I'll say it now: I am not a math person. Math fascinates me, but I have serious issues with basic arithmetic. That said, on July 22, we got the opportunity to listen to Chris Sangwin from the University of Birmingham talk to us about his work in geometry and maths. I think he's a really cool dude (I did get to eat dinner with him after the talk) and so I'm really excited to talk about what he talked about. Enjoy!
The question is: how do you know if something's round? While that may seem like a simple-ish question, it's really not. Think about it. If you were asked to judge a free hand circle competition (I know, its a weird competition; but bear with me on this hypothetical for a bit), how would you go about judging which was the best circle? Symmetry? Use a reference circle? Draw loads of diameters? There isn't a perfect way of judging the circularity of a circle. The best way is to us the width. In a circle the width is constant, so you go around and see if the width remains constant throughout the shape. But if it was that simple, there would be no more to talk about. Enter Reuleaux's Rotor. There exist shapes of constant width that are not circles that work very well as rollers. They're not great as wheels because you tend to bounce up and down a bit on rollers that are not circles; but they can move something laterally very well and in a very cool manner. Shapes of constant width have some very cool applications. For one, all UK coins are shapes of constant width and they aren't all circular (of course so are all US coins, but that's not very interesting because US coins are circles). This is nice of course because it means you can buy a soda from a vending machine without worrying about the orientation in which you insert the coin (the Australians didn't figure this out; some of their coins are not of constant width). Shapes of Constant Width can also be used to drill a square hole, which is really, really cool. This is really important for engineering because engineers sometimes need to drill square holes and because you can then make a Rotary Engine. That's exciting because they are significantly more fuel efficient, smaller, and produce fewer vibrations than a traditional engine. They're slightly less reliable, but that's really because less in terms of R&D has been allocated to their development.
But back to roundness. I have a feeling that the people reading this blog weren't alive in 1986 to witness the Challenger Disaster but it's actually a really important reminder of why we need perfect roundness. First a bit of background: at that time (I'm not really sure about now), the parts of a space shuttle were built separately and shipped to the Kennedy Space Center to be assembled there. In transit, one of the seals around one of the rocket boosters became a little less than round. Instead of fixing it, the suits told the engineers to bang it back into shape and keep using it. Long story short, that seal leaked and the fuel ignited causing the shuttle to explode. This tragic accident is an important example of why roundness is key to engineers, so they use both theoretical and practical tests to determine if a shape is truly round. If the width is constant, you have something that might be a circle; so further tests must be conducted. Engineers currently use a "V" test: if you place a shape in the V and rotate it while measuring three points, two on the V and one outside, you can determine if the shape is a circle or not. There's a reason not all shapes of constant width are used as wheels; if the shape is a circle, then it maintains three points of contact (i.e. the third point outside the V will not change position). I can't find a good youtube video of this one, so you'll just have to use your imaginations. There are 6 different angles that are used for the Vs and if the shape works in all of them, the shape is declared a circle. Or that used to be the case. Because Chris Sangwin is cool, he and his colleagues created a shape that disproves this test. Or rather a bunch of shapes. Well, really a way to make a shape that disproves any V test. It's complicated and it's called Sangwin's shape (or that's what I'm calling it). And it's really cool.
You really ought to get this guy's book. He was a great speaker, so I'm sure the book is fascinating. He also has a website, that is totally worth checking out since it covers some stuff that he didn't cover in his talk to us. All in all, this was a really cool talk about a really interesting subject that I can't pretend to fully understand. I'll leave you with a joke Chris told us while we were waiting for some audio problems to be fixed (a cricket match randomly started playing through speakers that were not being used):
How do you tell the difference between an extroverted and an introverted mathematician?
He stares at your shoes instead of his.
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