A couple of months ago in April, my buddy Tim and I went to visit Cliff Stoll at the acclaimed “Acme Klein Bottle Emporium.” I had been meaning to go ever since I found out he was located just a couple of blocks down from the Berkeley campus, but I wanted to drag one of my math friends with me before doing so. Seeing as topology is among Tim’s many esoteric fields of interest within mathematics, it did not take much convincing to get him to come along with me. I wasn’t even sure if we would be able to see him the day of, as I scheduled the visit after my ME 280B midterm and he wasn’t picking up my calls once I finished the class. Thankfully he got back to me within the following hour, and we took a quick drive down Telegraph to visit his home.
To those unfamiliar with the concept of a Klein bottle, it’s among the more well-known examples of a non-orientable surface. As any layman description would put it, this informally means that the surface is one-sided such that you could trace any curve on the surface and return to the (mirrored) starting position. The most rudimentary example of this is a Möbius loop which can be visually represented by sewing together the ends of a rectangular strip in a half-twist. The Möbius loop is great for building intuition as it can actually be embedded in three-dimensional Euclidean space.
And from the aforementioned description of non-orientable surfaces, it holds the curious property of being defined by a single boundary curve. This isn’t far off from what we would ordinarily expect as if we get two separate rectangular strips and attach them to one another you go from having 8 distinct edges to 4. If you take that rectangular strip and attach it to itself to form an annulus, you go from having 4 distinct edges to 2. Now what if you were to take two Möbius loops and attach them to one another? Would you get a surface with 0 edges? Yes! And that’s what a Klein bottle is, a non-orientable surface with no boundary. This still isn’t too unusual as a sphere also has no boundary, but a sphere is an orientable surface with two sides while a Klein bottle is one-sided and thus, there is no means to distinguish it from having an “inside” and an “outside.” This infamously suggests that the Klein bottle has zero volume, which is also true and hard to make much sense of unless you shift your perspective on what it means for an object to “enclose” a volume in the first place. Does a cup enclose a volume? If you put a lid on it, yeah I guess it does, but a cup contains volume insofar as gravity allows the cup to be filled within the physical world. It’s not too different from how some non-Jordan curves enclose no area and likewise, if a surface does not divide space into two distinct regions, then it does not enclose a volume.
Now a true Klein bottle is inherently four-dimensional and is not restricted by the usual three-dimensional visualization of having to self-intersect itself. And in reality, a Klein bottle is actually a non-oriented manifold such that it does not have a continuous pointwise orientation on a manifold $M$ and therefore does not have a consistent orientation. If it does not have an orientation, then it is not orientable and is therefore not oriented. To the esteemed differential geometrist, this effectively stems from translating the usual notion of defining orientations in an n-dimensional vector space $V$ to orientating a manifold $M$ by virtue of orienting the tangent space at each point in $M.$ I won’t elaborate on this much more but there’s a great reference to this in Chapter 21 of Loring W. Tu’s “An Introduction to Manifolds” textbook, check it out. The upshot is that a “pointwise orientation” on a manifold $M$ assigns an orientation $\mu_p$ at each point in the manifold of the respective tangent space $T_pM.$ If the orientation is continuous at every point in $M$, then it is continuous on the manifold and the manifold is thus orientable.
