From Pie to Pi once more...
You will recall that last week I had to apologise for a certain
brevity in my race report, partly occasioned by my time being taken
attending a lecture on "Algebraic methods for specification and
formal development of software". I confess I was really quite
hurt when not one of you asked me how it went, and could I
perhaps summarise the best bits for you.
But I forgive you, and to prove it, I'm going to teach you some
maths that will allow you to feel all superior to John Rentoul of The
Independent!
If you are a regular reader of the Indie, you will know that they
have a feature called "Mea Culpa" in which they hold up their hands
to whatever linguistic solecisms, lazy clichés or other journalistic
sins they may have committed each week. The most recent column
included this comment about one of their film reviews:
I did stumble
over “the end credits last a small eternity”. Perhaps the theory
of general relativity allows for different sizes of infinite.
Huh?! General relativity?! That's got nothing to do with it!
Any mathematician can tell you that there are different sizes of
infinity  indeed, an infinite number of different infinities! I'm
just going to tell you about two of them, though, and you will
understand it, I promise you!
Let's start counting: 1, 2, 3, 4... (I'll skip a bit) ...39793,
39794, 39795... (I'll skip a bit more) ...50387749993771777379...
well, you get the idea. You can count forever and never get to the
end because there is no end. It's infinite. Got that? The number of
numbers you can count is 'infinity'. Lots of things can be counted:
sheep in a field; imaginary sheep jumping over a gate in your head;
imaginary sheep arguing with real unicorns about how many ways
Brexit will be better for us... It doesn't matter how many you
count, there can always be more of them, at least in principle.
So that's one infinity. It's nice and simple and it's actually the
smallest infinity. So how can something be 'more' infinite than
infinite? Ah, because not everything can be counted!
Imagine you have a 12inch ruler in front of you. (A 30 cm ruler
would do perfectly well, except that it's a Frenchie plot to steal
Brexit from us!)
So you have 12 inch divisions, but each of those is perhaps marked
into 10 smaller divisions  0.1 inches. And if your eyes are good
and your hand is steady and you have a very finetipped pencil, you
could divide those 0.1inch bits into another 10: hundredths of an
inch, 0.01". And somewhere between two of those markers, you'll find
Pi inches: 3.14159256358 approximately. In fact, you can keep
dividing any interval on your ruler, no matter how small, into any
number of smaller intervals. You can do this  yes  infinitely many
times!
But this isn't the same infinity! It's not countable! It's beyond
counting because it's too big!
Why? Well I won't give you a formal mathematical proof, but it's not
hard to see when you look at it the right way. To count things, you
have to be able to put them in some kind of order and then just pair
them off with simple counting numbers: first, second, third, etc...
But you can't do that with positions on your ruler, because every
time you choose two to go next to each other, you can still fit as
many new divisions as you like between them! Ok, you put those
into your order, and you can still add new ones in between!
Positions on a ruler are not countable because every time you think
you've laid them out for counting, you can always find more that you
hadn't included. There are 'infinitely' many positions, but it's a bigger
infinity than countable infinity.
So there you are: just by thinking about sheep and rulers, you have
now grasped something that even professional mathematicians didn't
realise until the 19th century, and many at first refused to believe
it could be true!
(Actually, for Brexit, there's another kind of number which is more
appropriate, called (no joke) an 'imaginary' number, but you've all
worked hard enough for now.)
Well done! Your reward for being such good students is to see some
pictures...
It's a bit of a cold and frosty morning, although not as cold and
frosty as last year. The forecast is maybe as high as ten degrees
later.
Off they go, with Steve Winder taking the lead from the start. At
the time I'm taking the picture, I'm looking out for Amanda and
Merilyn and don't realise it's him. I'm slightly confused when I
hear somebody call out "Hi Steve" to me, and don't actually figure
it out until much later.
This year there's a new feature: King/Queen of the Sand Castle for
the quickest ascent of everybody's favourite hill. The idea is that
everyone has a wristband which can be registered by a phone app the
marshalls are using, so you tap in at the bottom and tap out at the
top. Merilyn is going for a time, but is compromised by the fact
that she has no clear space ahead and there are few overtaking
opportunities. Later it will turn out that her wristband hasn't
registered at the top anyway, so she doesn't get an official time.
Bah Humbug, I think is the technical term.
(It would, of course, be possible to work it out from GPS data, but
that's probably a bit too much faff.)
Amanda's goal is simply to get up without slipping or twisting
anything.
Whatever else you say about this race, you can't deny it's
atmospheric! The mist and the low sun make for brilliant photos.
Hi Matthew!
Amanda and Merilyn are just a little too late for the best of the
mist across the canal, so I've left my previous spot and I'm walking
the race route in reverse. This shot is right into the sun, but the
tree blocks it enough to make the picture work.
Steve W's on good form, well out in front. He's only doing three
laps, although I'm not sure I'd call nearly nine and a half miles a
sprint!
Apparently this was just a slightly damp spot earlier... People now
are not entirely sure what to do.
Some just go for it, although I'm sure I told her that the
shallowest route was folllowing straight along the line of the path.
Some are less successful than others.
Some try the longjump technique.
And almost make it cleanly without twisting an ankle. He's
definitely limping a bit as he runs off.
This is the aforementioned shallow (ish!) line. You can take it
cautiously and sensibly if you want, but the downside of that is
you're in the cold water for longer.
Our girls are trying to decide...
I don't think they're shouting, "Come on in, the water's lovely!"
This is the reason: one of the sluice gates on the weir has been
opened. We presume it's to prevent more serious flooding elsewhere,
although it would have been nice if whoever opened the gate had told
people what it might mean.
The race route is now redirected to avoid this section, but it's all
a bit adhoc and people have to partly make it up for themselves.
Some Clapham Chasers will later admit to me that they've been
naughty girls and deliberately run another lap through the flood,
but most people reckon once is plenty.
Mummy has done this climb several times now, and thought she'd
finished until young Hattie thought otherwise!
Quite a few people have decided they've done enough now, but not
these two!
Ok, given that this was never going to be a PB race even in ideal
conditions, might as well stop to pose for a picture against the
chapel ruins.
And while the second half has been pretty much random, they've now
made up a reasonably accurate halfmarathon distance. A certain
degree of improvisation would have been necessary regardless,
because 4π is about half a mile short of 13.1.
On a tangent, Steve Winder did 3π, and it occurs to me that while
even multiples of π are common in maths and physics, factors of
three are not... I can't immediately think of any common formula
that includes 3π; any suggestions from the floor?
Not exactly a lap of honour, but Matthew is apparently up for the
full ultra 10π, so we'll accompany him to the end of the field.
Ah, the Pi Pies!
As well as the usual Mr Kipling/Tesco's Finest/whatever, there are
some homemade mince pies that are really rather good. Don't know
who made them, but they get the allround seal of approval. I've not
had time to make any myself this year, but I would have been happy
to have made these.
Ok, maybe they don't get the award for most circular, but who cares.
And we say farewell to another crazy lady, Lydia, who says really
nice things about my pictures on Facebook so I like her!
And we say farewell to the 2018 race season too. We're off to hunt
tigers in India for Christmas, so love and best wishes to you all
and provided we don't get eaten, we'll see you in 2019.
Steve.
