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 Post subject: The TetrarhonsPosted: Mon Oct 07, 2013 1:12 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
My newest puzzle is presented here:

This is the complete set of 28 Tetrarhons. The Tetrarhons are a subset of the Polyrhons (Poly Rhombic Dodecahedra). In this case we are looking at all possible ways of connecting 4 Rhombic Dodecahedra. In this set there are 8 sets of chiral pairs. While the Tetrarhon's pack exactly like sphere do its interesting to note that there are only 25 Tetraspheres. Three of the chiral pairs which lack mirror symmetry correspond to a Tetrasphere which has it. Can you identify which 3? Hint: There are 11 planar (i.e. they can lay flat in one layer either hexagonally or squarely) Tetraspheres and 14 non-planar Tetraspheres. There are 14 in each set when the same division is applied to the Tetrarhons.

Here is a nice drawing of the set (ignore the first column in this image):

And the question I'm hoping to find an answer to soon... can this set of pieces be used to build a rectangular "pyramid" with a 6x7 base (6 layers, 1x2 at the top). If the answer is yes and you are the first to show me how to build it then I'll get you a \$25 discount off of any of my Shapeway puzzles.

I will be adding more pictures to this thread as I get the pieces dyed. This project was sparked by this thread which you can see is the source of my pyramid question. And here is Oskar's puzzle dying post that I referenced in the video. So you should have a good idea what 7 colors I'm interested in testing.

Enjoy,
Carl

P.S. If you say there are only 27 pieces in the video you are mostly correct. I didn't notice the 28th piece was just out of frame at the top right when I set up. You can see its shadow and most of that piece does come into frame as I pick it up toward the end of the video. Sorry about that...

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Last edited by wwwmwww on Sun Oct 13, 2013 2:23 am, edited 1 time in total.

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 Post subject: Re: The TetrarhonsPosted: Mon Oct 07, 2013 10:50 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Thanks so much for doing this, Carl!

Just to reiterate another puzzle from the previous thread - half these pieces are "planar" (the centers of the four dodecahedra lie in a plane). Can the planar pieces be used to make a size-6 tetrahedron? Or can the whole set make two, generally?

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 Post subject: Re: The TetrarhonsPosted: Tue Oct 08, 2013 3:56 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
Thanks so much for doing this, Carl!
You are very welcome.
Jared wrote:
Just to reiterate another puzzle from the previous thread - half these pieces are "planar" (the centers of the four dodecahedra lie in a plane). Can the planar pieces be used to make a size-6 tetrahedron? Or can the whole set make two, generally?
Yes... another very interesting question. I'm personally a better designer then I am a solver so I've been looking for programs that may pack these pieces for me.

I've looked at these so far:
Burr Tools
PolyForm Puzzle Solver

Burr Tools appears to be able to play with the Tetraspheres which are very closely related BUT I don't see how to deal with the chiral pairs where the mirror symmetry comes into play for the the tetrahedrons.

PolyForm Puzzle Solver at first doesn't appear to cover the tetrahedrons but I'm not sure. I emailed Japp and he sent me this suggestion:

Quote:
If you think of your rhombic dodecahedrons as cubes with pyramids on their faces, then you could probably use my Polysolver as is. Use the Cube grid, imagine it coloured like a checkerboard, and only use cubes of one colour. Those cubes correspond to the cores of your rhombic dodecahedra, the unused cubes correspond to spots where up to six pyramids come together. I haven't tried this, but I think it should work.

I'm still thinking about that. Not sure if this would behave properly or even if it would solve the symmetry issue mentioned above.

I've even considered writing my own program but my programs are notoriously slow... i.e. taking lifetimes to do something a real programer's program could do in 5 minutes.

Are there any other 'solvers' out there I should look into?

Carl

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 Post subject: Re: The TetrarhonsPosted: Tue Oct 08, 2013 6:52 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
Burr Tools appears to be able to play with the Tetraspheres which are very closely related BUT I don't see how to deal with the chiral pairs where the mirror symmetry comes into play for the the tetrahedrons.

You could build rhombic dodecahedrons out of rhombic tetrahedra in the rhombic space grid. I think once you build one you can use that as the discrete unit for building the pieces. Check out http://burrtools.sourceforge.net/gui-doc/Spacegrids.html#5_1 This must come up often enough because there is a tutorial for building one step-by-step.

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: The TetrarhonsPosted: Sun Oct 13, 2013 2:21 am

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Here are the shapes that have been discussed so far:

As you can see it is possible to form these shapes with a set of Tetrarhons. Here is a video of me taking these shapes apart.

Enjoy,
Carl

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 Post subject: Re: The TetrarhonsPosted: Sun Oct 13, 2013 10:20 am

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
Very cool! Did you end up using a computer to find those packings?

