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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 1:19 am 
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GuiltyBystander wrote:
Based on Brandon's numbers for the number of distinct positions and assuming a maximum branching size of 38, the god number for this puzzle is > 500. I think this means you need to keep scrambling for another 10-20 minutes.

Hmm, a branching of 38? The naive number would be 19 * 2 * 2 = 76.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 1:44 am 
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Brandon Enright wrote:
GuiltyBystander wrote:
Based on Brandon's numbers for the number of distinct positions and assuming a maximum branching size of 38, the god number for this puzzle is > 500. I think this means you need to keep scrambling for another 10-20 minutes.
Hmm, a branching of 38? The naive number would be 19 * 2 * 2 = 76.
Oops, you're right. I think I forgot the *2 from rotations. This puzzle has too many turnable layers. Still, puts the number at > 430 turns.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 2:43 am 
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I grabbed a cup of coffee and sat and watched the scramble all the way through, totally transfixed. Such a beautiful puzzle when scrambled, the colours look great.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 3:23 am 
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This puzzle is so smooth in turning! WOW!!

I would never be able to solve this.. but that is ok. I can drool over watching you scramble it :mrgreen:


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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 8:31 am 
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Am I correct in reading "jumbling adds no extra icasahedral states" as jumbling doesn't allow pieces to jump orbits?

Also, since how a puzzle is stickered has a huge impact on number of non-jumbling states(remove all the stickers and the Nebula has only one icasahedral state) and sticker mods are generally considered "trivial"(though I imagine a sticker mod of the Nebula might be as labor intensive as a 3*3*3 shape mod if not more so), wouldn't it make for a more meaningful comparison to consider a puzzle stickered so that every piece is unique and has visible orientation(not sure a stickering analogous to a supercube would be enough for the Nebula)?

So, the Super 17*17*17 or the Super Nebula: which has more states ignoring the latter's jumbling? Is the super Petaminx(or higher order super megaminxes if the base puzzle has been built) come close? Are there any other physical Twisty puzzles that might challenge these puzzles for most non-jumbling states on a physical twisty puzzle?

As for Jumbling states, I know no method for getting an exact count has been developed, but have better bounds other than a lower bound of 1 for jumbling puzzles and an upper bound of uncountable infinity been determined?

So, how many visible parts would the logical 3 layer per face/7 layers per axis counterpart to this and the star have?

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 1:25 pm 
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Jeffery Mewtamer wrote:
Am I correct in reading "jumbling adds no extra icasahedral states" as jumbling doesn't allow pieces to jump orbits?
Well lets be clear about what an orbit is here. Every piece type is in its own orbit. In casual twistypuzzle-speak we usually don't bother to call those orbits. We're usually only interested in orbits within a particular piece type.

When it comes to jumbling, different piece types can move out of their orbits into the spot of other piece types. This often happens in groups where a block of several adjacent pieces all move into the spot normally occupied by another piece.

In these cases, the outside of the puzzle (and the cut pattern) change shape.

So, jumbling can't reach more states when all the pieces are put back into their respective piece-specific orbit but it certainly can reach more states if you count the jumbled states. That is, as long as the puzzle is unjumbled, it doesn't matter if it took a jumbling path to get there, there is an equivalent path that doesn't use jumbling that can reach the same state.

Jeffery Mewtamer wrote:
Also, since how a puzzle is stickered has a huge impact on number of non-jumbling states(remove all the stickers and the Nebula has only one icasahedral state) and sticker mods are generally considered "trivial"(though I imagine a sticker mod of the Nebula might be as labor intensive as a 3*3*3 shape mod if not more so), wouldn't it make for a more meaningful comparison to consider a puzzle stickered so that every piece is unique and has visible orientation(not sure a stickering analogous to a supercube would be enough for the Nebula)?
The symmetry group of the icosahedron and the symmetry group of the dodecahedron is the same. For a given piece with sixty instances, on an icosahedron you have twenty groups of three identical pieces. On the dodecahedron the same piece type comes in twelve groups of five identical pieces.

When you look at the "boost" you get by stickering all of the pieces uniquely, the dodecahedral version gets a much bigger increase. That is, five factorial raised to the twelfth power is a much larger number than three factorial raised to the twentieth power.

Eitan's Nebula has a lot more states than a Teraminx so when you compare the super-sticker version of the teraminx to the super-sticker version of Eitan's Nebula, even though the Teraminx gains a lot more states from the super stickers, it can't reach even the normally stickered Eitan's Nebula.

