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 Post subject: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 3:26 pm 
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Mr Owl, how many pieces does it take to get to the core of a Type-ⅡEdge-Turning Multidodecahedron?

Let's find out...

Image

Looks like the answer is 2845.

Maybe most of you are asking... what is a Type-ⅡEdge-Turning Multidodecahedron? Those I'll send here:

http://www.twistypuzzles.com/forum/viewtopic.php?f=1&t=14875

were I looked at the Type-ⅡFace-Turning Multidodecahedron.

Now I guess we know why all the Big Chops made so far use magnets.

Enjoy,
Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 4:32 pm 
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Aaaaaaaaaaaaaaaaah, that animation is scary! :shock: :lol: :cry:

I don't really get it but it's still cool to watch! :lol:

Alex

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 5:19 pm 
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The Gelatatinbrain 1.4.2 fits Drew's hint list very nicely. Looks identical to the starminx and the dino dodecahedron, and it is surely worthy of Indiana Jones.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 5:48 pm 
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>_>


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Shhhh!

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 6:02 pm 
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A 1... A 2... CHOMP! 3. (the world may never know!) :mrgreen:

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 6:18 pm 
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APJ wrote:
I don't really get it but it's still cool to watch! :lol:

Alex


I guess I really should say a bit more about what the animation is...

In an Edge-Turn Dodecahedron there are 15 axes of rotation. This Multidodecahedron contains the family of all such Edge-Turn Dodecahedrons with 2 cut planes per axis. In this animation the cut planes are fixed at 1 unit apart such that if the scale factor is < 1 all you see is the core. As the scale factor grows the size of the Dodecahedron grows and you see more and more of the pieces.

Each piece type is named and labeled with a letter (English... then Greek). They are named in the order in which they appear from the inside out so the core is named piece "a". This puzzle has 41 piece types and each type is sorted according to its symmetry. There are 8 groupings of pieces. I named them core, face[f] center, edge[e] center, corner[c], [f-c] for a face piece that lies on the line between the face center and a corner, [f-e] for a face piece that lies on the line between the face center and an edge, [c-e] for an edge piece that lies on the line between the corner and the edge center, and asymmetric for those pieces that have no symmetry. Andreas pointed this sorting scheme out to me later which appears more standard:

http://www.speedsolving.com/wiki/index.php/Center
http://www.speedsolving.com/wiki/index.php/Edge

0 - core ()
C - corner
E - edge (precisely: midge)
F - face (precisely: central center)
W - Wings
T - T-center
X - X-center
L - Obliques

Note the symmetry group determines the number of copies of each piece. The last number is the number of copies of that piece that are moved in a single edge turn. And at the bottom I total up the number of pieces that are contained within and the number of pieces that are exposed on the surface for each puzzle. There are also 42 unique Edge-Turn Dodecahedral puzzles with 2 cut planes per axis. I'm counting the core by itself as the first puzzle which is trivial and Big Chop as the last puzzle which isn't really reached until the cut planes join... in this animation the cut planes are fixed so you never really get there. I should also point out the puzzles I've named ETD0.5 and ETD1 I consider the same puzzle as they have the same pieces exposed on the surface.

As with the Type-Ⅱ Face-Turn Multidodecahedron a puzzle could be made on Gelatinbrain that would allow you to solve all 41 piece types at the same time. In the other thread I show how that could be done by just showing two puzzles at the same time. Here you would need to view at least 6 puzzles simultaneously. Here is one such set:

Image

Image

Image

Image

Image

Image

Some pieces can be seen on more then one puzzle above but all can be seen on at least 1. I wish Gelatinbrain would make something like this... I wonder how many solvers here think they could solve 41 piece types at the same time?

Did that clear anything up?
Carl

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Last edited by wwwmwww on Sat Dec 05, 2009 6:31 pm, edited 3 times in total.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 6:24 pm 
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That did explain it. :D

I certainly wouldn't have a go at all 41 at once. The puzzles in the pictures look hard enough! :D

Alex

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 6:37 pm 
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Steryne wrote:
A 1... A 2... CHOMP! 3. (the world may never know!) :mrgreen:


BINGO!!! Maybe I'm giving away my age with that opening...

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 8:54 pm 
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Drewseph wrote:
>_>


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Shhhh!


Ohhhhhh! Can't wait!!!


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 9:20 pm 
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wwwmwww wrote:
In an Edge-Turn Dodecahedron there are 15 axes of rotation. This Multidodecahedron contains the family of all such Edge-Turn Dodecahedrons with 2 cut planes per axis. In this animation the cut planes are fixed at 1 unit apart such that if the scale factor is < 1 all you see is the core. As the scale factor grows the size of the Dodecahedron grows and you see more and more of the pieces.
Thanks for the fascinating animation and explanation. Please can you add an image of Gelatinbrain 1.4.2 + 1.4.3, which I think is also a member of this family? It only has 3 piece types but it is a puzzle I would very much like to see added to Gelatinbrain.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sat Dec 05, 2009 9:58 pm 
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Julian wrote:
Thanks for the fascinating animation and explanation.


You are welcome...

Julian wrote:
Please can you add an image of Gelatinbrain 1.4.2 + 1.4.3, which I think is also a member of this family? It only has 3 piece types but it is a puzzle I would very much like to see added to Gelatinbrain.


I can make the image... give it some time to render. But that puzzle isn't a part of this family. Gelatinbrain 1.4.3 is Big Chop, the deep cut edge turning dodecahedron. If you add its cut plane to Gelatinbrain 1.4.2 you end up with a puzzle that now has 3 cut planes per axis and not 2. You are correct that puzzle does have 3 piece types exposed on the surface... but none of those 3 piece types are any of the 41 in this family. They would be part of the family of the next higher order Type-ⅡEdge-Turning Multidodecahedron which has 3 cut planes per axis but the complexity of these families grow exponentially. If we say order is equal to the number of cut planes per axis we see the following...

The order=0 Type-ⅡEdge-Turning Multidodecahedron has just 1 piece. The core.
The order=1 Type-ⅡEdge-Turning Multidodecahedron has 120 pieces of the same type. It's Big Chop.
The order=2 Type-ⅡEdge-Turning Multidodecahedron has 2945 pieces of 41 types. As seen above.

As we are already at a level that is already so complex it may never be built even in bits and bytes I don't see much point in going higher. Something I find ironic is that people are building 12x12x12's and talking about 17x17x17's which using this definition are order 11 and 16 puzzles respectively but there are still these 40+ order 2 puzzles that have yet to be built. A handful exist in bits and bytes at Gelatinbrain but aside from Big Chop (an order=1 puzzle) the only other one I could find that was even named was the Copterminx.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sun Dec 06, 2009 3:24 am 
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As I have told Carl earlier (in private communication): Well done!

We have another thread which needs to be transformed into an article!
... and another 4 articles, too. ...


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sun Dec 06, 2009 4:23 pm 
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wwwmwww wrote:
Julian wrote:
Please can you add an image of Gelatinbrain 1.4.2 + 1.4.3, which I think is also a member of this family? It only has 3 piece types but it is a puzzle I would very much like to see added to Gelatinbrain.
I can make the image... give it some time to render. But that puzzle isn't a part of this family. Gelatinbrain 1.4.3 is Big Chop, the deep cut edge turning dodecahedron. If you add its cut plane to Gelatinbrain 1.4.2 you end up with a puzzle that now has 3 cut planes per axis and not 2. You are correct that puzzle does have 3 piece types exposed on the surface... but none of those 3 piece types are any of the 41 in this family.
Oops! I see. One of the things I really like about Gelatinbrain is his knack for including interesting puzzles that can be solved in a reasonable amount of time, while leaving out those that would take too long. I'd say 1.4.1+1.4.4, 1.4.4+1.4.5, and 1.4.5+1.4.2 are sitting right on the fence, and I like to think that "where there is doubt, there is no doubt", so he left them out!

wwwmwww wrote:
I wonder how many solvers here think they could solve 41 piece types at the same time?
Fortunately, I'm pretty sure that isn't necessary. Following your diagrams above, we could solve the puzzles one by one working up from the shallowest cut to the deepest cut. So we'd solve piece types 1-7 of ETD1.5, then types 8 & 9 of ETD2.5, types 10-17 of ETD4.5, types 18-26 of ETD9.5, types 27-32 of ETD13.5, and finally types 33-41 of ETD20.5. The commutators we use to solve the new piece types of each new puzzle will leave the previously solved puzzles unchanged. In case you are curious, my instinct tells me that the very last piece type to be solved would be 41. My guess from looking at the puzzle for a while and trying to imagine algos, is a solving order of 39, 37, 38, 36, 35, 34, 33, 40, 41. The first two puzzles would be quick wins but I'd probably need at least a couple of hours each to figure out algos for the other four.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 9:48 am 
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Julian wrote:
Oops! I see.