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Prior to using my real name I posted under the account named bmenrigh.

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 Post subject: Re: The TetrarhonsPosted: Sun Oct 13, 2013 2:40 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Yes, a computer was used. I asked Peter Esser (who runs a fantastic polyform site, www.polyforms.eu) to help look for them.

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 Post subject: Re: The TetrarhonsPosted: Sun Oct 13, 2013 3:50 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
Very cool! Did you end up using a computer to find those packings?
Yes, as Jared just pointed out. And Peter found these solutions modeling the pieces as simple edge connected cubes.

This is exactly what Japp had suggested and I wasn't sure it would work. It certainly does and does capture the symmetry concerns that I had. I had been playing with Burr Tools and I had been able to model a single Rhombic Dodecahedron. I was still playing with it and hadn't yet figured out how to copy it when Jared sent me Peter's solutions. Burr Tools models each Rhombic Dodecahedron as 24 cubes on a 5x5x5 Cubic Grid. The other 101 are voids. Looking at Peter's models and how simple it is in comparison I think its certainly the more efficient way to go. I'm still curious about the number of solutions for each of these shapes so I'm going to continue to play with several of these solvers and see what I can come up with.

One advantage I see that Burr Tools may have is (based on the name) I assume it looks at rather it's possible to actually assemble the solution. I don't think that is ever an issue with the Tetrarhons but it does become an issue with other sets. Jared has started a thread about what he named the Polytrocs and there you could certainly find solutions you could never assemble. Is Burr Tools smart enough to exclude those solutions? I'll post about this in that thread so as not to take this too far off topic.

Carl

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 Post subject: Re: The TetrarhonsPosted: Sun Oct 13, 2013 4:36 pm

Joined: Tue Aug 11, 2009 2:44 pm
Carl, these are very cool, but are you sure this hasn't been explored already? I would be pretty surprised if nobody had played with these 28 pieces before, and found these assemblies. Still, I could be wrong. I'll do some checking around. I think at the very least there is room for exploration in interlocking assemblies with these pieces (though there has been some work here, see below),

FWIW it's worth checking out the source of the image you used above, from The Puzzling World of Polyhedral Dissections, by Stewart Coffin. This is an awesome book, required reading, though it has nothing about twisty puzzles. It used to be hard to find and highly prized (though somehow I wound up with two copies!), but now it's available online.

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 Post subject: Re: The TetrarhonsPosted: Sun Oct 13, 2013 4:58 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bhearn wrote:
Carl, these are very cool, but are you sure this hasn't been explored already?
No. If it has been explored before the results of that exploration aren't easy to find. And even if this has been done before I don't know where someone can buy a set of the Tetrarhons to play with. I've seen instructions on how one might make their own set out of wood, for example. But not everyone has the woodworking skills and/or tools necessary so I thought a 3D printable set had some value. I'm in no way trying to claim that I invented the set of Tetrarhons.
bhearn wrote:
I would be pretty surprised if nobody had played with these 28 pieces before, and found these assemblies. Still, I could be wrong. I'll do some checking around. I think at the very least there is room for exploration in interlocking assemblies with these pieces (though there has been some work here, see below).
I too would think this would be an old topic and if you find something PLEASE share. Jared has also started a thread about what he named the polytrocs here as he wasn't even able to find a name for this set. In Stewart Coffin's book I only see them enumerated to size 3 and I'd be shocked if the set of size 4 polytrocs hasn't been counted and named before but I can't seem to find that info if they have been. There has probably been shapes built with them too but if so its not easily found.
bhearn wrote:
FWIW it's worth checking out the source of the image you used above, from The Puzzling World of Polyhedral Dissections, by Stewart Coffin. This is an awesome book, required reading, though it has nothing about twisty puzzles. It used to be hard to find and highly prized (though somehow I wound up with two copies!), but now it's available online.
Yes, I'm very aware of Stewart Coffin's book its the only place I see the polytrocs enumerated to 3. I've read large parts of it but I don't have a paper copy of it so I've never tried to sit down and read the whole thing on-line... and yes I know I should.

Carl

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 Post subject: Re: The TetrarhonsPosted: Sun Nov 17, 2013 9:04 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
I now have two sets. One is colored black and while, the planar pieces are white and the non-planar are black. The white pieces are sealed so they shouldn't get dirty. And one set is colored in seven colors. All of those parts are sealed.

Attachment:

Sets.png [ 567.19 KiB | Viewed 1066 times ]

Enjoy,
Carl

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 Post subject: Re: The TetrarhonsPosted: Thu Nov 28, 2013 5:05 am

Joined: Wed Oct 23, 2013 7:39 am
Thanks for the set. I have never heard of this name before. It seems new to me.

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