Note that the Petaminx has more states than Eitan's nebula and adding super stickers help the Petaminx more than it help Eitan's Nebula.

Jeffery Mewtamer wrote:
So, the Super 17*17*17 or the Super Nebula: which has more states ignoring the latter's jumbling? Is the super Petaminx(or higher order super megaminxes if the base puzzle has been built) come close? Are there any other physical Twisty puzzles that might challenge these puzzles for most non-jumbling states on a physical twisty puzzle?
NxNxN cubes gain a lot more from super stickers. The icosahedron is disadvantaged in terms of super stickers so adding super stickers usually makes it worse off compared to other puzzles.

Jeffery Mewtamer wrote:
As for Jumbling states, I know no method for getting an exact count has been developed, but have better bounds other than a lower bound of 1 for jumbling puzzles and an upper bound of uncountable infinity been determined?
Well we do know the method, in principle, found counting reachable states when including jumbling. The trouble is enumerating all shapes the puzzle can be in requires code nobody has written. You have to implement the geometry and check for things like overhang bandaging. This puzzle might have an unwieldy number of jumbled states because of its size and master layer but most other jumbling puzzles we have aren't so bad.

Jeffery Mewtamer wrote:
So, how many visible parts would the logical 3 layer per face/7 layers per axis counterpart to this and the star have?
There isn't an unambiguous way to identify the three-layer version of this puzzle. Depending on where you put the cut for the third layer you can create many different puzzles.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 5:46 pm 
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^To clarify, when I was talking about jumping orbits, I was thinking of how the face centers of the Helicopter Cube, despite being mechanically identicle are locked into multiple orbits when only 180 degree turns are allowed, but jumbling moves allow them to move between orbits. If I am reading things correctly, the Nebula lacks any pieces with this property.

But otherwise, my curiosity is satisfied.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Mon Feb 10, 2014 6:34 pm 
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Jeffery Mewtamer wrote:
^To clarify, when I was talking about jumping orbits, I was thinking of how the face centers of the Helicopter Cube, despite being mechanically identicle are locked into multiple orbits when only 180 degree turns are allowed, but jumbling moves allow them to move between orbits. If I am reading things correctly, the Nebula lacks any pieces with this property.
It is possible to make pieces on an icosahedron that come in orbits like a Helicopter Cube with the appropriate choice of grips. For example, if you have a piece with two face grips where the faces share a point but not an edge, then the resulting pieces come in 6 orbits of 10 pieces each. I don't see any natural way to achieve this on a normal FTI though, you'd need to use circles or some other trick like that.

The Slice-Only version of Eitan's Nebula would have some interesting properties. It seems easily doable too by designing a new core that fuses the core to the 20 inner centers and 20 outer centers so that the faces can't be turned but the slices can be.

Edit: looking at the slice-only version, it has the pieces in orbits I describe above, face-centers (orientation visible), and edges (only sides visible). It would actually be an interesting and not-very-challenging solve.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 11:04 am 
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What a magnificent addition to the FTI family! It's so wonderful... congrats. And a very, very sincere thanks for beating me to it. :)

Eitan, did you say this is a combination of Radiolarian 3 and Eitan's Star?

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 11:08 am 
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JasonSmith wrote:
What a magnificent addition to the FTI family! It's so wonderful... congrats. And a very, very sincere thanks for beating me to it. :)
It was my pleasure. :)

JasonSmith wrote:
Eitan, did you say this is a combination of Radiolarian 3 and Eitan's Star?
Yup!

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 12:30 pm 
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What would this puzzle look like if the two layers on each face were the same thickness?


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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 1:09 pm 
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Brandon Enright wrote:
Jeffery Mewtamer wrote:
So, how many visible parts would the logical 3 layer per face/7 layers per axis counterpart to this and the star have?
There isn't an unambiguous way to identify the three-layer version of this puzzle. Depending on where you put the cut for the third layer you can create many different puzzles.
There is a way to approach this question and get the limits as to what the visible (and invisible) piece counts would be. The face turn icosahedron has the same pieces as the corner turn dodecahedron. Here is the approach to count the pieces for the corner turn dodecahedron with 1 layer per face or 3 per axis.

See this post of Andreas here. Jump to the section called "DC - cornerturning dodecahedrons"

To answer your question model a copy of Eitan's Nebula with its 4 cuts fixed. Add another cut parallel to the existing cuts on each axis and include the a copy of the added cut which has been mirred about the plane through the origin and perpendicular to the axis. Now vary the depth of your new cut from the surface to the point it meets its mirrored copy at the origin. Count the surface pieces at each depth that new pieces appear and/or old pieces disappear. You'll find a whole lot of different puzzles and you'll also know how many surface pieces each has.