Here is your picture anyways.

Image

And this actually tells me something. I can extend this table to order 3...

The order=0 Type-ⅡEdge-Turning Multidodecahedron has just 1 piece of 1 type in 1 symmetry group. The core.
The order=1 Type-ⅡEdge-Turning Multidodecahedron has 120 pieces of 1 type in 1 symmetry group (Obliques). It's Big Chop.
The order=2 Type-ⅡEdge-Turning Multidodecahedron has 2945 pieces of 41 types in 8 symmetry groups. As seen above.
The order=3 Type-ⅡEdge-Turning Multidodecahedron has 4920 pieces of 41 types in 1 symmetry group (Obliques).

Without making another animation for the order=3 Type-ⅡEdge-Turning Multidodecahedron I can see that the 3rd cut, the deep cut, would never move and the other 2 cuts would follow the same as they do in the order=2 Type-ⅡEdge-Turning Multidodecahedron. As a result you'd still have just 41 piece types, but the face centers would be cut into 10 pieces, the edge centers into 4 pieces, the corners into 6 pieces, the core into 120 pieces, etc. In short each piece that wasn't already an oblique would be cut up to remove all symmetry. So the total number of pieces would be 41×120 or 4920.

Julian wrote:
Fortunately, I'm pretty sure that isn't necessary. Following your diagrams above, we could solve the puzzles one by one working up from the shallowest cut to the deepest cut. So we'd solve piece types 1-7 of ETD1.5, then types 8 & 9 of ETD2.5, types 10-17 of ETD4.5, types 18-26 of ETD9.5, types 27-32 of ETD13.5, and finally types 33-41 of ETD20.5. The commutators we use to solve the new piece types of each new puzzle will leave the previously solved puzzles unchanged. In case you are curious, my instinct tells me that the very last piece type to be solved would be 41. My guess from looking at the puzzle for a while and trying to imagine algos, is a solving order of 39, 37, 38, 36, 35, 34, 33, 40, 41. The first two puzzles would be quick wins but I'd probably need at least a couple of hours each to figure out algos for the other four.


Very interesting... thanks for all that. But finding those commutators which can be used to solve the deeper cut puzzles while leaving the previously solved puzzles unchanged is exactly what I'd consider being able to solve all 41 piece types at the same time. For example I suspect that if someone wanted to solve just ETD20.5 they'd be able to use simplier commutators then those that would be needed to solve the entire Type-Ⅱ Edge-Turning Multidodecahedron. My gut tells me that as there are 41 pieces you may need commutators atleast 41 moves long (maybe much longer) just to isolate the moving of a single piece type while solving the deepest cut pieces. Finding those commutators I would think would be a real challenge even for the best puzzle solvers. It's certainly well beyond me at the moment. I really wish Gelatinbrain (or anyone that would know how) would make something like this so people like yourself could play with puzzles like these. Maybe it isn't really as hard as I'm thinking.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 11:52 am 
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wwwmwww wrote:
Without making another animation for the order=3 Type-ⅡEdge-Turning Multidodecahedron I can see that the 3rd cut, the deep cut, would never move and the other 2 cuts would follow the same as they do in the order=2 Type-ⅡEdge-Turning Multidodecahedron. As a result you'd still have just 41 piece types, but the face centers would be cut into 10 pieces, the edge centers into 4 pieces, the corners into 6 pieces, the core into 120 pieces, etc. In short each piece that wasn't already an oblique would be cut up to remove all symmetry. So the total number of pieces would be 41×120 or 4920.


That means we now have two look-a-likes of type 41:
The first kind are the pieces of the now broken up core. These are affected only by the inner 2 slices.
The second kind are the omicron-pieces already in the animation. These are never affected by the inner 2 slices. Seems logical, since the corners of the 4x4x4 are duplicated in its virtual inner 2x2x2.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 12:46 pm 
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Andreas Nortmann wrote:
That means we now have two look-a-likes of type 41:
The first kind are the pieces of the now broken up core. These are affected only by the inner 2 slices.
The second kind are the omicron-pieces already in the animation. These are never affected by the inner 2 slices. Seems logical, since the corners of the 4x4x4 are duplicated in its virtual inner 2x2x2.


Interesting observation...

Here is another thought I just had... is the order=4 Type-ⅡEdge-Turning Multidodecahedron even well defined? If I fix 4 equally spaced cut planes per axis of rotation and allow the puzzle to grow I could make an animation similiar to the above. However I don't think I would capture all possible pieces that would be present in all possible Edge-Turn Dodecahedral puzzles with 4 cut planes per axis. The two deeper cut planes must be the same distance from the center of the core and the two shallower cut planes must be similiarly placed but the relative position of the two types shouldn't be fixed. This leads to two problems...

(1) I don't think its possible to construct a model where each piece types occupies some positive volume inside a single dodecahedron. For example as I show the order=2 Type-ⅡEdge-Turning Multidodecahedron is a set of puzzles (ETD0 through ETD20.5) I think the order=4 Type-ⅡEdge-Turning Multidodecahedron could be a set of different order=4 Multidodecahedrons where the cut plane spacing is different in each and each of those is a set of different puzzles. Boy that could get very very messy. How many different cut plane spacings would be needed to be sure all piece types were represented?

(2) Should the depper cut planes be allowed to murge independantly of the shallower cut planes? I don't think so or you could end up with an issue similiar to having a 3x3x3 inside a 4x4x4. If you turn 1 slice of the 4x4x4 but not the other what do you do with the slice layer of the 3x3x3? So I think a "real" order=4 Type-Ⅱ Edge-Turning Multidodecahedron must have all cut planes some positive distance from the center of the core and all 4 cut planes would merge only if the dodecahedron is of infinite size.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 5:57 pm 
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Finally!!!!
I researched this EXACT thing about 8 months ago. I determined that there were 2844 pieces of 40 unique types not including the core, sound familiar? I should have known that it would be Carl to repeat the same thing. I believe I posted long ago that the Deep cut rhombic tricontahedron was by far the most difficult single layered puzzle to make because it would take almost 3000 pieces to build... Now everyone knows why. Excellent job on the animation Carl, I only used Jaap's sphere and it was quite tedious.

Although, I think it might interest some people to know that I organized my model somewhat differently. Instead of how many pieces of a given type move in one twist, I defined how many ways ecah piece can move in one twist. Organizing the pieces in a complicated tree graph with the level down equal to the number of ways a given piece can move and assigning each nodes parents as the adjacent pieces on the puzzle that belong in the level above a given piece results in a diagram of HOW TO BUILD THE PUZZLE. Each given piece must be designed to support its childrenin the graph (which is interestingly enough non-planar btw). The big chop pieces are the very bottom piece on the chart, the only piece type that can be moved 15 ways (note that 30faces/2 = 15 :wink: )

Although I should point out that Matt Shepit discovered a way of eliminating certain pieces that are redundant in the mech. I haven't figured out exactly how many pieces fall in this category but I would bet if the right mech "path" was chosen, about 1/3 of the pieces could be eliminated. 8-)

Peace,
Matt Galla

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wwwmwww wrote:
Some pieces can be seen on more then one puzzle above but all can be seen on at least 1. I wish Gelatinbrain would make something like this... I wonder how many solvers here think they could solve 41 piece types at the same time?
I bet I can :wink:


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 6:26 pm 
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we need some edge-turning dodecahedrons now!

