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 Post subject: Another neglected polyform - doesn't even have a name!Posted: Thu Oct 10, 2013 2:13 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
After getting totally stoked about polyrhons again thanks to Carl I've been thinking about other 3D polyforms.

There aren't many polyhedra which can tile space alone, and of the ones that can there are three which are highly symmetric. The first of course is the cube, which provides us with polycubes. The second is the rhombic dodecahedron, which makes polyrhons. The third is the truncated octahedron.

Stewart Coffin did a little work with these pieces but didn't enumerate them past size 3 (there are 6 of that size, and 2 of size 2). On this page there is a diagram of those. They don't have a name to my knowledge. "Polytrocs" seems like as good a name as any other...

So, how many tetratrocs are there? And more importantly, can they make nice shapes?

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Thu Oct 10, 2013 3:17 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
So, how many tetratrocs are there? And more importantly, can they make nice shapes?
Good question. Get me an image of them and I'll be happy to model a set.

However something here is bothering me. If you just look at the center of mass of the truncated octahedrons don't these pack just like spheres do?

I'm sure the Rombic Dodecahedrons pack like spheres so if you plot the center of masses of a bunch of spheres and then assigned all the regaining space to closest center of mass don't you end up producing the shape of a Rombic Dodecahedron? This tells me truncated octahedrons must pack differently. Do they? I must be missing something.

Carl

P.S. Thinking about this some more. Truncated Octahedrons have 14 (6+8) neighbors where as Rombic Dodecahedrons have 12. Also the distance between neighbors is all the same with Rombic Dodecahedrons and there are two neighbor distances with Truncated Octahedrons. So clearly the packing of the Truncated Octahedrons isn't the same as that of spheres or at least the same packing as that of the Rombic Dodecahedrons. I shouldn't say spheres. And this image tells me why AND answers the question I had stuck in the back of my head.

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Fri Oct 11, 2013 9:36 am

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Haha, you made the same mistake I did. To put it in twisty terms, you confused the DaYan Gem III for the Gem VII.

There is one thing I realized here though - there are some issues with interlocking that need to be addressed when solving puzzles made with polytrocs. A basic example is that the square tetratroc made of four units connected by the square faces, has a hole that pieces could logically fit through, but couldn't physically.

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Sun Oct 13, 2013 3:59 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Jared wrote:
There is one thing I realized here though - there are some issues with interlocking that need to be addressed when solving puzzles made with polytrocs. A basic example is that the square tetratroc made of four units connected by the square faces, has a hole that pieces could logically fit through, but couldn't physically.
I think there are solvers that take these restraints into consideration. Burr Tools may be one of them. I'm just starting to play with it so I'm not certain. Either way I don't see these "issues" as problems. I view them as opportunities to make more interesting puzzles.

Would someone be interested in making an image of the set of tetratrocs? I could take a shot at it by hand but I wouldn't want to miss any if I were going to model them for 3D printing.

Carl

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Sun Oct 13, 2013 4:21 pm

Joined: Thu Dec 31, 2009 8:54 pm
Location: Bay Area, California
wwwmwww wrote:
Jared wrote:
There is one thing I realized here though - there are some issues with interlocking that need to be addressed when solving puzzles made with polytrocs. A basic example is that the square tetratroc made of four units connected by the square faces, has a hole that pieces could logically fit through, but couldn't physically.
I think there are solvers that take these restraints into consideration. Burr Tools may be one of them. I'm just starting to play with it so I'm not certain. Either way I don't see these "issues" as problems. I view them as opportunities to make more interesting puzzles.

WSF is very flexible. The Quintessence series of 120-Cell segments that can be assembled in many amazing ways relies on the flexibility of the parts. If the parts were rigid it would be impossible to assemble any of the self-supporting shapes.

I suspect that with flexible pieces and a lot of patience lots of "impossible" assemblies of the polytrocs can actually be assembled.

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

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Sun Oct 13, 2013 4:34 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
bmenrigh wrote:
I suspect that with flexible pieces and a lot of patience lots of "impossible" assemblies of the polytrocs can actually be assembled.
The square hole that Jared mentions I think would be much too small to be able to fit a single polytrocs through even considering the flexibility of SW&F. I had thought about printing the piece with the square hold as a U-Shaped piece which one of the connections intentionally left open to allow trying to force a piece into it but I'd be surprised if even that worked. I'm not sure if there are any other restrictions like this with a set of polytroc pieces or not where flexibility might help. I as yet don't even know the count of pieces in the set. And even if there was a way to remove these restrictions... should the puzzle take advantage of that? Shouldn't the root puzzle be the same regardless of rather the set is made our of SW&F, or wood, or steel?

Carl

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Mon Oct 14, 2013 2:58 pm

Joined: Sun Nov 23, 2008 2:18 am
Not really sure how to go about systematically counting the polytrocts, but I think I have found a way to break the problem up and have found the solution to one part.

Since there are two types of faces(squares and hexagons) on the monotroct, than the polytrocts of size n can be divided into three groups based on type of joint:
1. Those that are joined only square-to-square.
2. Those that are joined only hexagon-to-hexagon.
3. Those that contain both types of joint.

Since the squares correspond to the faces of a cube, I believe subset 1 has direct parallels to the polycubes, so their would be 8 tetratrocts having only square-to-square joints.

No idea how to count, much less elegantly enumerate subsets 2 and 3 above.

Also, while its much less symmetric, I think the square pyramid with height equal to half the length of a base edge would make for a pretty interesting set of polyforms, especially since it would form a superset of oth polycubes and polyrhombs(the cube being six such pyramids joined such that their apexes meet at the cube's center and their bases form the cube's faces, and the rhombic dodecahedrong can be built from 12(six forming a cube, plus one attached to each face of the cube).

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Sat Oct 19, 2013 4:06 pm

Joined: Thu Dec 02, 2004 12:09 pm
Location: Missouri
Peter Esser (of http://www.polyforms.eu) took a look at these and he had this to say:

"I had a look at pieces composed of truncated octahedrons and decided to use the program for pseudo cubes with adjusted neighborhoods.
In this case touching hexagons are displayed as corner connections and touching squares look a bit strange as separated cubes of distance 2 in orthogonal direction."

It looks like there are 44 of them. I haven't tried to verify this yet but I expect this is correct. Peter knows his polyforms.

Carl

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 Post subject: Re: Another neglected polyform - doesn't even have a name!Posted: Thu Oct 31, 2013 6:08 pm

Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
44 pieces makes 176 total volume, which does not allow for many nice shapes... a box 4.5 x 4.5 x 6 units should be able to hold them at least, with 11 offset layers of 16 units each.

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