Carl

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 1:39 pm 
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Jared wrote:
What would this puzzle look like if the two layers on each face were the same thickness?

If you mean lowering the second cut, and you used curvy cuts, and you fudged the depth a bit to eliminate lots of extra pieces, then approximately like:
Attachment:
5adbb2ba6f02514d48b58d9a7a6de83c.png
5adbb2ba6f02514d48b58d9a7a6de83c.png [ 104.71 KiB | Viewed 1103 times ]

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 2:00 pm 
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Eitan, did you cut the tiles by hand or did you use a laser cutter or something?

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 2:10 pm 
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I cut the Cubesmith tiles using my Cameo sticker cutter. It took a while to find the right settings, but after that, it worked very well.

The other 5 colors (4 metallic colors and the gold-rimmed black color) are 0.022in acrylic. I cut these on my university's laser cutter, which allowed me to raster in the logos. The material I used was 2-color acrylic, which is mostly one color of plastic, but with a layer of a different color on one surface, so you can raster through it. Fun stuff to play with. Great for making hacked-together engineering projects look more polished. https://www.inventables.com/categories/ ... 5D=2+color

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star)
PostPosted: Tue Feb 11, 2014 5:24 pm 
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BelcherBoy2000 wrote:
:shock: * picks jaw up off floor* I thought Eitan's Star was the hardest puzzle ever, and then this thing. Good luck if you ever scramble it. Wow, I'm just speechless right now. This blows every other puzzle ever out of the water with ease.

But if you do scramble it, you know how to take it apart, right?. I am simply blown away by this.

EDIT:You did scramble it. That's when you don't catch it on camera and say "oops!"

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Tue Feb 11, 2014 6:01 pm 
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So the Slice-Only version of Eitan's Nebula is equivalent to Gelatinbrain's 2.1.10 minus the outer petal pieces which I've partially grayed out:
Attachment:
slice_master_fti.png
slice_master_fti.png [ 109.83 KiB | Viewed 1019 times ]


It's interesting that the slice layer is so thin that there are only 1 and 2-grip pieces and nothing more. [1,1] commutators are all that are needed to solve it.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Wed Feb 12, 2014 1:32 am 
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That slightly fudged version with the curved cuts looks great. Much larger sticker surface area on the majority of parts. I wonder how much the fudging would affect turning quality. I take it that the different piece shapes wouldn't affect jumbling, but I'm not 100% sure on that?

Brandon, are you serious.. [1,1] commutators for the entire slice layers? I wouldn't have picked that. But that doesn't take into account the insane unjumbling.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Wed Feb 12, 2014 2:32 am 
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Burgo wrote:
That slightly fudged version with the curved cuts looks great. Much larger sticker surface area on the majority of parts. I wonder how much the fudging would affect turning quality.
Well, when I said "fudged" I meant that I set the cuts to what looked like the resulting layers were exactly the same thickness. Then I noticed that it was pretty close to having them just touch which gets rid of a ton of parts, so I adjusted them to do so. I'm using curved cuts and I haven't explored what could be done with straight cuts. There actually are some tiny pieces on the surface near where the lines cross but those pieces would have to be left out of the design. I don't think this could cause any issues but with the depth of that second cut, I suspect designing the puzzle at all, regardless of the tiny parts, would be very hard.

Burgo wrote:
I take it that the different piece shapes wouldn't affect jumbling, but I'm not 100% sure on that?
I don't know. I don't think the second cut is deep enough to enable additional jumbling of a smaller portion of parts. Even if it is, I'm pretty sure the unjumbling task would be straight forward.

Burgo wrote:
Brandon, are you serious.. [1,1] commutators for the entire slice layers? I wouldn't have picked that. But that doesn't take into account the insane unjumbling.
Yeah it's pretty cool. Pick two faces that share and edge and call them A and B, then the commutator [A,B] (A B A' B') is a pure 3-cycle of edges. Then pick two faces that share only a point, call them C and D, then the commutator [C,D] performs a pure 3-cycle of the strange new piece that can be seen on Gelatinbrain's 2.1.10

But there's something interesting. Jeffery Mewtamer's questions about orbits and jumbling got me thinking about this is the first place. I noticed that if a piece had two grips that shared a point but not an edge then it would come in several orbits. Jumbling would move these pieces out of their orbits so a jumbling-scramble of the slice-only Eitan's Nebula would require jumbling to fix the orbits just like a jumbling scramble of a Helicopter Cube requires jumbling in the solve. That [1,1] commutator doesn't look all that useful when you need jumbling to move pieces between orbits...