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 6:52 pm 
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Thanks for the image! I find 1.4.2 + 1.4.3 a pretty puzzle, and I think it would be interesting to solve too.

wwwmwww wrote:
Julian wrote:
Fortunately, I'm pretty sure that isn't necessary. Following your diagrams above, we could solve the puzzles one by one working up from the shallowest cut to the deepest cut. So we'd solve piece types 1-7 of ETD1.5, then types 8 & 9 of ETD2.5, types 10-17 of ETD4.5, types 18-26 of ETD9.5, types 27-32 of ETD13.5, and finally types 33-41 of ETD20.5. The commutators we use to solve the new piece types of each new puzzle will leave the previously solved puzzles unchanged. In case you are curious, my instinct tells me that the very last piece type to be solved would be 41. My guess from looking at the puzzle for a while and trying to imagine algos, is a solving order of 39, 37, 38, 36, 35, 34, 33, 40, 41. The first two puzzles would be quick wins but I'd probably need at least a couple of hours each to figure out algos for the other four.
Very interesting... thanks for all that. But finding those commutators which can be used to solve the deeper cut puzzles while leaving the previously solved puzzles unchanged is exactly what I'd consider being able to solve all 41 piece types at the same time. For example I suspect that if someone wanted to solve just ETD20.5 they'd be able to use simplier commutators then those that would be needed to solve the entire Type-Ⅱ Edge-Turning Multidodecahedron. My gut tells me that as there are 41 pieces you may need commutators atleast 41 moves long (maybe much longer) just to isolate the moving of a single piece type while solving the deepest cut pieces. Finding those commutators I would think would be a real challenge even for the best puzzle solvers. It's certainly well beyond me at the moment. I really wish Gelatinbrain (or anyone that would know how) would make something like this so people like yourself could play with puzzles like these. Maybe it isn't really as hard as I'm thinking.

Carl
It definitely gets trickier towards the end. I think the key is to base all commutators PQP'Q' on a conjugate P, in the middle of which is one of two sequences:
(1) ABA, where the overlap does not include any piece types already solved;
(2) ABAB, where there is no overlap at all for the previous puzzles.
This way everything should cancel out for the shallower cut puzzles while solving the deeper cut puzzles.

ETD2.5 without spoiling ETD1.5:

Guaranteed, I think. The fairly obvious (3,1) and (5,1) commutators to cycle the 8/h and 9/i pieces respectively are of type (1) above and have no effect on ETD1.5. (By the way, I think Gelatinbrain 1.4.5 can be in place of this puzzle, as the 3/c pieces are already in ETD1.5.)

ETD4.5 without spoiling ETD2.5:

Here we have a huge range of choices -- for any A we pick, I count 10 different possible B edges to use for type (2) algos. No point in solving the corners at the moment though.

ETD9.5 without spoiling ETD4.5:

There's a suitable distance apart for a type (1) algo affecting diamond groups of pieces, each centered on a corner, to solve p, r, z, and x. There's also a suitable further distance apart for a type (2) algo to cycle three slivers of 17 "new" pieces - a central 20/t surrounded by s, u, v, w, and y.

ETD13.5 without spoiling ETD9.5:

I can't really visualize this one properly, as it's both complicated and too far away from 1.4.6 and 1.4.3. I'd guess that it's another combination of type (1) and (2) algos, probably finishing with type (2) cycling of slivers of piece types 30-32.

ETD20.5 without spoiling ETD13.5:

This is the really tricky one. I think every piece type must be solved with type (2) algos, all based on edges that are as far apart as possible while still intersecting. I think I could use 1.4.3 to find the last or second-last algo, though, as whatever pure-cycles 1.4.3 pieces should only affect piece types 40 and 41.

But, as you say, it would be nice to have these puzzles in simulated form to give the challenge a try. If people like Matt, Michael, Evgeny, and I compared notes and teamed up on this one, we could probably come up with a very good solution.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 8:20 pm 
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(F (CDE ABAB EDC, A) F, E)

Above is a screenshot of Gelatinbrain 1.4.3 (Big Chop), showing a (24,1) commutator that has no effect on EDT1.5, 2.5, 4.5, 9.5, or 13.5 (because A and B don't overlap at all with those puzzles, everything cancels). I'd expect just a cycle of 3 pieces of type 41 in puzzle EDT20.5, with or without moving some type 40 pieces around too. So this algo with setups (which are surprisingly short) would be the second-last or last stage of the Type II edge-turning challenge.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 11:18 pm 
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Allagem wrote:
Finally!!!!
I researched this EXACT thing about 8 months ago. I determined that there were 2844 pieces of 40 unique types not including the core, sound familiar?

It sure does.
Allagem wrote:
I should have known that it would be Carl to repeat the same thing. I believe I posted long ago that the Deep cut rhombic tricontahedron was by far the most difficult single layered puzzle to make because it would take almost 3000 pieces to build... Now everyone knows why. Excellent job on the animation Carl, I only used Jaap's sphere and it was quite tedious.

Congrats! And here I thought I might have been the first to count these. Looks like I missed that honor by 8 months unless we were both beat to the punch by someone else.
Allagem wrote:
Although, I think it might interest some people to know that I organized my model somewhat differently. Instead of how many pieces of a given type move in one twist, I defined how many ways ecah piece can move in one twist. Organizing the pieces in a complicated tree graph with the level down equal to the number of ways a given piece can move and assigning each nodes parents as the adjacent pieces on the puzzle that belong in the level above a given piece results in a diagram of HOW TO BUILD THE PUZZLE. Each given piece must be designed to support its childrenin the graph (which is interestingly enough non-planar btw). The big chop pieces are the very bottom piece on the chart, the only piece type that can be moved 15 ways (note that 30faces/2 = 15 :wink: )

Do you have your model/complicated tree graph in a format you could share here? I'd love to take a peak?
Allagem wrote:
Although I should point out that Matt Shepit discovered a way of eliminating certain pieces that are redundant in the mech. I haven't figured out exactly how many pieces fall in this category but I would bet if the right mech "path" was chosen, about 1/3 of the pieces could be eliminated. 8-)

Would the eliminated pieces leave holes/voids inside the puzzle? That might effect the stability of the puzzle. Also if you are just wanting the deep cut puzzle... Big Chop... many of the pieces could be fused together as the slice layer pieces have to be fixed to one side or the other. Still I'm sure you are up well over 1000 pieces and I don't know if something that complex could be built and expect to function.
Allagem wrote:
PS
wwwmwww wrote:
I wonder how many solvers here think they could solve 41 piece types at the same time?
I bet I can :wink:

I'd love to see it done... I really hope someone builds an applet that allows people to play with this thing. How long would it take to solve? How many moves? Who would be the first to solve it? Etc. I need to get a job first... but if someone were to build an applet that allowed this puzzle to be played with I'd be willing to offer a prize to the first solver.

If I had the skill to make the applet myself I think I would make an interface that looked something like Gelatinbrain which showed the front and the back of the puzzle. I would then add a slide bar that could be clicked on to drag the view back and forth between ETD1 and ETD20.5.

Once that's solved I could up the ante by somehow coloring the puzzle such that each piece had a unique position such that you could super-solve the Type-Ⅱ Edge-Turning Multidodecahedron. I must admit this is a fun puzzle to think about.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 07, 2009 11:40 pm 
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Julian wrote:
(F (CDE ABAB EDC, A) F, E)

Above is a screenshot of Gelatinbrain 1.4.3 (Big Chop), showing a (24,1) commutator that has no effect on EDT1.5, 2.5, 4.5, 9.5, or 13.5 (because A and B don't overlap at all with those puzzles, everything cancels). I'd expect just a cycle of 3 pieces of type 41 in puzzle EDT20.5, with or without moving some type 40 pieces around too. So this algo with setups (which are surprisingly short) would be the second-last or last stage of the Type II edge-turning challenge.


