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 Ɐ
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 Ɐ
x ϵ
A, z
≤
x and
y
ϵ A such
that Ɐ
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.
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