What I've got here: I have got a
Super Egg! This is from the 1970s, in this
wonderful leather pouch that, you know, is
very very Game of Thrones and out it pops.
Even though it's an egg it's an egg that
stays up, absolutely beautiful.
Yeah and it's a mathematical object and
the history of this object is
fascinating, it's a great story and it
also leads to some really quite good fun
maths so why don't I tell you the story?
It all starts with Swedish town planners.
And Swedish town planners- Sergels torg
is a square, or it's actually not a square,
this is the problem, it was a rectangle
in the middle of Stockholm. And they
wanted to put a roundabout in it, okay? If
they were going to put a circle, you know,
you're wasting this space - no circle. If
you put an ellipse, this is an ellipse,
here it's too narrow, the traffic is
going to get squashed - couldn't do an
ellipse. So they were thinking about what
shapes they could use and they end up
asking this Danish guy, he was actually a
poet and a mathematician like an amazing
Renaissance man called Piet Hein, and
Piet Hein came up with a solution and
the solution was something called a
super ellipse. I'm going to just start
with the equation for a circle; x squared
plus y squared equals 1, ok, if this is 1
this is a circle. Let's play around with
this, but let's enlarge the exponent. So
let's go to x, say, to the 4 plus y to
the 4 equals 1. What this turns into, and
these are actually called Lamé curves, is
that it gets a little bit fatter, it's a
little bit fatter, okay? It's like it's
becoming a square. So if we did the
square, this is the square; it's becoming
a little bit towards it so - and I apologise
for my rubbish writing - let's go to x to
the 8 plus y
to the 8 equals 1; this looks-
basically it looks like you know a
BabyBel cheese stuck in a box. If we were to
go to, say, x to the infinity plus y to the
infinity equals 1 it would be the
square, okay? So by playing around with
the exponent it sort of puffs out and gets-
approaches in the limit the square. - (Brady: Squaring the circle!)
Exactly! You could call it a squircle.
Let's now start with an ellipse, this is
a really rubbish ellipse. The equation
for this is x over 3 squared plus y
over 2 squared equals 1. And let's do
exactly the same thing. If we were to
turn this to a 4 it gets a bit chunkier.
If we were to then turn it to an
8 you'd get even- it would be almost
looking like the rectangle. So what Piet
Hein who had to try and find a shape for
the traffic to go round in Sergels torg
is that he started so- just say here, this
is the aerial view of Sergels torg, and
he started with the ellipse;
which, as I said before, couldn't do an ellipse because it's too narrow
here, and he actually increased the
exponent to get what he called
a superellipse. So the superellipse is halfway
in between the actual ellipse and the
rectangle. And the exponent that he felt
was both most aesthetic and the most
practical was 2.5. And it is true that
this was then implemented and Sergels
torg which is- it's got this kind of
horrible kind of obelisk thing
in the centre and at the moment it's right
in the center of Stockholm; but more than
that this shape it became an icon of
1970s design and the Azteca Stadium,
where the final of the World Cup was held,
you look at the roof of the Azteca Stadium
that's a superellipse. And there's loads
of other Danish design that uses this
shape. And what Piet Hein discovered was
that if he was going to make his
superellipse three dimensional it looked like
this; and actually it was an ellipse.
Because obviously an ellipse would just
fall over, it's got a peak- the
auperellipse actually stands. Lamé, a French
mathematician, also did work on this.
Sometimes they're called Lamé curves but now
because Piet Hein's so famous
they would be called superellipses.
This Numberphile video is made possible by the
Mathematical Sciences Research Institute;
but we'd also like to thank a special
sponsor for this video, audible.com. They
are the leading provider of audiobooks
and other spoken word publications. They
have loads of stuff on their website, it's
seemingly endless, and among them is
actually the book Alex's adventures in
Numberland by Alex Bellos, who is the
person you just watched in this video!
And one of the best things about this
audiobook is not only is it a brilliant
best-selling book but it's read by Alex
himself so you can hear him tell you all
about it. [Alex] 'Entering the world of maths as
an adult was very different from
entering it as a child..' - And yes there is
a whole bunch of stuff about superellipses
in this book. And if you want to
look for some other mathematical reading,
well there's loads of stuff on audible.
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