WOW!!! That is surprisingly short. I was initially afraid this was a puzzle so complex that no one would ever want to play with it but the more I see the more I think it could actually be quite fun. I certainly can't think of any other twisty puzzle that lets you play with anything close to 41 piece types. If I counted correcty the 12x12x12 has 21 piece types which I think is the record for puzzles that have actually been made.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Tue Dec 08, 2009 2:32 am 
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Allagem wrote:
I believe I posted long ago that the Deep cut rhombic tricontahedron was by far the most difficult single layered puzzle to make because it would take almost 3000 pieces to build... Now everyone knows why. Excellent job on the animation Carl, I only used Jaap's sphere and it was quite tedious.


Allagem wrote:
Although I should point out that Matt Shepit discovered a way of eliminating certain pieces that are redundant in the mech. I haven't figured out exactly how many pieces fall in this category but I would bet if the right mech "path" was chosen, about 1/3 of the pieces could be eliminated. 8-)

Peace,
Matt Galla


It seems we are on the similar track, Matt.
Adding layers in a deep cut puzzle is much more simpler than adding an axis of rotation. And I agree that deep cut vertex turning rhombic triacontahedron has too many axes and is by far hopeless to be built.


wwwmwww wrote:
If I counted correcty the 12x12x12 has 21 piece types which I think is the record for puzzles that have actually been made.


Carl, I'm not good at solving cubes but physically there are 10 more mirrored pieces, altogether 31 visible piece types. Maybe the curse of eliminated hidden layer. Anyway I don't know why. Is there more restrictions on swaps? :?

Your illustrations are very impressive. I've also been thinking about nested puzzles e.g. 222 in 444. Physically they are not easy to be built. But we can rearrange stickers to visualize it on the external surface. For example, a set of colored numbered stickers for 444 cube can illustrate its internal 222 cube. Maybe also indicating the hidden redundant pieces(hidden 333 cube) if stickers are appropriately designed.

Enjoy!
Leslie


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Tue Dec 08, 2009 9:43 am 
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Leslie Le wrote:
Carl, I'm not good at solving cubes but physically there are 10 more mirrored pieces, altogether 31 visible piece types. Maybe the curse of eliminated hidden layer. Anyway I don't know why. Is there more restrictions on swaps? :?


Those 10 pieces are the Obliques. They come in pairs that are mirror images of each other. In the 12x12x12 you have:

10 Types of Obliques with 48 pieces of each type (Gray in attached pic)
5 Types of X-Centers with 24 pieces of each type (Red in attached pic)
5 Types of Wings with 24 pieces of each type (Green in attached pic)
1 Type of Corner with 8 pieces of this type (Blue in attached pic)

http://www.speedsolving.com/wiki/index.php/Center
http://www.speedsolving.com/wiki/index.php/Edge

Add that up 10*48+5*24+5*24+1*8=728 which accounts for all the surface pieces in a 12x12x12.

Granted if you were casting the parts you'd need 2 molds for each of the Obliques or 31 molds in total but the same is true for the Obliques/Asymetric pieces in this Type-ⅡEdge-Turning Multidodecahedron.

Did that answer your question about the restrictions on swaps? Not sure I understood the question.

Leslie Le wrote:
Your illustrations are very impressive. I've also been thinking about nested puzzles e.g. 222 in 444. Physically they are not easy to be built. But we can rearrange stickers to visualize it on the external surface. For example, a set of colored numbered stickers for 444 cube can illustrate its internal 222 cube. Maybe also indicating the hidden redundant pieces(hidden 333 cube) if stickers are appropriately designed.


I don't think you can "see" the 2x2x2 inside a 4x4x4 by just re-stickering the 4x4x4. I think you need something like the Crazy 4x4x4's seen here:
http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=14856
to do that.

Carl


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Tue Dec 08, 2009 12:08 pm 
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wwwmwww wrote:
Those 10 pieces are the Obliques. They come in pairs that are mirror images of each other.
...
Did that answer your question about the restrictions on swaps? Not sure I understood the question.


Clear enough. I thought 21 counts the number of all distinctive parts, misunderstanding.

wwwmwww wrote:
I don't think you can "see" the 2x2x2 inside a 4x4x4 by just re-stickering the 4x4x4. I think you need something like the Crazy 4x4x4's seen here:
http://www.twistypuzzles.com/forum/viewtopic.php?f=15&t=14856
to do that.

Carl


Carl, i'm afraid i have to run into details.

Suppose we have a cube with distinctive and oriented avatars in replacement of stickers. Let's name it 'avatar cube'(AC). Clearly, every possible state of the AC has a different projection, i mean a projection like this,
o
o o o o o
o
don't know how to name it. With an AC, we have

Lemma 1.(Deformation Uniqueness)Every move(90 deg rotation) of the cube in any state corresponds to a unique mechanical deformation determined by all avatars. The movement of every hidden block, no matter it is floating or attached to the core so long as it moves unambiguously under external commands(moves), including the core, is determined.

L1 basically says that verything happened inside is determined by external commands(moves). The command itself is described by avatar projection together with the slice being rotated(90 deg).

L1 originates from physical realizations, i.e., no blocks out of control(over visible pieces). Otherwise we will be unable to handle internal blocks. This explains why L1 involves mechanism. L1 is nothing if we hold the precondition that a cube must be fully and unambiguously commandable by rotations of visible pieces when talking about physical cubes.

Now with a physical puzzle with nested structure, everything happened inside can be fully monitored by its avatar version. Thus to "see" the 222 cube inside a 444 cube(physically there is a 222 in 444, e.g., R4), appropriate stickers should be enough(to tell states of the 222).

It will be crazy if avatars of a nested puzzle degenerate into colored stickers - we lost clue of the internal. In this case, we'll probably go back to seek for a way to "see" the internal, e.g., half transparent stickers, crazy 444 etc.

Further more, if we forget about external-driven mechanism(i.e., software simulation), ambituities of external commands(rotations of visible slices) rise. Moreover, subtle driven mechanism(virtual) can be designed, like that of 4D cubes.

Please be patient with my poor English, I'd like to explain if necessary. :D

Enjoy!
Leslie


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Tue Dec 08, 2009 3:02 pm 
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Leslie Le wrote:
Suppose we have a cube with distinctive and oriented avatars in replacement of stickers.


Will this do?
Image

Each surface cubie now has a unique position in the solved stated. This is called a Super 4x4x4. However the first program that I'm aware of that allowed you to play with the 2x2x2 inside the 4x4x4 is the CubixPlayer2 program where it stuck a Super 2x2x2 inside a Super 4x4x4. To solve the puzzle it was called a Super-Super solve so I initially called the Super 4x4x4 with a Super 2x2x2 inside a Super-Super 4x4x4. If you just want a normal 2x2x2 inside a normal 4x4x4, I tried to call that puzzle a [s]Super[/s] Super 4x4x4. Where in some forums [s] [/s] is used to strikeout text. This was meant to imply the second meaning of "Super" without the original meaning.

http://twistypuzzles.com/forum/viewtopic.php?f=1&t=14868

Jared came up with the great idea of calling these Multicubes and calling them Super Multicubes with the special sticker set. Much less confusing and its that terminology that I'm using with the Multidodecahedrons as well.

But just because you have solved or even super solved the outer 4x4x4 you still haven't super-super solved it? This means the inner 2x2x2 could still be scrambled. This question has been asked before.

http://twistypuzzles.com/forum/viewtopic.php?t=5719
http://twistypuzzles.com/forum/viewtopic.php?t=10922

In short the cubies of the inner 2x2x2 are not locked in any way to the outer cubies of the 4x4x4 so stickers attached to those outer cubies still don't give you "much" insight into the state of the inner 2x2x2 cubies. It will convey some parity information I believe so I'll stop short of saying "no" insight.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 12:54 am 
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wwwmwww wrote:
In short the cubies of the inner 2x2x2 are not locked in any way to the outer cubies of the 4x4x4 so stickers attached to those outer cubies still don't give you "much" insight into the state of the inner 2x2x2 cubies. It will convey some parity information I believe so I'll stop short of saying "no" insight.


