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Sydney Opera House: It’s All About The Math

The peeling of an orange and an in-depth understanding of mathematics led a team of architects and engineers to discover how to construct one of architecture’s iconic images: the Sydney Opera House in Sydney, Australia.

Danish architect Jørn Utzon won the 1957 world-wide design competition for the Sydney Opera House.  His simple but mesmerizing sketch proved inscrutable for six years to the team from the world’s foremost engineering firm charged with structural design.  Its task was to find a repeatable form that would allow Utzon's design to be constructed in a cost-effective manner.  Mathematics and nature helped them to solve the problem.

The architecture-engineering team tried numerous designs using geometric shapes.  They considered more than twelve iterations including parabolas and ellipsoids in their search for a common mathematical denominator that would describe the three-dimensional curves of the building’s distinctive sails.

Then, as the story goes, during a lunchroom conversation the team observed how nature forms a simple geometric shape from segments of an orange.  When the team analyzed the sails in this new light, as segments of a perfect sphere, their tumblers clicked into place and the key to solving the problem was seen.  They now knew how to design the sails! 

Each of the “sails” would be constructed of several identical concrete pieces, or segments.  Just as cake layers can be made by pouring batter into a two or more square or round pans and then pieced together to make a larger cake, these sail segments could be formed by pouring concrete into a mold.  The sails then could be made by piecing together the segments.  And because they were identical, the exterior surface could be covered by mass-produced ceramic tiles identical in shape to each other and with the same curve as the concrete segments. 

Each of the sails appears to be formed by two curved triangles, each standing on one of its three points and leaning against the other for support.  While simple in concept, this requires extremely complex mathematical computations.  Each half of a sail consists of a series of these concrete segments.  If enough of these segments were joined together, they would form a complete circle because they were purposely formed as pieces of a sphere.

Think again of the orange and how its peel covers the spherical orange.  These concrete segments form the peel for the Sydney Opera House, only arranged to form the sails rather than a sphere.  While Nature can produce simple, elegant forms with no apparent effort, man’s attempts to copy her require highly developed math skills.

Finding the mathematical common denominator made all the difference in constructability.  It meant that all of the pieces — from the basic structure to the decorative tiles forming the outer skin — could be mass produced.  Reducing the number of unique pieces and parts to a minimum made the project financially feasible.  And Nature, as is often the case, provided a proven example to follow.

Article by Duncan Abernathy AIA, Virginia Center for Architecture

If you would like to find out more about math and architecture check out these resources:

Sydney Opera House Website

Utzon's original drawings submitted to the competition

Utzon's Design Principles

Wikipedia -Sydney Opera House

Geometry and the Sydney Opera House- work by students

Photo credits:

Matthew Field, PhillipC, Greg O'Beirne

Video photography by: Johnny Yip