We were lucky enough to have a visit from Cliff Stoll, geek celebrity and proprietor of Acme Klein Bottle. Acme is the finest source of Klein bottles on the internet.
Cliff came with an esoteric dilemma: how to engrave a glass Klein bottle. Acme Klein bottles are blown from borosilicate (Pyrex) glass, which has a low coefficient of thermal expansion, which means that the usual way of engraving a curved glass surface—laser engraving—doesn’t actually work. With more common types of glass, you can use a laser engraver to etch anything you want into the surface. But with Pyrex, the surface simply melts unevenly rather than creating the microfractures that give an etched appearance.
So how would you etch the curved surface of a Klein bottle? It turns out, to our surprise, that it is remarkably easy to do it with an Ostrich Eggbot fitted with a diamond engraver attachment.
There was one complication, which is that a Klein bottle is a funny shaped object! In order to fixture the Klein bottle in the Eggbot, we made a couple of extra large couplers—much larger than the tiny pads normally used to hold the ends of an egg—with EVA foam rubber pads on their surfaces. The extra large couplers held the Klein bottle securely for rotation.
We did some initial tests with Sharpie and a medium sized Klein bottle to make sure our fixturing worked well.
And then we hooked up an engraver for a real test.
Here’s what the Klein bottle looked like after engraving. Not being particularly creative, we etched the word “KLEIN” into the side. Because the Klein bottle is made from thick borosilicate glass, it takes engraving remarkably well. It is a much more sturdy object than the fragile Christmas ornaments that we have engraved in the past.
While we can’t imagine that it is a major market segment, the Eggbot seems to be ideal for working with Klein bottles (insomuch as anything can be perfect for working with a closed, non-orientable, boundary-free manifold). But regardless, it’s quite wonderful to find an unexpected application like this, where our little robot can solve a real-world problem that we had never even considered.
This is one of the coolest things that you can actually buy. It is a Klein Bottle Opener by Bathsheba Grossman. It is made in the shape of a Klein Bottle, a 3D representation of a single sided shape. And it opens bottles. It’s a 3D printed stainless steel sculpture that fits nicely in the hand, giving you just the right kind of leverage; an absolute pleasure to use.
But— and this is where we were caught off guard— there’s a second great, yet completely independent, kind of Klein bottle opener out there: the Beverage Tool by Klein Tools. Klein tools is quite truly (as they say), “the #1 choice among professional tradesmen.”
We happened upon this gem at Hand-Eye Supply, the Core77 store, while visiting Portland, Oregon. As far as we can tell, it was there because they like well-made tools, including those from Klein, not because they like mathematics.
The tool has convenient “Tip-Ident” mark in the shape of a bottle cap so you can quickly find it among other tools.
And now, dear reader, you know where to get a complete set of Klein bottle openers.
Klein bottles are an entertaining mathematical idea–a shape with no volume. A Klein bottle is basically a tube where the inside is connected to the outside. Making a Klein bottle in our 3D world requires a bit of cheating to work, by adding a hole in one of the walls of the tube to provide a place for an intersection.
The most common physical realization is a glass Klein bottle, which you can ogle and buy at Acme Klein Bottles. They also sell wonderful knit Klein bottle hats which can be bought with a matching mobius strip scarf. I was lucky enough to be given a set as a gift, and it is cozy and bright and wonderful. My only complaint is that the narrow neck of the Klein bottle makes it hard to pull it inside out (or right side out, since it is the same thing) to play with it.
I have found that the concept of a mobius strip is more understandable when you can hold it in your hands and turn it around and around, and I thought the same would be true with a Klein bottle, if only it were a little more flexible than my hat. With that in mind, here’s how to make a simple fabric Klein bottle you can play with from two sleeves of a worn out shirt. Continue reading Fabric Klein Bottle