Fun with twisted toruses (or tori, if you must)

 Upload 2010 08 Math Monday The Twisted Torus Ambrasobject

 Upload 2010 08 Math Monday The Twisted Torus Torzle-Part

George Hart of Make: Online says, "A twisted torus is a donut which twists as it closes on itself." Besides making me hungry for a Krispy Kreme, this twisted torus makes me want to ask one of my MakerBot-owning pals to print out a bunch of parts so I can try to build one myself. How do you close the loop?

[Above] is an elegant solution from the 16th century: a twisted torus in the Schloss Ambras Museum, assembled from seventy-six identical wood components.

Each piece has a slot for the next to pass through and there is enough play in the slots for the chain to be twisted slightly before closing into a loop. It is a visual puzzle to figure out how the ring was closed. If you make many copies of the basic unit sketched below, you can try to solve the puzzle and assemble them into your own twisted torus.

Math Monday: The twisted torus


  1. The only way I can see to do this is to carve the whole thing out of a single piece of wood. Doable, but difficult as all hell.

    1. Or – one of the pieces is missing the “barb” at one end. Hard to tell without seeing it from all angles or handling it…

  2. My guess is that one piece would have a tab subtly shallower and another piece’s slot would be subtly wider than the others. If the fit was tight, but passable, it would hold easily and be hard to detect even with some inspection. Especially so given the description: “there is enough play in the slots…”

  3. No, Promethean Sky cannot be right: this is not an expensive prestige puzzle for rich people, it is simply a tool for keeping your strain of yeast alive until next time you brewed beer in a Northern European village. They were in regular use till some 100-150 years ago.

    We call them “yeats wreaths” here in Denmark, and they used to be pretty popular antiques a couple of decades ago, leading to quite a few forgeries from unscrupulous dealers – who obviously did not have any trouble making them.

    I did not look long for it, but I found a picture on a Danish web page about brewing. I am not sure how long the link will work. Here goes:

    Yours sincerely, Bodil G

    1. Shoot, I was hoping for some kind of 4th dimensional manipulation, but I’ll bet boiling water and clamping is the answer.

  4. If you look at the link about 4 repeats around from the front you can see that it doesn’t have a full barb.

    It’s at about 4o’clock if it was a clock face.

  5. My thanks to Bodil G for the tremendously informative post. The gærkrans certainly seems like a useful tool for maintaining a specific type of yeast for future loading. All those nooks and crannies would increase the surface area nicely, and I imagine that agitating the thing would dislodge the culture fairly well. I’m not sure that this resolves the question of original application, though. (The torus dates from the later 16th century, so it predates the 19th-century use of the gærkrans.)

    My first doubt pertains to shape. Most of the contemporary examples of a gærkrans that I have been able to find on the web are incomplete loops, employing string or leather straps to complete the chain. An example from the Odense museum is especially noteworthy in this regard. No doubt there’s an issue of ease here: a loop will be easier to retrieve from the vat than a simple chain; both would harbor the culture well, in addition to allowing for easy transfer of the yeast. But while looping makes the logistics of brewing easier, it need not result in a complete torus such as we find in the Schloss Ambras Museum. In fact, wouldn’t using strings or straps for the final connection ease the process of, say, replacing links that break? After all, if you’re repeatedly soaking, agitating, and then hanging this thing, it’s only a question of when, not if. This is not to say a complete torus cannot serve as a gærkrans, but just to wonder if it’s not necessarily the optimal form. Of course, brewers have long had a number of methods at their disposal. If I recall correctly, Belgian operations specializing in lambic, or top-fermented, ales are careful not to clean their interiors too carefully, lest they kill off their local/proprietary strains. So, it would seem that optimal form depends on context.

    In addition to shape, though, there are other reasons to wonder about original application. A 1596 inventory of the Schloss Ambras collection calls the torus a trivet (“Pfannenkhnecht”). Of course, what makes this object such a good candidate for being a gærkrans – its flexibility, for example – makes it less suitable for holding pots filled with boiling-hot liquid. But to take the inventory at face value presumes this object served a utilitarian purpose in the first place, and I don’t believe it did. Notice that the wear patterns and staining on it have more to do with handling than with contact with hot, heavy kitchenware. (There’s also no evidence that it has been dipped in beer at any point, sad to say.)

    Strangely enough, the torus also retains a lot of the signs of woodworking: gouge marks, tear-outs, and so forth. I think that’s the key, though. The components of the torus are, in essence, mortise pieces, as if someone got a notion in the woodshop and then cobbled this thing together. What’s especially interesting is that this rather roughly made object stood alongside examples of impeccable craftsmanship: seamless, beautifully cut and finished goblets, desk sets, and the like. In fact, it was part of a ducal collection dedicated to precisely that sort of showing off. (I agree, in a way, with the idea that the torus wasn’t expensive, though I still think it was a kind of prestige piece, not least because there are examples listed in other inventories from around the same time.) So, why keep a relatively crude trivet – or a gærkrans, for that matter – among such gems? The only answer I have been able to offer is its physical completeness, coupled with that curious twist. That is, I believe it was in the ducal collection precisely because of how it was made, not any practical application.

    On this point it’s worth bearing in mind that Archduke Ferdinand of Tirol, the original owner of the torus, was an amateur materials scientist. He maintained his own metals studio and, according to one visitor, manufactured functioning rifles with barrels made of – I kid you not – wood. For someone with that kind of hobby, the roughness of manufacture + the *seeming* inexplicability of its assembly would have been a source of delight, not to mention a nice way to mess with visitors.

    As for contemporary function, it’s worth noting that you can buy rings like these in some Danish tourist shops, and other people have made their own versions, too. Consequently, we might just as well hypothesize that the torus makes a fine wall hanging or, for that matter, an especially stylish hat:

    What unifies these examples, as well as the discussion in which we’re all engaged, is the curiosity such an object provokes. And that, I think, is the crux of the matter.

    Bret Rothstein

Comments are closed.