I find it interesting that if you only turn the shallow layers without slices, this is the same as the Radiolarian 3 and it has edge pieces. Also, if you only turn the slice layers an not the outer layers it also has the exact same edge pieces but they show up in a completely different way. But if you turn the outer layer and slice layers at the same time (or independently of each other) then the resulting puzzle doesn't have the edge pieces.

Note that the slice-only version of 2-layer puzzles like this one are not always simpler. For example, the slice-only version of Eric's Master Starminx is an insane beast -- way harder than Master Starminx with both layers available.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Wed Feb 12, 2014 10:56 am 
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Brandon Enright wrote:
Note that the slice-only version of 2-layer puzzles like this one are not always simpler.
Just to make sure we are on the same page. By two layer puzzle I believe you are talking about "per face" so there are either 4 or 5 layers per axis. The two puzzles mentioned here both have 5 layers. So if we number these 1, 2, 3, 4, and 5 across the axes then by "slice-only version" I believe we are talking about only being able to turn layers 2 and 4 on each axis. On any given axis layers 1, 3, and 5 are fixed relative to each other. The reason I'm asking is I also consider layer 3 a slice layer but I don't think you are allowing it to turn independantly. Please correct me if I'm wrong. If I'm correct I just think "slice-only" isn't the best term to describe these puzzles though I don't really have a better term to offer... maybe sub-face turn only.
Brandon Enright wrote:
For example, the slice-only version of Eric's Master Starminx is an insane beast -- way harder than Master Starminx with both layers available.
What is this based on? Have you solved this puzzle? I don't believe a physical one has been made but maybe its on Gelatinbrain? Or are you able to work this out in your head without a reference puzzle?

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Wed Feb 12, 2014 11:35 am 
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wwwmwww wrote:
Brandon Enright wrote:
Note that the slice-only version of 2-layer puzzles like this one are not always simpler.
Just to make sure we are on the same page. By two layer puzzle I believe you are talking about "per face" so there are either 4 or 5 layers per axis. The two puzzles mentioned here both have 5 layers. So if we number these 1, 2, 3, 4, and 5 across the axes then by "slice-only version" I believe we are talking about only being able to turn layers 2 and 4 on each axis. On any given axis layers 1, 3, and 5 are fixed relative to each other. The reason I'm asking is I also consider layer 3 a slice layer but I don't think you are allowing it to turn independantly. Please correct me if I'm wrong. If I'm correct I just think "slice-only" isn't the best term to describe these puzzle though I don't really have a better turn to offer... maybe sub-face turn only.
I knew I should have been more specific :lol:

So yes, I'm specifically referring to the outer slice layers on order-4 or order-5 puzzles. Turning the only the middle slice layer (on puzzles that have one) always result in "bar" pieces which are interesting but not what I'm talking about here.

So on Eitan's Nebula, I think we agree it's an order-5 puzzle (5 layers per axis, 4 of them independent). If you list the layers from one side to the other, then the two outer layers on opposite faces are 1 and 5 and the middle layer is 3. The two layers I'm restricting the puzzle to are 2 and 4 just like you say above.

On order-4 and order-5 puzzles I casually call the slice "sub-face" layers "slice layers" and if its an order-5 puzzle I call the middle layer the "middle-slice layer". When I say "slice-layer-only" I'm really just talking about the sub-face layers and not the middle slice. I think this is what people usually say casually on higher-order, non-cubic puzzles. It's good to be specific though.

This is what I mean:
Attachment:
eitan_nebula_metal_slices.jpg
eitan_nebula_metal_slices.jpg [ 4.92 MiB | Viewed 893 times ]


wwwmwww wrote:
Brandon Enright wrote:
For example, the slice-only version of Eric's Master Starminx is an insane beast -- way harder than Master Starminx with both layers available.
What is this based on? Have you solved this puzzle? I don't believe a physical one has been made but maybe its on Gelatinbrain? Or are you able to work this out in your head without a reference puzzle?
I have several prints of the Master Starminx. I've performed a full "grip analysis" of the subface-slice-only version of it.

It would be trivial to make a subface-slice-only version of it with a new core where the corners are attached to the core. Pretty much exactly like your Doctor Skewb core.

