If M.C. Escher had owned fidget toys, he would have worn "One ring to rule the ball" as a pendant. A single, continuous line contains a small ball that makes orbit after mesmerizing orbit but cannot escape, always held down in four directions. Available in solid, durable stainless steel with optional antique bronze and gold plated finishes to fit any context, "One ring to rule the ball" now also includes a free rubber necklace so that you can confuse your mind any time the fancy strikes you! One customer echoes many others in saying, "I couldn't put it down. This thing is just too fun!"

Purchases by those gracious customers have now paid for the design time on this piece so it is available on a pay-what-you-want basis. Ordering from this page contributes 5$ towards continuing work such as the new, even more impossible "One ring to pwn the ball" (just released October 18th!) which can be found at http://www.shapeways.com/model/145172/one_ring_to_pwn_the_ball.html and the open-source "Ultimate Desk Toy" at http://www.shapeways.com/model/109166/ultimate_desk_toy.html . To download the original 3d file for your own use under a CC-BY license ( http://creativecommons.org/licenses/by/2.5/ca ) click "Download" at top right . To learn more about this design or other work from Terra Cotta, visit our website at http://www.terracottapf.com

There are six regular convex polytopes in 4D, which are analogous to the five Platonic solids in 3D. This is the fifth, the hyperdodecahedron, a remarkably beautiful object brought to my attention by George Hart.

Here it's shown in a Schlegel diagram so you can see all 120 dodecahedral cells, though most are transformed by perspective: in this projection, the only regular dodecahedra are the biggest one on the outside and the tiniest one at the center.

A bigger model is here.

A pendant for metal printing.  I think this may be the most adorable thing ever.

A smaller version of Triple Gear is available here. A baseplate and axle for using a motor to move the triple gear is available here. Also see 15 cm axle for Triple gear and 30 cm axle for Triple gear.

In this unusual mechanism three gears mesh together in pairs, and yet they can turn!

If you take three ordinary gears and put them together so that each gear meshes with the other two, then none of the gears can turn because neighbouring gears must turn in opposite directions. Triple gear avoids this problem by having the three "gears" arranged like linked rings - the gears then rotate along skew axes, and the opposite direction rule no longer applies (although see also Oskar van Deventer's Magic Gears for another possible solution).

This is joint work with Saul Schleimer. We were inspired by another of Oskar's designs, his Knotted Gear, which consists of two linked rings that gear with each other, and of course we wondered if it would be possible to do three linked rings!

Here is a paper on the mathematics behind the Triple gear, and how we designed it.

A note on materials: I have so far printed it in White Strong & Flexible only. It may arrive with the rings slightly fused together, but gently moving them back and forth will loosen them up, and the mechanism gets smoother with use. I'm not sure what effect the polishing process would have on the gearing mechanism, since it would polish the exposed gear teeth but not those which are meshed as it comes out of the printer. So I have disabled the polished material options. If you really want to try it, let me know. I also haven't tested any of the "frosted detail" materials, but I imagine that they should work fine. It isn't printable in the other "detail" materials because of cleaning problems.