This is absolutely true. Solving the inner 2x2 and the exterior normal 4x4 does not guarantee the 4x4 centers are oriented. Likewise solving the 4x4 and the orientation of the 4x4 does not guarantee the inner 2x2 is solved, as is proven by the attached screenshot. It's a program I wrote a while ago and I did a legit scramble on it and then solved it into this state. Obviously all 6 sides of every cube are colored, making it very confusing, but also perfectly defining a unique position for each cubie.

Also Carl, technically you're correct about saying "no" insight because it is not possible for the inner 2x2 to be in any state when the outer 4x4 centers' orientations are solved. I'm not sure exactly how many positions are eliminated from this but it's at least half. Also if the entire 4x4 itself is solved, not including center orientations, already some of the inner 2x2 positions are impossible (hence the 4x4 parity :wink: )
But really it doesn't mean much because many positions that still look fully scrambled are still possible for the inner 2x2. (ex, the image below)

Peace,
Matt Galla


Attachments:
File comment: As you can see the outer 4x4, including center orientation is solved, but the inner 2x2 is nowhere close
Super Super 4 Cube Unsolved Center.jpg
Super Super 4 Cube Unsolved Center.jpg [ 69.7 KiB | Viewed 6030 times ]
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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 12:56 am 
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Your example is perfect and thank you for pointing me to those threads. I'll give it in nut.

Assumption on mechanism of super-class cubes: we are talking about a cube of N^3 cubies, a rotation of slice drives the whole layer of N^2 cubies accordingly.
1. According to the assumption, thus L1, solving a supermulticube is no more than a supercube;
2. A multicube can be superstickered to have an easy physical representation of the nature of multicubes. The motivation to destroy(by superstickering) a multicube is to make it physical.
3. If the driving mechanism of crazy 444 is exactly the same as super-class cubes(and pretend not to see those circular facelets), then in order to solve the crazy 444, just hypersticker all facelets except those circular ones. Solve the hyperstickered facelets then you'll solve all.[the gray area to be hyperstickered in pic below]

Attachment:
SuperCubes.png
SuperCubes.png [ 17.59 KiB | Viewed 6029 times ]


The driving mechanism is the key to all-in-one solvability.







Below is a reference, refer to it if necessary.

In this thread,
http://twistypuzzles.com/forum/viewtopic.php?t=5719
I think Tony is talking about the same idea: an invisible physical nested cube in a larger cube. In this case, he said
Tony Fisher wrote:
That would mean there are 8 little cubes hidden in the center. If they were coloured the same as the 4x4x4 exterior then we would have a 2x2x2 puzzle waiting to be solved. Only slice moves on the 4x4x4 would affect it of cause but simply solving the 4x4x4 exterior would not be enough.

When a normal 444 cube is solved, it is not "solved".

And I believe that stardust4ever also noticed this nested nature in
http://twistypuzzles.com/forum/viewtopic.php?t=10922
as he said,
stardust4ever wrote:
I observed that the 6x6x6 has a fully functional 7x7x7 mechanism built into it, albeit totally invisible on an assembled puzzle.

And
stardust4ever wrote:
A perfect N-cube would actually consist of N^3 cubes, each with its own unique position/orientation.

It seems we are still talking about the same thing: a cube with N^3 cubies(There's a direct analogy in the case of dodecahedron.)

And according to L1, the answer to
stardust4ever wrote:
My question is, assuming that the Professor is also a supercube (all pieces have an exact position/orientation), once solved, what would the state of the hypothetical inner 3x3x3 be like?

is the inner 333 is not necessarily solved because we lost clue to the inner. Actually, due to the fact that the 555 is generally not solved perfectly(facelet swaps).

I guess we'd better include Jared's def,
Jared wrote:
I guess I was confused by your notation. I was thinking:

Supercube: regular cube with stickers which force 1 solution.
Super-supercube: cube-in-cube with regular stickers.
Super-super-supercube: cube in cube with super stickers.

I think we need better terminology though - perhaps "multicube" instead of "super-supercube" would be better. That way it would be:

Supercube: same as before.
Multicube: cube-in-cube with regular stickers.
Supermulticube: multicube with super stickers.


With this definition, what I was trying to make in previous posts is that: in order to solve supermulticube, solve the very external super stickers is enough. It is due to the unambiguous mechanism(rotating one slice rotates all N^2 cubies). L1 is not trivial if no such mechanism is defined, which is equivalent to say that the mechanism is what matters in this all-in-one solvability.

And building a supermulticube is not quite necessary because external super stickers can give every hint to those nested cubes.

In the case of multicube(-dodecahedron), it is crazy because we have no visible clue of the inner cubes. Software is then needed. What I mean by 'appropriate stickers' is to give multicube a set of superstickers. In this case, solve the superstickered external solves everything inside no matter what kind of stickers are used in inner cubes.


A crazy 444 is different from multicube because the mechanism to drive the inner cubies(in this case, 24 sectors) is different from that of N^3 stacked cubies.


EDIT: remove italic text -- Leslie

Finally if we stick to multicube(-dodecahedron), then software is generally needed. (However, we can put it into reality using superstickers)


Leslie


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 1:11 am 
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Allagem wrote:
This is absolutely true. Solving the inner 2x2 and the exterior normal 4x4 does not guarantee the 4x4 centers are oriented. Likewise solving the 4x4 and the orientation of the 4x4 does not guarantee the inner 2x2 is solved, as is proven by the attached screenshot. It's a program I wrote a while ago and I did a legit scramble on it and then solved it into this state. Obviously all 6 sides of every cube are colored, making it very confusing, but also perfectly defining a unique position for each cubie.

Peace,
Matt Galla


Matt I suddenly come up with a simple question: if a hyperstickered 444 cube is solved & oriented, is it all solved? I mean, every detail of the inner physical mechanism bound to be in its initial state?

If it is yes, then we are in no contradiction but some misunderstandings concerning terminologies i think. :D

Leslie


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 2:38 am 
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I'm not 100% sure what you mean by hyper stickered (are we talking 4 dimensional here?).... In whatever case you mean, every piece in the outer 4x4x4 that is not in the inner 2x2x2 (that's a total of 56 cubies) can be solved perfectly into its uniquely defined position and orientation, and the inner 2x2x2 is still not guaranteed to be solved.

There is NO way to sticker a standard 4x4x4 to show the state of the inner 2x2x2 puzzle. However, consider the new crazy 4x4 with the small circle on each face. Colored normally this uniquely defines every piece EXCEPT the centers of the exterior 4x4 puzzle (the pieces immediately outside the circle on each face) so if you add some sort of indicator as to which exterior 4x4 center piece is which (such as super stickers) then the resulting puzzle is perfectly equivalent to a super super 4x4x4 that has exactly one solved state. Does that make sense?

Also I realized that I may be confusing you when I was saying center orientation before. In my head I was thinking the center orientation of the 1x2x2 block of exterior center pieces on each face. However calling it oriented is not really accurate at all because each of the exterior center pieces has only one orientation each. What you really have to get right is the permutation of these pieces perfect, not only to the right face, but to the right spot out of the 4 in that face. The super stickering is one way of marking down this necessary permutation but it could be acheived in several other ways.

Also, this should help clear some things up, the sequence u'r'd'rur'dr (all inner slice moves) cycles 3 pieces in the inner 2x2x2 without affecting ANY of the exterior 4x4x4 pieces in any way. So if you think you have another way of showing the inner 2x2x2 puzzle, this sequence should have an effect. If it has no effect, the inner 2x2x2 puzzle is still hidden.

Peace,
Matt Galla


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 9:28 am 
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I mean a common superstickered 444 cube.

