Calculating an Impossible Number
Investigate how algebra was used to calculate the exterior tiling of the Opera House sails. Eddie Woo is joined by Ken Kobayashi, a BIM Integration Specialist who has digitally reconstructed the Sydney Opera House using modern computers.
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.
Copyright Sydney Opera House Trust.
- The sails of the Sydney Opera House are probably its most distinctive feature. They solved some important problems when it came to manufacturing the building's pieces, but they created some problems too. It's easy enough to tile a flat building. But how are you supposed to come up with a way to tile this? Answer, with some ingenious mathematics. The mathematics of covering surfaces is called tessellation. This simplest kind of tessellation uses copies of a single shape. Simple but effective. We can also combine additional shapes at varying angles to create a more varied pattern. Like the ones you find here on the forecourt of the Sydney Opera House. The consistent geometry of a flat surface, called a plane, is relatively easy to tame. When we move into three dimensions, things can still stay straight forward, if we combine planes into shapes like pyramids and prisms. Where things really get tricky is when we add curves. Some curves aren't too difficult. A cylinder, for instance, we can tile in much the same way as a plane. That's because we can take the curved surface and wrap it easily with a rectangle. But as anyone will know, who's tried to give a soccer ball or a basketball as a present, you can't use the same trick with a sphere. That's because it's mathematically impossible to wrap a sphere with a rectangle, unless you cut or tear the rectangle in some way. To understand how these sales could be tiled, we need to talk to someone who knows their geometry inside out. Like an architectural designer, who's rebuilt the entire opera house digitally. Ken, the Sydney Opera House was open more than 45 years ago. How has architectural design changed since then?
- The Opera House was actually one of the first building that used one of the ancestors of computers to design the building. And we cannot imagine how difficult it was back then to do those structural calculations. Had they done all the calculations by hand, they would have needed an extra 10 years to calculate the whole structure. Nowadays, it's very easy for us to calculate by even model in 3D using our most basic computers.
- It must have been a challenging task to digitally construct the fine details of the roof. How did you ensure that your model was accurate?
- The Sydney Opera House is made from primary shapes. So it's all portions of spheres. Every roof structure is a portion of spheres and spheres have an equation. So I had to basically calculate those points according to the equation of the sphere. So that's just an example of how I used mathematics to confirm the accuracy of the model.
- Utzon had to have all of the Sydney tiles manufactured in Sweden and then brought over by ship. He couldn't afford to vastly overestimate the number of tiles or the project costs would skyrocket. But on the other hand, it'd be dangerous to underestimate the number of tiles because if he ran out, it would be months before new tiles could be made and shipped over. He needed to be precise in calculating how many tiles be needed. So how did he do it? Utzon used one of my favorite pieces of mathematics, algebra. Let's look at Utzon's design and see how he used algebra to solve this puzzle. The beautiful chevron pattern is visible from afar. But it's only when you get close that you can see it's composed of individual matte and glossy tiles. Since the chevrons get wider as you climb the structure, you need a larger number of matte tiles for each successive chevron, which is given by this equation. But, the chevrons are all essentially the same height. So the side tiles alternate between 24 and 26 tiles, depending on whether it's an odd or even chevron. Therefore, we can calculate the number of side tiles with this equation. A similar process can be used to calculate the number of glossy tiles. And when all the number crunching is done, we end up with 960,005 tiles to cover all the sails. However, Utzon wanted a margin for error in case any tiles were broken in transit or during laying. So he ordered 10% extra. When this calculation is done, we end up with 1,056,006 tiles. And when you put them all together, this is the majestic effect that they create. As Louis Khan, the American architect put it, "the sun did not know how beautiful its light was, until it was reflected off of this building." Looking at the Sydney Opera House from a distance is awe inspiring. But it's when you come in close that you realize how many tiny details come together to produce this singular experience of wonder. Mathematics is the only tool that allowed Utzon and his team to create the beautiful chevron pattern for the shells of the Sydney Opera House.