Twisted Torus

George Hart

This is a nylon model of a twisted torus made by assembling identical units. It is a "speculative reconstruction" of the historical artifact below, a twisted torus in the Schloss Ambras Museum, assembled in the 16th century from seventy-six identical wood components.

The wood artifact above fits in the same tradition of celebrating geometry as the following 1567 woodcut by the German graphic artist Johannes Lencker.  This is the earliest printed image I know of a twisted torus:

My rapid prototyping model of the twisted torus is based on units of the following shape:

Three units assemble at 90 degrees to make the module below. Note that the unfilled slot is parallel to the base plane of the rightmost wedge. So one can make an arbitrarily long straight chain of these modules.

If the number of units is a multiple of three, the modules can be curved around to close into a cycle with no twist.  Below is a no-twist cycle with 75 units (25 modules).

If the number of units is one more than a multiple of three, it is natural to give it a slight twist before closing.  Below is an assemblage of 76 units. This is the same number, twisted in the same manner, as the wood artifact above:

Below is an image of an analogous construction with 77 units. It has the reverse twist from the above.

Bret Rothstein, in the Art History department at Indiana University, has studied this object to understand its possible purpose. Bret told me about this object and I made the computer models to get a sense of how much play was needed in the slot in order to close the loop with a twist but not so much play that it would feel sloppy and loose. He made a walnut reconstruction of the units, shown below just before closing the cycle.

Below is the closed loop after Bret was finished. Can you figure out how to make the final connection?

My stl version is below.  I didn't make enough units to close the loop, so I can only make an arc. Playing with these parts gives you a hands-on opportunity to think about why this was made and how the loop was closed. Was it a mechanical puzzle challenging the user to disassemble it and reassemble it?  Was it just a visual puzzle, challenging the viewer to think about how it was assembled?  Was it a "masterpiece" displayed to prove the skill of the creator?  Was it functional, perhaps a trivet or laurel to be worn on the head like a mazzocchio? Were the parts leftover material in some workshop, perhaps a wooden furniture or carriage maker, which someone casually put together into a chain?  I have no knowledge about any of this.  Let Bret and me know if you find out anything illuminating about these questions.

If you have access to a 3D printer and want to make your own version of these models, here are the stl files.  The first two are for making individual parts that you can link together.  The last three files are closed cycles; each is a rigid model in which the parts intersect their neighbors slightly.


Adam Mayer uploaded my stl file to Thingiverse and Charles Pax made this complete ring.

The MakerBot folks kindly gave it to me at the NY Maker Faire. 
Coincidentally, this was just four days before I happened to be going to Hawaii!

Then Phil Renato made one also.

Beer Yeast Storage Device??

   Surprisingly, it seems that this design may be what is known in Danish as a "Gærkrans", which was apparently used to collect and store yeast between beer brews.  Thank you to Bodil G., commenting on a Boing-Boing post of my MakeZine blog, for suggesting this. Here are some examples.  (But most wood floats, so how does it collect yeast that sinks?)

Scholarly Analysis

Bret Rothstein wrote an article about the historical object, arguing it was intended to be visually and intellectual difficult (long before any possible use in beer-making):  Bret Rothstein, "Making Trouble: Strange Wooden Objects and the Early Modern Pursuit of Difficulty," Journal for Early Modern Cultural Studies, Volume 13, Number 1, Winter 2013, pp. 96-129.

Hungarian Examples

Search for images of "kutyagerinc" which seems to be a Hungarian word.