Orientation refers to the center(which can be invisible, even not exist) which is subject to eliminate parity. Solving the center involves recording steps to keep track of the center. If in this case, the inner 222 cube can still be scrambled, I could be terribly wrong.

Anyway I'm still in doubt with larger cubes...

Leslie


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 9:43 am 
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(Sorry for not commenting on this topic for the last few days, but being busy with many preparations,
I wanted to spent some sufficient time reading such a mesmerizing topic).

I definitely agree with Andreas that this topic deserves to be transformed into an article.

The way Carl presented the entire description, and his arguments accompanying it, is the best proof
that something complex can be presented in a very understandable way.

I am sure many of us are thankful and impressed by the time you have spent to make those thoughts
as well as creating the appropriate images. Definitely CFF material! But what is more amazing, is that
others have come close to the same conclusion via other paths.

The "side effects" of those wonderful thoughts (e.g. hidden orientations) will happily baffle us for a long time!

:)


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 10:02 am 
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Leslie Le wrote:
Your example is perfect and thank you for pointing me to those threads. I'll give it in nut.


Here it is in a nut shell? Maybe? Something about “I’ll give it in nut” makes me want to wear a cup before I proceed. :wink:

Leslie Le wrote:
The driving mechanism is the key to all-in-one solvability.


Not sure I agree… a puzzle is what a puzzle is… regardless of the mechanism. More on that below.

Leslie Le wrote:
A crazy 444 is different from multicube because the mechanism to drive the inner cubies(in this case, 24 sectors) is different from that of N^3 stacked cubies.


Allagem wrote:
There is NO way to sticker a standard 4x4x4 to show the state of the inner 2x2x2 puzzle. However, consider the new crazy 4x4 with the small circle on each face. [This is the Crazy 4x4x4 cube Ⅰ] Colored normally this uniquely defines every piece EXCEPT the [face] centers of the exterior 4x4 puzzle (the pieces immediately outside the circle on each face) so if you add some sort of indicator as to which exterior 4x4 center piece is which (such as super stickers) then the resulting puzzle is perfectly equivalent to a super super 4x4x4 [or Super 4x4x4 Multicube] that has exactly one solved state. Does that make sense?


The stuff in [] above are additions by me. But I need to point out that the Crazy 4x4x4’s are NOT different from that of N^3 stacked cubies (N=4 in this case). See my post here:

http://twistypuzzles.com/forum/viewtopic.php?p=182904#p182904

Sure it appears there are more then 4^3=64 pieces in these puzzles but some of the pieces are locked together in pairs or triplets. See the pictures in the post I link to above. Using normal stickers we have the following:

Crazy 4x4x4 cube Ⅰ IS a 4x4x4 Multicube.

Crazy 4x4x4 cubes Ⅱ Ⅲ and Ⅳ ARE all Super 4x4x4 Multicubes. They look like different puzzles but they are all in fact the same. They differ only in what sides of those 4^3 or 64 cubies are stickered. In all three, enough faces are stickered to give each cubie a unique position in the solved state making it a Super 4x4x4 Multicube.

As Matt points out even the Crazy 4x4x4 cube Ⅰcan be turned into a Super 4x4x4 Multicube with the use of super stickers.

I agree… this observation isn’t easy to see and it you look at my other posts in that thread and this one:

http://twistypuzzles.com/forum/viewtopic.php?p=180336

You will see that it took me several steps to be able to see it myself.

Leslie Le wrote:
In this case, solve the superstickered external solves everything inside no matter what kind of stickers are used in inner cubes.


Now this raises a question I haven’t thought about… assume we have a Multicube with Super Stickers on the external cubies and Normal Stickers on the inner cubes. When we solve this Multicube is it really solved (or super-super solved)? In the case of a 4x4x4 Multicube the answer I know is yes as even with normal stickers each piece of the inner 2x2x2 has a unique position as 3 faces of each of those cubies is stickered. So let’s look at a 5x5x5 Multicube. The inner 1x1x1 has normal stickers, the inner 3x3x3 has normal stickers, and the external cubies of the 5x5x5 have super stickers. All of these cubies have enough information on them to give then unique places in the solved state except the 3x3x3 face centers which just have a single color on them so their orientation could be off. If you solve this puzzle will parity guaranty these pieces are correctly oriented? I honestly don’t know. I don’t think so… Matt could you try this and make another picture like you did above? Even if parity does fix those face centers as you go to higher and higher order Multicubes the number of pieces parity would have to fix I think grows too fast and this would have to break at some point. I’m almost certain this wouldn’t work for a 6x6x6 Multicube.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 10:06 am 
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Hi Pantazis, I refer to this post:
[quote="Leslie Le"]
Matt I suddenly come up with a simple question: if a hyperstickered 444 cube is solved & oriented, is it all solved? I mean, every detail of the inner physical mechanism bound to be in its initial state?
[\quote]

Matt's pic enlightened me to raise this problem. It's clear that I've missed some points in more previous posts.

Leslie


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 10:28 am 
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kastellorizo wrote:
The way Carl presented the entire description, and his arguments accompanying it, is the best proof
that something complex can be presented in a very understandable way.


Wow! Thanks for all that. It's funny what I can do being unemployed. Texas Instruments laid me off in 2003 and in the 6 months it took me to find a job then I taught myself POV-Ray. Currently I've been out of work for less then 2 months but I HAVE to do something. The job hunting keeps me busy but I still have more time for stuff like this then I would while employed. You on linkedin.com by any chance? As this stuff isn't my job I don't think you could recommed me. But the line above reads like my ideal recommendation. Thank you!

Carl

P.S. I just gave you the honor of being the first quote in my signature of any forum. In fact this may be my first signature period.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 10:48 am 
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wwwmwww wrote:
kastellorizo wrote:
The way Carl presented the entire description, and his arguments accompanying it, is the best proof
that something complex can be presented in a very understandable way.


Wow! Thanks for all that. It's funny what I can do being unemployed. Texas Instruments laid me off in 2003 and in the 6 months it took me to find a job then I taught myself POV-Ray. Currently I've been out of work for less then 2 months but I HAVE to do something. The job hunting keeps me busy but I still have more time for stuff like this then I would while employed. You on linkedin.com by any chance? As this stuff isn't my job I don't think you could recommed me. But the line above reads like my ideal recommendation. Thank you!

Carl

P.S. I just gave you the honor of being the first quote in my signature of any forum. In fact this may be my first signature period.



LOL thanks, but in this case I should be thanking you. I was not making compliments, just stating the truth.
I admire people who can pass to other complex thoughts. The schematics animation was simply brilliant!

It is something many teaching places are missing, people who can transfer knowledge with ease.
Plus, it is not the first time you present something nice and well thought.

I am not on linkedin, well at least not yet. But I remember being through unemployed times.
Those times, computers can be very helpful, as it gives the right tool to creative people to
unleash their real power. I am sure many people here have such experiences. :)

I think now we should go back on topic... I can see some tomatoes heading my way!

:mrgreen:


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 11:44 am 
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wwwmwww wrote:

Now this raises a question I haven’t thought about… assume we have a Multicube with Super Stickers on the external cubies and Normal Stickers on the inner cubes. When we solve this Multicube is it really solved (or super-super solved)? In the case of a 4x4x4 Multicube the answer I know is yes as even with normal stickers each piece of the inner 2x2x2 has a unique position as 3 faces of each of those cubies is stickered. So let’s look at a 5x5x5 Multicube. The inner 1x1x1 has normal stickers, the inner 3x3x3 has normal stickers, and the external cubies of the 5x5x5 have super stickers. All of these cubies have enough information on them to give then unique places in the solved state except the 3x3x3 face centers which just have a single color on them so their orientation could be off. If you solve this puzzle will parity guaranty these pieces are correctly oriented? I honestly don’t know. I don’t think so… Matt could you try this and make another picture like you did above? Even if parity does fix those face centers as you go to higher and higher order Multicubes the number of pieces parity would have to fix I think grows too fast and this would have to break at some point. I’m almost certain this wouldn’t work for a 6x6x6 Multicube.

