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 Post subject: Pentultimate designs
PostPosted: Sat Mar 22, 2003 9:11 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
I have long been fascinated by the pentultimate. I've come up with a general design which I'm fairly confident is workable.

Imagine two blocks which you wish to slide along each other in a straight line. If you cut a slot along their boundary, you can put a disc in the middle which will rotate as they move. You can add teeth to the disc and notches to the edges of the slot, thus forcing the gear to rotate and move exactly half as much as each block does relative to the other one. A more esoteric trick is to add circular notches to the faces of the disc and grooves to the sides of the slot for those notches to move in. This will have the same effect in terms of how the disc moves, but has the additional property that the two blocks can't be pulled apart while the disc is inside.

If we bend things a bit we can make the slot be circular, and the blocks then rotate. My trick for making a pentultimate is to cut grooves of this form along the hidden sides of each visible piece and have a disc sit under each boundary between a triangular and pentagonal piece. The discs keep the puzzle from being taken apart, but allow rotation along the cuts. Every time a rotation is done around a cut, it moves a fifth of a complete rotation, making each of the ten discs beneath move half that distance, resulting in each of them moving to the next position along and the puzzle once again being in a rotatable position.

The details of course must be hashed out, but I'm confident a real physical pentultimate which is robust enough to play with could be made using this technique.

An interesting possibility this internal mechanism raises is that the outside of the the puzzle could be made transparent, and the puzzle would then be to position and orient the internal discs.

Am I being even vaguely understandable here? I very badly want to get to play with a pentultimate some day, and am curious if other people think my design makes sense, or have ones of their own.


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 Post subject: Re: Pentultimate designs
PostPosted: Sat Mar 22, 2003 10:21 am 
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Joined: Sat Mar 22, 2003 10:21 am
Location: Orlando, Florida
I'm kind of lost about what you're saying. but I think there is an easier more simpler method to building a Pentultimate.

The pentultimate seems to have a structure similar to that of a skewb. So I would take a skewb (probably would be a mini type), and attach branches to 4 of the corners( The ones that make up the base of the skewb). And all 6 of the middle squares would be have branches coming off as well.Connected to each of the 10 branches would be 10 of the corners on the pentultimate. The rest of the pieces are placed, just as you would place the rest of the pieces on a skewb. You would place the pentagons between the corners. The rest of the holes , which would be between the pentagons would have the other 10 corners of the pentulitimate in them.

The skewb that would be in the middle would probably have to be a skewb ball though, so all the other pieces could rotate around it.

I'm pretty sure this design would work, but there's no way that someone like I could build it. It would be nice if someone could build one of these.


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 Post subject: Re: Pentultimate designs
PostPosted: Sat Mar 22, 2003 11:32 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
If only it were so easy!

The symmetry of what you're talking about is all wrong, but other approaches based on simple sliding blocks run into much deeper problems.

The pentultimate fundamentally differes from the skewb. In the skewb there are four pieces whose relative positions never change relative to each other. In the pentultimate that simply doesn't happen. Even worse, if you take the approach of the 2x2x2 of picking a single piece as the one which is always immobile, there is always at least one slice which doesn't border on the particular piece which you wanted to be stationary (six ways to slice, five edges to the pentagonal pieces).

My design works around the myriad problems of not having anything stationary in a deep cut puzzle by simply having nothing stationary at all. When you rotate on a slice, neither side of that slice is the priveledged one which is stationary. They both move, and to keep them attached there are discs put in the middle which move half the distance relative to either side.

The discs approach is kind of a cheap trick as a mechanism, but I can live with some overly technological machinery in order to have a working pentultimate!


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 Post subject: Re: Pentultimate designs
PostPosted: Sat Mar 22, 2003 11:33 am 
Jeff, I might not be fully understanding what you say, but I don't see how it could be done with a skewb.

There are only 5 platonic solids, tetrahedron, cube, octahedron, dodecahedron and icosahedron these have 4, 8, 6, 20 and 12 vertices. These are all of them because essentially these are the only numbers of points which can be distributed evenly over the surface of a sphere. ie the distance between any two nearest points is always the same.

There isn't a platonic solid having 10 vertices, so you can't choose 10 vertices from a dodecahedron so they are evenly distributed. It wouldn't have the right symmetry.

Max


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 Post subject: Re: Pentultimate designs
PostPosted: Sat Mar 22, 2003 11:51 am 
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Joined: Sat Mar 22, 2003 10:21 am
Location: Orlando, Florida
I still think my way would work.

Max, you can't say that my design would have 10 axes because all the axes are connected to a skewb, and the skewb has 4 axes.