I plan on modding one of my prints to fix the corners in place to the core but I haven't worked out the best way to attach them to the core yet.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Thu Feb 13, 2014 1:26 pm 
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Brandon Enright wrote:
So the Slice-Only version of Eitan's Nebula is equivalent to Gelatinbrain's 2.1.10 minus the outer petal pieces which I've partially grayed out
Ok... now that I'm sure we are on the same page with regards to what the "Slice-Only version of Eitan's Nebula" is/looks like, I have a question. How does Gelatinbrain's 2.1.10 turn? It appears to be a face turn puzzle which moves 12 pieces and the 6 pieces in the circle center on that face stay fixed. Gelatinbrain hasn't worked correctly on my PC for years so I can't easily check. I count 36 pieces in a slice layer on "the Slice-Only version of Eitan's Nebula". Half of these pieces are formed by the intersection of two slice layers so I assume only those can be scrambled, but that still leaves 18 pieces moving. So either my understanding of Gelatinbrain's 2.1.10 is wrong or I'm missing something else.

Hmmm... can the other 18 pieces in the slice layer of Eitan's Nebula be thought of as a 19th piece? Sort of like a 3x3x3 face center it can't move its location but I guess it could change its orientation with respect to the other 18 pieces.

And would the "Slice-Only version of Eitan's Nebula" jumble? I don't see how Gelatinbrain's 2.1.10 could jumble if it were a physical puzzle... I'm pretty sure the Gelatinbrain version doesn't jumble.

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Thu Feb 13, 2014 3:04 pm 
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wwwmwww wrote:
Brandon Enright wrote:
So the Slice-Only version of Eitan's Nebula is equivalent to Gelatinbrain's 2.1.10 minus the outer petal pieces which I've partially grayed out
Ok... now that I'm sure we are on the same page with regards to what the "Slice-Only version of Eitan's Nebula" is/looks like, I have a question. How does Gelatinbrain's 2.1.10 turn? It appears to be a face turn puzzle which moves 12 pieces and the 6 pieces in the circle center on that face stay fixed.
Yeah it turns exactly like that. It's a shallow-cut FTI and the pieces inside of the circle don't turn with the face. It looks like 12 pieces are moving but the three wedges in the circle part around a face are all connected and are the sides of the "invisible center" and show its orientation:
Attachment:
gb_2.1.10_center.png
gb_2.1.10_center.png [ 30.57 KiB | Viewed 773 times ]


wwwmwww wrote:
I count 36 pieces in a slice layer on "the Slice-Only version of Eitan's Nebula". Half of these pieces are formed by the intersection of two slice layers so I assume only those can be scrambled, but that still leaves 18 pieces moving. So either my understanding of Gelatinbrain's 2.1.10 is wrong or I'm missing something else.

The pieces come in pairs. Here is an example of the circle-wedge piece on Eitan's Nebula. The red sides are the slices the teal piece turns with:

Attachment:
master_fti_new_piece.jpg
master_fti_new_piece.jpg [ 946.02 KiB | Viewed 773 times ]


wwwmwww wrote:
Hmmm... can the other 18 pieces in the slice layer of Eitan's Nebula be thought of as a 19th piece? Sort of like a 3x3x3 face center it can't move its location but I guess it could change its orientation with respect to the other 18 pieces.
Indeed :-) Just like the center on 2.1.10

wwwmwww wrote:
And would the "Slice-Only version of Eitan's Nebula" jumble? I don't see how Gelatinbrain's 2.1.10 could jumble if it were a physical puzzle... I'm pretty sure the Gelatinbrain version doesn't jumble.
Actually 2.1.10 should jumble. If you put the purple wedge in the teal wedge spot via a partial-turn then you enable a jumbling turn:
Attachment:
gb_2.1.10_jumble.png
gb_2.1.10_jumble.png [ 30.56 KiB | Viewed 773 times ]

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Sun Feb 16, 2014 11:27 am 
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Ok... I see it now. Your 12 pieces is really 10 parts as 3 of the pieces are the same part. That 3 piece part becomes an 18 piece part on the Slice-Only version of Eitan's Nebula. And the remaining 9 pieces become 2 piece parts on the Slice-Only version of Eitan's Nebula. Very interesting. And I now see how 2.1.10 could jumble too.

Thanks,
Carl

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 Post subject: Re: Eitan's Nebula (aka Master Eitan's Star) VIDEO!
PostPosted: Sun Feb 16, 2014 11:30 am 
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Man, this trip is fun!
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File comment: Brandon playing with the Nebula, while Nan Ma, Derek Bosch, and Bob Hearn watch.
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