Carl

super stickers on the outer pieces would not be needed, all the outer pieces already have absolute orientation and permutation because they are stickered on multiple sides.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 12:01 pm 
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elijah wrote:
super stickers on the outer pieces would not be needed, all the outer pieces already have absolute orientation and permutation because they are stickered on multiple sides.


Look at the 6x6x6 for example. The outer edges and corners all have multiple sides stickered... but not the face centers. They only have one sticker. So to fix their position in the solved state a super sticker would be needed.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Wed Dec 09, 2009 12:08 pm 
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Time to resurrect some rather old point in this discussion:
wwwmwww wrote:
(1) I don't think its possible to construct a model where each piece types occupies some positive volume inside a single dodecahedron. For example as I show the order=2 Type-ⅡEdge-Turning Multidodecahedron is a set of puzzles (ETD0 through ETD20.5) I think the order=4 Type-ⅡEdge-Turning Multidodecahedron could be a set of different order=4 Multidodecahedrons where the cut plane spacing is different in each and each of those is a set of different puzzles. Boy that could get very very messy. How many different cut plane spacings would be needed to be sure all piece types were represented?
You need at least two different cuts. Think about the faceturning dodecahedron. There are two types of face pieces (12 pieces each). In space, they look like a pentagonal pyramid directly attached to the core and (above that) like a pentagonal pyramid without ground face since it is endless.
If you take this Multidodecahedron and add another layer of cuts there are three ways to place these cuts:
1. The cuts go through the point of intersection between those two aforementioned pieces.
2. The cuts are nearer to the core.
3. The cuts are farer away from the core.
In the 2nd and 3rd case you always get positive volumes you won't get in the other case.

Still no notion about how many cut have to b made exactly. But at least two.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sun Dec 13, 2009 4:59 pm 
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Re: Carl's edge-turning multidodecahedron challenge
wwwmwww wrote:
I wonder how many solvers here think they could solve 41 piece types at the same time?
I have worked out a solution using Wouter Meesen's excellent Ultimate Magic Cube (UMC) freeware simulator. Thanks Carl for the great animations and diagrams, and a really fun challenge.

My method is to solve the 6 puzzles separately, moving from the shallowest cut to the deepest cut. The exposed core pieces (type 1) are already solved, which leaves 40 types to solve. Apart from when solving types 10 and 27-29, algos for the 2nd to 5th puzzle are built around a simple 3- or 4-move "protective wrapper" to shield earlier piece types from change (see earlier in this thread for more explanation). The exceptions and the EDT20.5 pieces need careful observation and experimentation to leave the previous puzzles intact. For example, if you find an efficient algo to cycle one of types 27-29, you'll probably find that type 23 has been affected in the previous puzzle, so you'll need to adjust and/or expand the setups.

Below is my solving order without actual algos, where the length of the PQP'Q' commutators is written in (P,Q) format.
Note 1: I'm using WCA-style counting, so a Q of 2 means a slice move.
Note 2: Special setups are needed to orient the corners at the same time as permuting them.
Note 3: It's a good idea to work out solutions to Gelatinbrain 1.4.1 - 1.4.6 before attempting this one!
Note 4: My algos are notated in the background color next to the algo lengths; highlight text to see. _ means slice move.

Image

EDT1.5 (UMC depth 44)
1 - Already solved
4 - (1,1) (Q, R)
3 - (3,1) (RQR, M)
5 - (3,1) (TOT, Q)
2 - (1,1) (P, R)
7 - (3,1) (RPR, G)
6 - (5,1) (MRGRM, L or P or Q)
EDT2.5 (UMC depth 55)
8 - (3,1) (RPR, E)
9 - (5,1) (NRPRN, E)
EDT4.5 (UMC depth 81)
10 - (5,1) (OTATO, Q)
11 - (4,1) (AQAQ, S)
15 - (6,1) (K AQAQ K, O)
17 - (6,1) (K AQAQ K, H)
16 - (1,1) (ASAS)
12 - (6,1) (H ASAS H, E or F or G or N or R or U)
13 - (6,1) (O ASAS O, L)
14 - (6,1) (H ASAS H, Y)
EDT9.5 (UMC depth 107)
18 - (4,1) (ASAS, C)
26 - (6,1) (ɣ ASAS ɣ, D)
24 - (6,1) (ɣ ASAS ɣ, V)
22 - (4,1) (ARAR, H)
21 - (10,1) (YαN ARAR NαY, W or β or ɣ or δ)
20 - (8,1) (CO ARAR OC, T)
19 - (10,1) (OCB ARAR BCO, K)
25 - (10,1) ((ARAR, H), β)
23 - (6,1) (β ARAR β, X)
EDT13.5 (UMC depth 132)
27 - (12,1) (G (CFCF, T) G, R)
29 - (10,1) (RGQ AMAM QGR, V)
28 - (6,1) (E AMAM E, G)
31 - (1,1) (A, G)
30 - (6,1) (S AGAG S, H or I or J or K or O or T or X or Y or ɣ or δ)
32 - (6,1) (T AGAG T, H or I or J or O or X or Y or ɣ or δ)
EDT20.5 (UMC depth 160)
39 - (6,1) (O AGAG O, J)
34 - (8,2) (HP AGAG PH, A)
38 - (4,2) (AGAG, M)
36 - (8,2) (GO AGAG OG, H)
35 - (6,2) (P AGAG P, O)
33 - (10,2) (ɣEO AGAG OEɣ, G)
37 - (6,2) (N AGAG N, H)
40 - (12,1) (G (AGAG, I) G, X)
41 - (18,1) (NGMH (AGAG, I) HMGN, I)

Edit: Played some more with EDT20.5 and tightened up the last two stages from their previous (16,1) and (24,1).


Last edited by Julian on Tue Nov 09, 2010 3:39 pm, edited 4 times in total.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sun Dec 13, 2009 6:10 pm 
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Julian wrote:
I have worked out a solution using Wouter Meesen's excellent Ultimate Magic Cube (UMC) freeware simulator. Thanks Carl for the great animations and diagrams, and a really fun challenge.


You are very welcome and thank you. I need to check out that Ultimate Magic Cube simulator. I see it can use faded stickers to give each piece a fixed position. Could you comment on how solving a normally stickerd Edge-Turn Multidodecahedron would differ from super-solving a Edge-Turn Multidodecahedron with these faded super-stickers? And I'm not sure if this is possible or practical but would it be possible to make a video of a solve using that simulator? If possible, I'm guessing its not practical for many reasons.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Sun Dec 13, 2009 8:06 pm 
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wwwmwww wrote:
I need to check out that Ultimate Magic Cube simulator. I see it can use faded stickers to give each piece a fixed position. Could you comment on how solving a normally stickerd Edge-Turn Multidodecahedron would differ from super-solving a Edge-Turn Multidodecahedron with these faded super-stickers?
Apart from the centers, super-solving would use exactly the same algos but would just take longer. Instead of being able to solve any of 5 indistinguishable pieces to a particular position, you'd have to set up the exact piece. The usual opportunities to cycle 3 pieces at a time instead of 2 would be reduced to almost nil. The centers can be misoriented so that's extra setups (e.g. F/U U/R F/R to rotate the F center 144 degrees).

wwwmwww wrote:
And I'm not sure if this is possible or practical but would it be possible to make a video of a solve using that simulator? If possible, I'm guessing its not practical for many reasons.
As a ballpark estimate before I do a calculation sometime, we would probably be looking at around 30,000 moves and 72+ hours of solving time for me (not counting sleep or breaks). I'm quite a slow solver; it took me 12+ hours plus extended breaks to solve Gelatinbrain 1.4.6, which has fewer pieces than EDT9.5 by itself! This is one I will never actually solve by myself for lack of time and motivation. However, I can type up all my algos and post them in the background color, to provide more info without being a blatant spoiler.

