angle again. So, you cannot simply characterize a Helicopter Cube as a particular instance of a circle puzzle on a sphere, with given center points, R, and N. But you *can* (I believe) so characterize a Fractured Cube. Thus, in order to speak of the Helicopter Cube as jumbling, it's necessary to define exactly what it is in a somewhat more complicated way.
Ahhh.... I understand now. Thanks for clearing that up for me. Say a 3x3x3 which allows 45 degree face turns, which we believe jumbles, is inherently different then the Helicopter Cube. No... I don't believe this issue has ever been discussed before. I suspect you are the first to notice.

But to me this also seems to imply there may be bigger differences between planar 2D puzzles and 3D, or puzzles which exist on 2D spherical surfaces. Does this behavior seen on the Helicopter Cube, an unexpected turn being allowed, ever happen in your 2D circle puzzles? In the Helicopter cube the cuts are made with just the 120 degree turns being considered. The other turns are more or less happy accidents. If you cut up a circle puzzle to allow turns of X degrees... do the cuts ever happen to also allow turns of Y degrees where Y isn't some multiple of X? I don't think so... not sure I can prove that but I don't see a way for that to happen.

Going back to this statement:
bhearn wrote:
I'm not sure I see any essential difference here between 2d and 3d puzzles.
I think there may be some things 3D allows that 2D doesn't and vice versa. And now that I understand the issue you were pointing out I think you may be coming to the same conclusion.

2D doesn't seem to allow the Helicopter situation.
3D looks like it could have issues producing bandaged puzzles which require a finite but large number of iterations to fully unbandaged.

Carl

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Last edited by wwwmwww on Thu Nov 21, 2013 1:56 pm, edited 1 time in total.

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 1:06 pm 
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Jared wrote:
it alllmost looks like with a little more unbandaging you'd get a regular octagon. Could the shape and/or mechanism be fudged to make it happen, or am I pipe dreaming?
The QuadStar puzzle actually is closer to having a regular octagon appear in the center. Here's a picture of one face of the QuadStar with extra cuts. It's close to octagonal symmetry, but as you can see, not quite:
Attachment:
quadstar face with extra cuts.jpg
quadstar face with extra cuts.jpg [ 18.91 KiB | Viewed 1201 times ]

Here's a picture of the Fractured Cube face with the same type of cuts. As you can see, it's further than QuadStar from having an octagon appear in the center:
Attachment:
fractured cube face with extra cuts.jpg
fractured cube face with extra cuts.jpg [ 20.08 KiB | Viewed 1201 times ]


Fractured Tetrahedron is actually the closest of these puzzles to having a regular polygon at the center of the face. Of course, it can be fudged to make a pentagon at the center, and you end up with Timur's beautiful Sky Globe puzzle.

Another way to achieve regular symmetry on the faces of these puzzles is to fudge the base solid form, as Timur did for his Biaxe puzzle.

Yet a third way to turn this type of puzzle into a non-jumbling geometry is to employ curved cuts. This is what I did to create the Diamond Delight puzzle. This unbandaging method results in puzzles that are very easy to solve since the face centers are eliminated.

I haven't experimented with this, but I'd guess that "de-jumbling" (or unbandaging) this type of puzzle might also be achieved with pillowing (or more likely anti-pillowing) the solid form and using planar cuts. If I'm right (and I may well not be), the result would be similar to using curved cuts. In other words, the centers would be eliminated, making for a simpler puzzle to solve.

There must be other approaches to unbandaging these puzzles. Can anyone else think of any?

bhearn wrote:
BTW I really like this puzzle concept. To me the Fractured Tetrahedron seems the most elegant in this line. I may just have to get one!
Thanks! :D

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 1:41 pm 
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There is a HUGE difference between 2D and 3D puzzles when it comes to angles and jumbling potential. In 2D puzzles all of the turning axes are parallel to one another. Thus, when two moves that share some piece in common are executed in succession, the overall rotation of the common piece is merely the sum of the two angles of rotation. The same is not at all true for 3D puzzles and irrational numbers are almost guaranteed to pop up unless very specific axis arrangements are used. The cumulative effect of multiple rotations around non parallel axes in 3D can be tracked by 3x3 rotation matrices and very quickly introduce irrational numbers for every pair of rotations except those chosen from very specific sets (ie. cube, dodecahedron, etc.)

Imagine a flat 2D puzzle where every move rotates some portion of the puzzle by 60 degrees. We don't even need to think of the exact geometry. If we imagine an arbitrary piece of any shape anywhere on the puzzle and perform two clockwise moves in succession on THAT piece, the end result is it will have rotated exactly 120 degrees and possibly translated some distance across the surface of the puzzle. Now with the geometry of the fractured cube we essentially have moves in multiples of 60 degrees around axes perpendicular to the faces of an octahedron. Regardless of the exact shape of some piece in this puzzle, two succesive clockwise moves centered around adjacent axes (let's say URF followed by URB) on a single piece results in a rotation of that piece around an axis that is about 26.565051771 degrees off of vertical through an angle of 114.0948425521 degrees. There. Does THAT sound like a jumbling puzzle? :roll: If you have any sort of 3D manipulation program and would like to see this for yourself, draw any shape, any shape at all inside or outside an octahedron (or cube) to represent the piece. Then rotate it 60 degrees around one face (or cube corner) and rotate the resulting image 60 degrees around an adjacent face (or cube corner).