Bram, the 10 axes would be able to change position since they would all be attached to the skewb. The pentultimate does have 4 pieces who's relative postion doesn't change, and these would be the same pieces that are part of the skewb in the middle.

Jeff


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 Post subject: Re: Pentultimate designs
PostPosted: Sat Mar 22, 2003 1:16 pm 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
> The pentultimate does have 4 pieces who's relative postion
> doesn't change

That is incorrect. In the process of twisting a pentultimate, any edge of any visible piece can briefly come in contact with any edge of any other visible piece.

The pentultimate page links to some simulators which can help with visualization of this.

I've spent a lot of time thinking about this, and I'm convinced that an esoteric technique, such as my discs or a ratcheting mechanism, must be used to build a pentultimate.


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 Post subject: Re: Pentultimate designs
PostPosted: Sun Mar 23, 2003 11:33 am 
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Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA
There are a few more ways of distributing points evenly on a sphere. One can space any number of points evenly around the equator, and one can put points at the midpoints of the edges of the cube or dodecahedron. The midpoints produce 12 and 30 points, respectively. Still no 10, and even if you had 10, it still wouldn't work because the way the pentultimate rotates would interfere with umbrella-shaped pieces which tried to sit stationary and hold everything in.

By the way, the different vertex arrangements result in a few other deep cut puzzles. Ignoring hockey pucks, there's the skewb, 2x2x2, pentultimate, 12 axis cube puzzle, and the thing described in patent #4600199. Those last two would probably require construction just as tricky as the pentultimate, if not more so, and only have one type of piece, so they're likely to actually be easier to solve. Plus their slicing action would be counterintuitive and klunky, even if the physical action of it could be made smooth.

The pentultimate appears to be by far the most compelling of the completely unexplored rubik-type puzzles. I'm sure others feel this way as well, and the only reason one hasn't been built already is that noone's figured out how. I'm trying to remedy that situation.


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 Post subject: Re: Pentultimate designs
PostPosted: Sun Mar 23, 2003 1:16 pm 
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Joined: Tue Feb 11, 2003 8:18 pm
Location: Melbourne, Australia
I came up with an almost identical idea for the pentultimate only a few weeks ago! Yes, the trick is that the connectors between parts move at half the speed of the twist, and always line up again at the end of a full twist.

Instead of having a spinning cog between pieces and teeth in the groove though, I was thinking of it the opposite way. That is, the connector which fits into the grooves has teeth, and there's a cog in each groove in each piece. To be practical you might need two cogs in each groove, not sure.

Bram's way is simpler though. My connectors would be bigger at the ends, to fit snuggly into bigger gaps inside each groove, so that the pieces couldn't come apart. The teeth could be along this larger part. I would need 60 connectors, plus at least 120 cogs (one either side of each edge), whereas Bram only needs 60 cogs.

I'd thought of having the cogs between parts, but couldn't see how to make the pieces hang on to it. Actually, I'm still not clear on this. Bram, how do the pieces hang on to the cog? You could have a raised circular part on the cog, but the pieces either side are moving in opposite directions, so how can something hold on?

Yes, I'm sure this method would work. My version would probably have been too impractical, with too many parts, but Bram's version sounds more doable.

I might try to scan some of my sketches and post them here (they're pretty bad, but they might help people to see what we're talking about :-)).

I also thought of another way the pentultimate could be made, but I'm not sure if it's more or less practical. I might keep that to myself for now :-) It does have fewer parts.

I'm not yet in a position to be able to physically make any of these possibilities though. I didn't want to mention them earlier because it seemed like a bad idea without a patent. But since Bram started it, I've shared (some of) my thoughts :-)

By the way, fantastic idea about having transparent faces and solving the rotation of the cogs! You could market it as a two-level puzzle. Just solve the external colours, or match the orientation of cogs as well. Although transparent, the outer faces could still have a colour so they can be solved. Either a faint colour in the transparent part, or a colour around the edging.

If anyone does make the pentultimate, I can't wait to see the mods! Icosahedron, icosidodecahedron, rhombic triacontahedron, compound of dodeca + icosahedron, etc! Anthony Greenhill will probably build one back up into a cube shape too :-)

Rob.


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 Post subject: Another thing to keep in mind...
PostPosted: Mon Apr 21, 2003 11:14 am 
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Joined: Sun Mar 17, 2002 1:51 am
Location: USA
The Skewb's rotation of axis is corner-based, whereas the Pentultimate is face-based. This indicates that if a spindle mech for the puzzle could be built (which I highly doubt) it would be attached to the faces, somewhat like a deep-cut dodecahedron. Of course, the center spindle itself would require mobility, since the pentagonal centers move in retrospect to one another. I am picturing a six-sliced center sphere, if this is feasible. I hope you can understand what I am trying to say.


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