So what does any of this have to do with Cliff Stoll? Well for over three decades, Cliff has been running an actual shop through his website selling glass Klein bottles in all kinds of odd shapes and sizes. I first heard of this when I was in high school through the videos Numberphile put out 9 years ago on Klein bottles featuring Cliff at his workshop. He is immediately known among every one of my math friends and I kind of assume that every mathematician is familiar with him. What I didn’t know was prior to his family-run Klein bottle business, Cliff was a radio engineer, turned astronomer, turned author, and finally a father. He mentioned this when we got to his place, and we were passing along brief introductions to one another. And he presumed we were already familiar with the idea of a Klein bottle since we were visiting Ph. D students, so we skipped the usual conversation I’m sure he’s reiterated hundreds of times. He went on to provide further context on his life and details not typically known outside of his Klein bottle business, some of which you might not find anywhere online. I wish I remembered more of the intricate details to his history, but he had done a lot of related work in RF engineering and telescope optics prior to his better-known accomplishments. Among which, Cliff was the guy who had pinned down the KGB hacking spy, Markus Hess, through a honeypot while he was working at the Lawrence Berkeley National Laboratory. Kind of bizarre to hear in passing but considering this is the same person who has solutions to math puzzles on a quilt and complex analysis equations written in every other board around his place, it’s not that hard to believe. Cliff is a man whose heart shines a tremendous light onto the world, fueled solely by his joyous passion for math. Unfortunately, a lot of his usual jovial spirit has dwindled due to the recent passing of his wife and a solemn air filled the rooms we entered with him. It’s hard to provide words of comfort to someone you don’t know, so I only hoped our company and interest in Cliff’s work was enough to give him some warmth.
I previously mentioned that the Klein bottle shop was a family business, and it was something that Cliff worked on intimately with his wife throughout the course of its operation. It seems much of his life’s projects were a combined effort with his wife over the years. Further back behind Cliff’s workshop, he showed us a small house that he worked on with her, including the quilt I referred to earlier and a beautiful porcelain bathroom decorated with koi fish and Klein bottle patterns. These remnants of her touch were left everywhere, from her garden, to her post-it notes, and the piles of mortem related paperwork. I cannot fathom how these reminders have been gnawing at Cliff, and I imagine is the primary reason why he’s finally trying to wrap up the Klein bottle project. And I think it’s commendable he’s trying to move forward with something new to take the place of this 30-year endeavor. I think everything he’s poured himself into that he calls “ways to waste your time” are all commendable. If not for the purity of subject matter, then for the meaningful ways it allows you to engage with the people you love, especially in their memory.
The way Cliff interacts with the world with pure curiosity is super admirable, and I wish more people followed in his footsteps of finding ways to imprint your mark wherever you go. If you know of Cliff, then I’m sure if you’re familiar with his miniature forklift robot to help transport and store Klein bottle packages kept under his house. Let me tell you, it’s as awesome in person as it is in the Numberphile videos, especially because of how it’s cobbled together from junkyard scrap. The whole arrangement of the robot being pieced from unmatching parts just looks like it came from the mind of a cartoonish genius making do with what they have. The hanging wires, the cylindrical lifting mechanism, the plywood base and walls — it really puts my non-existent engineering hands to shame. If I’m not mistaken, I believe Cliff was telling us how he got the space underneath his house structurally retrofitted just for this warehouse space for the robot and his Klein bottles. But the flow to just about everything in his workshop is like this: he gets an idea, encounters problems along the way, builds something to help work around it, and continues from there. He was telling us how signing off the packages for Klein bottles is tedious to do manually, so he built a device with attached markers that trace along Arduino-input vector graphics to automatically draw on the packages for him.
You can find tons of fun easter eggs from the Numberphile videos that he has laying around his house and workshop too. There’s the triple Klein bottle within a Klein bottle within a Klein bottle, the hole in a hole in a hole, the Coca-Cola Klein bottle, and some others whose topological wizardry completely evades my memory. One that I haven’t seen anywhere is an argon-filled Klein bottle loop that hangs over his computer desk space. He lit it up after I asked, “I wonder what this looks like when it’s filled up,” after seeing a particularly disorienting glass blown shape. I don’t have any pictures of it with me on hand, so I guess you’re just going to have to take my word on it.
I cannot emphasize enough how fortunate I feel to have been able to meet and chat with the Klein bottle mastermind himself for those brief 2-3 hours. It was momentous, though it pained me to see Cliff so grief-stricken and I wish there was something I could have said or done to have quelled his heartache. Maybe what I can say is this: Cliff if you’re reading this, I hope you are able to find some form of peace with the anguish that plagues you. Thank you for your kindness, your hospitality, and for allowing Tim and I to enter your home to peek into all the topological wonders it has to offer. You are a cherished soul among the math community and your local community in Oakland. I’m dedicating this post to you and your loving wife, to digitally live on forever.