In order for a puzzle to NOT jumble, not only does every move have to produce a rational angle, every COMBINATION of moves together has to produce an overall rational angle, and this is a VERY hard criteria (criterion?) to satisfy in 3D.
Fractured Cube absolutely jumbles because two moves in succession produce an irrational angle. Unless you start fudging something somewhere, it can never be fully unbandaged

Peace,
Matt Galla


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 4:26 pm 
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Thanks, Matt. I see you removed the bit about isotoxal polyhedra -- I was going to ask you to explain your reasoning there?

If I get a chance early next week (I may well not) I will try to adapt my unbandaging program for puzzles on spheres. I am curious now.


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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 5:56 pm 
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bhearn wrote:
Thanks, Matt. I see you removed the bit about isotoxal polyhedra.
Yes... thank you Matt. My gut was telling me to expect some big differences between 2D and 3D but I couldn't really tell why. Your post does a great job of addressing the "why" and states things much better then I could. I think I missed the statement about isotoxal polyhedra though it sounds like I may not have missed much. You have now got Bob curious and as I learned at the last IPP that is a VERY good thing to do.

Thanks,
Carl

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 6:26 pm 
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Allagem wrote:
Regardless of the exact shape of some piece in this puzzle, two succesive clockwise moves centered around adjacent axes (let's say URF followed by URB) on a single piece results in a rotation of that piece around an axis that is about 26.565051771 degrees off of vertical through an angle of 114.0948425521 degrees. There. Does THAT sound like a jumbling puzzle? :roll:
Oh I certainly agree, this sounds like a jumbling puzzle. But let me play devils advocate for a second. Is that enough to PROVE that it jumbles? Are you certain those numbers are irrational? I'm a bit outside my element here, but I know there are still numbers where their exact statis (rational or irrational) is still unknown. Some examples: Pi + e, 2 to the power e, Pi to the power square root 2, etc...

What if these angles are actually rational? Could Fractured Cube be fully unbandaged with some HUGE number of unbadaging iterations? Maybe there is some simple reason that makes it obvious these are irrational but (being honest) I'm too lazy to try and find it at the moment.

Carl

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 Post subject: Re: Does the Fractured Cube really jumble?
PostPosted: Thu Nov 21, 2013 7:28 pm 
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wwwmwww wrote:
bhearn wrote:
Thanks, Matt. I see you removed the bit about isotoxal polyhedra.
I think I missed the statement about isotoxal polyhedra though it sounds like I may not have missed much.

Yeah... if you're curious I had originally said that the only sets of rotations that don't produce irrational angles are isotoxal polyhedra. I know from studying lots of strange geometries that the key to making a puzzle out of a given shape is that the dihedral angles between faces (both adjacent and non-adjacent) need to be equal between faces that interact. Following this train of thought also helps with figuring out when puzzles have unexpected jumbling moves (i.e. icosahedra - it occurs when faces have the same dihedral angle to the rotating face without having rotational symmetry about the rotating face). The dihedral angle is the angle formed by two faces meeting at an edge. Thus the edges have to match -> the shapes are edge-transitive -> the fancy word for that is isotoxal. I thought I was on to something, but then I realized that prisms break the pattern. Prisms can easily produce non-jumbling puzzles and the sets of axes produced by then, when combined, still produce rational angles, even though prisms are NOT edge transitive - they have two types of edges. So with that big hole in my claim, I decided to delete it, hoping no one had read it yet :wink: I still think there is some subtle truths there, because the edge pieces in prism puzzles always come in two different orbits: corresponding to the two different dihedral angles - don't get too excited; this apparently profound statement can be made less amazing by noting the two types of edge pieces are forced to have different geometry. In any case, I wasn't ready to back up that claim and it wasn't necessary for my argument so I tried to do the clear thing and simply edit it out :)
wwwmwww wrote:
Oh I certainly agree, this sounds like a jumbling puzzle. But let me play devils advocate for a second. Is that enough to PROVE that it jumbles? Are you certain those numbers are irrational? I'm a bit outside my element here, but I know there are still numbers where their exact statis (rational or irrational) is still unknown. Some examples: Pi + e, 2 to the power e, Pi to the power square root 2, etc...

What if these angles are actually rational? Could Fractured Cube be fully unbandaged with some HUGE number of unbadaging iterations? Maybe there is some simple reason that makes it obvious these are irrational but (being honest) I'm too lazy to try and find it at the moment.

You are absolutely right Carl, I have not proven anything. Proving that number is irrational is an excercise in performing the calculation properly without rounding to get the exact mathematical expression (I believe it will involve a product of inverse cosines with a few square roots thrown in). Once an irreducible expression is obtained, we will have proven that the number is in fact irrational. I just... rrrreeeeaaaallllllllyyyy dont want to :roll: :lol: Maybe it could be made a little simpler by placing the first rotation on the x axis and just mking the second rotation around whatever angle it is... dihedral angle of an octahedron I believe... hmm wikipedia says that is pi-arccos(1/3) ok so take cosine and sine of that to get the components of the 3D rotation for the second angle, make your rotation matrices and perform some good ol matrix multiplication... then you have to normalize it to get the projected rotation.... yeah it's gonna be irrational... like... 7 times over again irrational.... :lol:

Peace,
Matt Galla


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