## Thursday, November 3, 2016

### Hey Bridge, Bridge, Bridge, Bridge, Swing! – Erich

In mathematics, often we are interested in uniqueness. It isn't always enough to say that something exists, but we want to show it is the only one of its kind. Here's an example.

Say A R and Ǝ z ϵ A such that x ϵ A, z x or in English we are saying set A is a subset of the real numbers and there exists (that's the upside down E) an element z of A, such that for any element x of A, z is less than or equal to x. In other words, z is a minimum element of A. We might want to show that this minimum is unique.

The easiest way would be to do a proof by contradiction. We will assume the opposite, that there is more than one minimum. And then we will show that any two minima (or minimums if you, like spellchecker, prefer) must actually be equal.

Say z ϵ A such that x ϵ A, z x and y ϵ A such that x ϵ A, y x . (Those upside down As mean “for any” or “for all”.) Then we know from the property of z that z y and we know from the property of y that y z. Therefore, z = y. The minimum must be unique.

In life outside of mathematics (and yes, even I accept that there is life outside of mathematics) we are also often interested in uniqueness. I guess an interest in uniqueness isn't unique.

Well, we got to see something unique, the last of its kind. In Sale, Victoria, Australia, there is a swing bridge.
 The Swing Bridge of Sale
If you don't know, a swing bridge is an alternative to a drawbridge. A drawbridge has a deck that rises to an angle to allow boats to pass. A swing bridge is also designed so boats can pass, but it rotates. The deck moves from perpendicular to the river to being parallel to the river's flow.

The swing bridge of Sale is not the only swing bridge left in the world. But it is the only swing bridge left in the world that rotates a full 360°. This swing bridge rotates on a central pillar. And it can, if needed, keep rotating all the way around.
 Part of the rotation mechanism on the central pillar
The mechanism to accomplish this requires many parts. Not only were there the rollers on the central pillar, but a complex gear structure is required.
 Gear structure
And the underside of the bridge needs to be able to roll on the supports.
 Rollers where the end of the bridge meets the support
The swing bridge isn't really needed these days. There is no longer boat traffic that requires it. But, normally the bridge is still rotated on Saturdays, Sundays, and the second Tuesday of every month, just so people can see. It's a national engineering heritage landmark.
 Recognized for its unique and historic character
Sadly, right now, people aren't rotating it. And why?
 A view of the catwalk from above
See that catwalk. Of course, it is designed so that workers have a place to stand while they maintain or repair the mechanism. But it is also apparently a favorite place for vandals to do their work. And because of recent vandalism, the swing bridge is not in operation until repairs can be completed.

It's a shame. It was neat to see the bridge regardless, knowing how it functions and that it is the only one of its kind left in the world. But it would have been even cooler to see it go.

Unfortunately, in this life outside of mathematics, Ǝ (there exists) vandals. And that problem is not unique.