And if you or someone else writes a sim that allows the saving of the puzzle state and move stack to exchange a puzzle file as part of a team collaboration, I would be happy to take some shifts. The solve could have markers placed at the moment each piece type is solved, so when viewing the solve at the end, playback speed could alternate between jumping dozens of moves each frame to individual moves for the last 30 moves of each section.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 9:09 am 
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Julian wrote:
And if you or someone else writes a sim that allows the saving of the puzzle state and move stack to exchange a puzzle file as part of a team collaboration, I would be happy to take some shifts. The solve could have markers placed at the moment each piece type is solved, so when viewing the solve at the end, playback speed could alternate between jumping dozens of moves each frame to individual moves for the last 30 moves of each section.


I'm pretty sure UMC already does this. If you save a scrambled puzzle, it will be scrambled teh same way when you load it.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 10:37 am 
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Aren't the edge-turning multidodecahedrons more complex than the face-turning multidodecahedrons?
I don't think there's ever been a discussion on FT multidodecahedrons.
Also, I don't think I understand the issue with the 4th order edge-turning dodecahedrons, could you make a rendering to further explain this issue?

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 11:20 am 
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elijah wrote:
Aren't the edge-turning multidodecahedrons more complex than the face-turning multidodecahedrons?
I don't think there's ever been a discussion on FT multidodecahedrons.


From the first post in this thread:
wwwmwww wrote:
http://www.twistypuzzles.com/forum/viewtopic.php?f=1&t=14875
were I looked at the Type-ⅡFace-Turning Multidodecahedron.


elijah wrote:
Also, I don't think I understand the issue with the 4th order edge-turning dodecahedrons, could you make a rendering to further explain this issue?


Yes... but not easily. In short it boils down to this. An order=1 MultiDodecahedra has now degrees of freedom. The puzzle is what the puzzle is regardless of the puzzles size so it can be shown with a picture. With order=2 and order=3 MultiDodecahedra there is 1 degree of freedom. You can fix the cut planes and allow the puzzle to grow so the puzzle and all its pieces can be shown with an animation. With order=4 and order=5 MultiDodecahedra there are 2 degrees of freedom. With orders 6 and 7 there would be 3 degrees of freedom, etc. These additional degrees of freedom come from the fact that the relative spacing of the different types of cut planes don't have to be fixed. With an animation you typically take advantage of one variable... time. To properly show an order=4 MultiDodecahedra there are two variables that would need to be changed and that would get complicated in a single animation. Making more then one animation is one way to deal with that but there are others... I'm not sure what's really the best way.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 12:07 pm 
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Andreas Nortmann wrote:
You need at least two different cuts. Think about the faceturning dodecahedron. There are two types of face pieces (12 pieces each). In space, they look like a pentagonal pyramid directly attached to the core and (above that) like a pentagonal pyramid without ground face since it is endless.
If you take this Multidodecahedron and add another layer of cuts there are three ways to place these cuts:
1. The cuts go through the point of intersection between those two aforementioned pieces.
2. The cuts are nearer to the core.
3. The cuts are farer away from the core.
In the 2nd and 3rd case you always get positive volumes you won't get in the other case.

Still no notion about how many cut have to b made exactly. But at least two.


Hmmm... elijah post made me think about this a bit. The answer I think is 21. I think the General Order=4 Edge-Turning Multidodecahedron is made up of 21 specific Order=4 Edge-Turning Multidodecahedron that can be presented with a single animation. They are the following:

(1) Order=4 ETmultiD0.5 with Order=2 ETD0.5 as its core.

Start the animation at ETD0 as above and continue with just the deeper cut plane until you get to ETD0.5. At this point add the shallower cut plane at the surface of ETD0.5. This cut... cuts up the pieces just added at the surface of order=2 ETD0.5.
Continue to grow the puzzle until no new pieces are added. At infinite scale factor this puzzle becomes a Big Chop.

(2) Order=4 ETmultiD1.5 with Order=2 ETD1.5 as its core.

Start the animation at ETD0 as above and continue with just the deeper cut plane until you get to ETD1.5. At this point add the shallower cut plane at the surface of ETD1.5. This cut... cuts up the pieces just added at the surface of order=2 ETD1.5.
Continue to grow the puzzle until no new pieces are added. At infinite scale factor this puzzle becomes a Big Chop.

(3) Order=4 ETmultiD2.5 with Order=2 ETD2.5 as its core.
.
.
.
(21) Order=4 ETDmultiD20.5 with Order=2 ETD20.5 as its core.

As pieces are just added at the ".5" puzzles I believe this guarantees you'd capture all the piece types present in the General Order=4 MultiDodecahedron.

Hmmm... thinking a bit more I'm not sure. There are 20 Order=4 MultiDodecahedal puzzle not captured in the above. Those being the ones with the interger order=2 ETD puzzles as their cores though all their piece types may already be captured. The other tricky part that I'm just now seeing is by doing the above I guarantee to cut each piece exposed at the surface of the core with the new(shallower) cut but in the context of the Order=2 MultiDodecahedron itself does this account for all the pieces that don't touch the core?

For example... jumping from Order=4 ETmultiD11.5 with Order=2 ETD11.5 as its core to Order=4 ETmultiD12.5 with Order=2 ETD12.5 as its core is it possible the cut plane introduced at the red face jumped over a piece that hasn't been exposed yet that it would have cut in half had I taken smaller steps? I'm not sure.

I guess all I can say is that a minimum of 21+20 or 41 animations are needed to capture all possible Order=4 puzzles. If you just want to capture all piece types... I think the minimum is 21 with alot of redundancy between them.

Carl

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 6:11 pm 
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AndrewG wrote:
I'm pretty sure UMC already does this. If you save a scrambled puzzle, it will be scrambled teh same way when you load it.
Thanks, I started using UMC over the weekend so I'm still discovering what it can do! But I don't think UMC can make dodeca E44 + E55 + E81 + E107 + E132 + E160; I get "Error: Maximum vertex limit reached" when trying to combine them. Even if UMC could do this, the result would be horribly convoluted to look at, which is why Carl suggested displaying the 6 puzzles separately.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 6:18 pm 
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Also, I have a further question.
I think I disagree that these nor the FT multidodecahedrons can really be called multidodecahedrons in the same way that cubes are called multicubes.
Here is the reason...
multicubes contain all lower order cubes of either evens or odds.
These multidodecahedrons that you are proposing are completely different, as you are not mixing orders tofether to make an order 1, order 3 and order 5 dodecahedron, but you are mixing types.

The reason this is not comparable to the same way you might consider cubes is because cubes can only be defined in orders and type(Face, edge, or vertex turning) whereas dodecahedrons have many many other options. They have order, type and then they have multiple puzzles for each order that are quite different because they have a lack of perpendiculur cuts, which causes seperate overlaps with slight distance changes.
For this reason, you cannot even attempt to catagorize these dodecahedrons in the same way as multicubes.

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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Mon Dec 14, 2009 8:14 pm 
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Carl, I have edited my earlier post to include actual algos.


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 Post subject: Re: A Type-ⅡEdge-Turning Multidodecahedron...
PostPosted: Tue Dec 15, 2009 11:39 am 
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elijah wrote:
The reason this is not comparable to the same way you might consider cubes is because cubes can only be defined in orders and type(Face, edge, or vertex turning) whereas dodecahedrons have many many other options. They have order, type and then they have multiple puzzles for each order that are quite different because they have a lack of perpendiculur cuts, which causes seperate overlaps with slight distance changes.
For this reason, you cannot even attempt to catagorize these dodecahedrons in the same way as multicubes.

I am afraid I don't understand that. Do you want to say there are more doctrinaire puzzles beyond faceturning, cornerturning, edgeturning dodecahedrons and their hybrids?
Can you give an example?

There might be a way to present Multihedrons of order=4 in one animation:
One could use a exploded view drawing for the completed multihedron of order=2.
The additional set of cuts then starts with the radius of the first set and this radius then grows during the animation. Sadly 2945+ pieces in one exploded view will be to much.


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