# Sydney: The Unsuccessful Hunt for Parabolas

My international hunt for shapes has brought me to Sydney, Australia
where I'm looking for some parabolas.
I'm starting here, at the house where opera happens in Sydney,
Of course, the Sydney Opera House.
Australians, once again, outdoing themselves at naming things.
I mean, you wouldn't believe what they call this bridge which goes over the harbour in Sydney.
[intro music]
When construction began on the Sydney Opera House in 1957,
they actually hadn't finalised what shape the sails were going to be.
The Danish architect wanted them to be based on parabolas, or strictly speaking, a paraboloid.
If you take a parabola, which is a plot of the x² function,
and then you rotate that shape through 3D space, you end up with a paraboloid.
The idea was the shapes of the sails would be based on sections of the paraboloid.
The problem is, the ribs underneath the tiles you can see,
the kind of concrete skeleton of the structure, each bit would have to be different.
because a paraboloid changes curvature as you move around it.
Even the shiny tiles you can see would be an issue, because the curvature changes underneath it,
you'd need different tiles for different parts of the structure.
It turned out a paraboloid simply wasn't practical.
If you're in Sydney, and you want a parabola,
then you have to turn over here to the Sydney Harbour Bridge and be disappointed.
It's not a parabola.
Seen here in silhouette against the sunset, the shape of the bridge is actually a catenary.
A ker-TEE-ner-ree or KAH-ter-nah-ree is named after a chain.
Chains are completely flexible. If I hold this from the top, it always hangs straight down.
Even if I try to change the angle at the top, always straight down under gravity.
A chain is always in pure tension.
So if I support it from both ends, I get this shape.
Looks a lot like a parabola, a lot of mathematicians over the centuries have argued it's a parabola,
it's actually a subtly different shape.
This is the shape that a piece of string or chain takes when it's solely under tension.
You've got gravity going down, each link is effectively completely flexible,
And so this is a shape where the force is run directly along the line of the shape.
Were we to flip this upside down and have gravity going the other way,
you end up with the perfect shape that's only in compression.
All forces are sent exactly along the structure, and that's why it's the ideal shape for a bridge.
If you're now expecting me to make a joke about being in Australia and being upside down...
Prepare to be sorely disappointed.
So Sydney Harbour Bridge, no parabolas.
Sydney Opera House, also no parabolas.
When the paraboloid proved to be too difficult, they switched to using a sphere.
All the sails of the Sydney Opera House are pieces cut out of exactly the same sphere.
Which means not only do they have the same curvature as each other,
but they have consistent curvature across their surface.
This way, the concrete ribs underneath the sails can all be just a handful of shapes.
And then those same shapes can be used over and over again.
You can easily manufacture loads of the same piece and then use it throughout the entire building.
So there you have it. Sydney Opera House, all sphere, no paraboloid.
[theme music starts]
Apparently, there is a plaque somewhere which celebrates the spherical nature of the Sydney Opera House.
And my final challenge is to find that.
Fun pointless fact: to offset the curvature of the Sydney Opera House,
They wanted to have nice geometric parallel lines underneath.
As you can see, these straight lines go all the way down the wall.
In fact, they then come straight across the floor in front of me
And then they carry on over into Sydney Harbour.
It got dark, but I found it!
So here it is. It's the plaque, it's bigger than I expected. It shows the original sphere that the bits were based on.
It shows the sphere with the sections cut out, and finally, them arranged in the shape they are in the Sydney Opera House.
If you wanna find the plaque, if you go up the main stairs on the south end of the Sydney Opera House,
It's right in front of the doors that go into the box office and information booth.
[music]
Okay so, funny story. I was about to upload this video, I did my final fact check,
and I couldn't find a single definite reference to the Sydney Harbour Bridge being a catenary,
nor could I find any authoritative source that confirmed it was any other shape.
So, I simply don't know.
I mean, don't get me wrong, the catenary is still the absolute best shape for an arch
that is solely under compression, I just can't confirm that's definitely what the Sydney Harbour Bridge is.
If anyone has any better information, please do let me know.
[more music]