Maths of the Sydney Opera House: The Purity of Geometry


Copyright © Sydney Opera House Trust.

About this video

Did you know that the Opera House was almost never built? Maths solved the problem! Find out how geometry and Utzon’s ‘Spherical Solution’ resolved the construction dilemma that the unique design of the arched sails caused with Eddie Woo and Peter Mould, former NSW Government Architect. 

Maths of the Sydney Opera House is a four part video series with award-winning maths teacher, author and Wootube star Eddie Woo uncovering the mathematical concepts behind the construction and design solutions of Opera House.

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- The Sydney Opera House is one of the most iconic buildings in the world. It's utterly unique, instantly recognizable, and described on the UNESCO World Heritage listing as a work of creative genius. But it was almost never built, why? Jorn Utzon's magnificent vision for the Sydney Opera House was selected as part of an international design competition, which received over 200 entries from all over the world. However, in Utzon's plans, he hadn't yet settled on a method of construction. He knew what he wanted to make, but he hadn't yet worked out how he was going to build it, and this caused major problems as the project progressed. Sir Ove Arup was the structural engineer brought onboard to make Utzon's dream a reality. Utzon's competition entry contained fanciful freehand sketches that were impressive, but weren't based on any specific mathematical shape. Arup guided his younger friend towards geometric figures that could be planned out precisely at large scales, but none of them were quite right. For instance, the strongest shape would be a catenary dome. This design perfectly distributes all of its weight along the structure. You can make your own catenary by taking a chain or a piece of string, and suspending it from two points. Turn the shape upside down and you get a catenary arch, rotate that through three dimensions and you get a catenary dome. This kind of shape is marvelously strong and stable, but it couldn't be used for the Sydney Opera House because Utzon's design called for sharp ridges along the top of the structure, and catenary domes are smooth. The engineers working with Utzon went back to the drawing board, and suggested that parabolic curves might be the answer. You can create your own parabola by throwing a ball through the air. The path traced out by the ball is always a parabola, and if you take this curve and rotate it in three dimensions, you get a parabolic shell. This idea was really appealing to Utzon, but a parabola is very difficult to construct because its curvature changes at every point along its surface. With some parts curved sharply while others are curved gently. This would be enormously expensive to build, because every section would require a unique mold to manufacture it. With time running out, it was looking increasingly unlikely that Utzon would be able to arrive at a workable solution. And then all at once it came to him, while peeling an orange. Rather than use complicated catenaries and parabolas, he needed to turn to a simpler shape, the sphere. Someone who can help us see the power of spheres is an architect. I wanted to talk to you about Utzon's spherical solution. What is it about a sphere that makes it so useful and unique for architecture and construction?

- Well, a sphere is like ball. A sphere is a complete shape, and it's the same shape all the way around. So Utzon, in making the shells, keep that curve consistent so that each of the shells was the same curve, and that allowed him to use bits and pieces of those spheres to build up the blocks that made the Sydney Opera House.

- Every single person who walks past the Sydney Opera House really is taken in by it's design and how unique it is. I wonder, from your perspective, what is it that makes this design so striking?

- I think the geometry is part of the purity of the opera house. And I think that purity is recognizable when you sail past it, you might not understand it, but it's like so many things that are wonderful that we don't fully understand, when you dig in, there's often a reason behind that, and I believe it's the geometry of the solution that is one of the great breakthroughs that give it it's coherence.

- Building the shell from sections of a sphere made construction possible through an industrialized process. But how do you create the different sails out of one simple shape? Utzon's key insight was that you could take the uniform curvature of a sphere, and nonetheless cut out many different shapes from it. Just like a single musical instrument can play many different songs, so too can a single sphere make many different shells. And this is how the iconic sails were ultimately created. This isn't just visually striking, it's also the secret of how each of the shells components was mass produced. You can demonstrate this by thinking in two dimensions. If you take a sphere and flatten it, you get a circle. You can see, if I take this section of a circle circumference, what we call an arc, I can replicate this over and over to get a full circle. Then to get longer or shorter sections, all I need to do is adjust the length of the arcs that I assemble. This is exactly how the different sails, even though they vary in size, were all built from pieces of an identical sphere. Simple building blocks can create wonderful shapes when combined in the right fashion. And Utzon had exactly the right kind of genius to achieve that effect. Something that's so many said it was impossible became a reality through the mathematical elegance of a sphere.

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