It occurred to me that these are probably the closest possible 3D equivalent to polyhexes.

Col. Sicherman has done a bit of work with compatibilities of the four-piece ones, which he calls "tetrarhons". "Polyrhons" sounds nice but I would have called them "Polyrhodes".

Anyway, I can't find any enumerations of them specifically. I did find this page on polyspheres, though, which stack together in the same way.

I dream about buying a good set of polyrhodes in the future. Maybe nifty ones with magnetic faces so you don't have to rely on gravity to hold them together!

Mega-bump! But I'm hoping I can get some replies this time.

I guess I totally missed this page the first time around - it's an enumeration of both "free" (flippable in 4D space) and "one-sided" (realistically embedded in 3D space) up to the decarhons.

I also found this picture from the online version of Stewart Coffin's The Puzzling World of Polyhedral Dissections:

It shows all the pieces of order 2 through 4, including chiral pairs that the other page does not.

There are 28 tetrarhons in that picture, with a total volume of 112 units. Coincidentally, a rectangular "pyramid" with a 6x7 base (6 layers, 1x2 at the top) has 112 units. Can they all fit into such a shape?

(The other pieces in the picture total 17 units. They can fit into a 19-unit octahedron with two opposite corners removed (square layers of 4, 9, and 4) quite nicely - and it's simple enough that I could work it out in my head.)

Jared, I remember seeing a new kind of construction toy at the UK Toy Fair in London some years ago, similar to lego, except the blocks were like squashed cubes (squashed from opposite corners), and 4 of them could stick together to form a rhombic dodecahedron, which could be assembled further as shown in your links. I wish I could remember what they were called...

Edit: Just found these:

http://www.google.com/patents/US3611620 ... &q&f=false
http://www.google.com/patents/US6173538 ... &q&f=false
http://www.google.com/patents/US5009625 ... &q&f=false

Thanks for the links!

(112 is also two size-6 tetrahedrons. Wonder if that's possible with the tetrarhons too?)

Are there any Shapeways users who could help me here? Perhaps a small set could be 3D printed. It'd be nice to know how much such a thing would/could cost.

Not quite what you are after but this discussion reminds me of the Ball of Whacks. Its base shape is a rhombic triacontahedron. I have a few of these and really enjoy them myself. Back when these first came out I emailed the designer and suggested he also make a rhombic dodecahedron version with the same size rhombus faces such that the two sets would be compatible.

And I do have a small dice collection that contains one of these:


Though they are hard enough to find your Shapeways idea is probably better. I can look into trying to make a set of these if you like.

Carl

The magnetic pieces were just a musing of sorts. Of course ordinary pieces are fine too.

I have two Ball of Whacks, along with 1 each of the X-, Y-, and Star Balls from the same designer. I also have four sets of the mag blocks sold in the TP.com shop, and while two sets is sufficient to build a rhombic dodecahedron, the size and lack of power in the magnets makes it impractical to simulate a set of polyrhombs(or whatever they are called).

A shapeways set, or simply a source of buying rhombic dodecahedron dice in bulk would be cool.

A bit more on the tetrarhons - exactly half of them are "planar" (can lay flat in one layer either hexagonally or squarely). Can the planar or non-planar tetrarhons, volume 56 units in either case, make a size 6 tetrahedron?

Might be interesting for a physical set to have the planar and non-planar pieces colored differently.