Mysterious mathematics 2: The Sydney Opera House

I received a response to my recent piece about mathematical genius in architecture, specifically addressing the story of the Sydney Opera House.  The comments are too long, too interesting, and indeed too important to bury as mere comment under an innocuous article of mine.  So I thought I'd reproduce it here as a full post. 

The author, Nino Bellantonio completed his masters degree quite a few years ago, using the Opera Houes as his major case study.   At a time when some of the protagonists were still easier to reach than now, he spent a good deal of effort untangling the story from the myths that already surrounded the building and its architect.

Nino writes:

"Although not generally stated, I think there can be little doubt that Utzon’s competition design was strongly influence by the work of Spanish expatriate Felix Candela whose work in Mexico was in the air in the 50’s and 60’s.

Félix Candela worked to prove the real nature and potential reinforced concrete had in structural engineering. Reinforced concrete is extremely efficient in a dome or shell like shape. This shape eliminates tensile forces in the concrete.

Candela did most of his work in Mexico throughout the 1950s and into the late 60s. He was responsible for more than 300 works and 900 projects in this time period. Candela was influence by Gaudi, and many of his thin shell projects used the hyperbolic paraboloid geometric form (hypar).

Eero Saarinen, amongst others, was also using thin shell construction at the time. Saarinen served on the jury for the Sydney Opera House commission and was crucial in the selection of the now internationally known design by Jørn Utzon.  Saarinen arrived late, and a jury which did not include him had discarded Utzon's design in the first round. Saarinen reviewed the discarded designs, recognised a quality in Utzon's design which had eluded the rest of the jury and ultimately assured for him the winning entry.

The story goes that Saarinen was so influenced by Utzon’s design that he went back home and finalized his design for the the TWA Flight Center at John F. Kennedy International Airport.  Saarinen's design featured a prominent pointed wing-shaped thin shell roof over the main terminal where massing and shape of forms is similar to those first proposed by Utzon.  Of course Saarinen’s scheme is only two similar wing-shaped thin shell roofs balanced against each other, with side shells (also reminiscent of Utzon’s) linking them.  There the similarity ends.

Unlike Saarinen’s highly definable curvatures, where the visual pointiness of the arch is realized only at the edges, Utzon’s curvatures were initially freeform.  In Utzon’s original design the curves in space were much more complex and geometrically undefined.  The engineers at Ove Arup’s could not find a geometry to define them without destroying the initial intent.   

Parabolic forms were explored, more like Candela’s, but Utzon was unsatisfied, and model testing showed up structural deficiencies even with a double shell structure reinforced by ribs.  It was only after the shell structure was abandoned in favour of an internally folded roof structure that the breakthrough to a precast ribbed structure was made.

It was at this time that Utzon started to use the sphere of an orange to explain the solution.  Whether the orange inspired the solution is open to some speculation, but Utzon himself has often denied it.  The solution had great geometric rigour; an assembly of precast elements would replace the long contemplated in situ shell formation, and scaffolding would become redundant.  The breakthrough is recorded in correspondence from both Arup and Utzon, and was presented to the client in the famous Red Book.  A young Rafael Moneo assisted in developing the necessary mathematical calculations which resulted in a series of drawings and models of the final version prepared for presentation to the client.

Once Utzon had the geometry, he and Arups designed the individual elements, all cast from the same moulds.  The hard part in assembling the

jigsaw puzzle in space, was to find the exact position demanded by the architect’s geometry.

So, the real mathematical geniuses, if they need to be found, are the young site engineer Peter Rice, who was sent from the London office as one of the team endeavouring to calculate the shells and Hornibrook’s Peter Bergin , who had joined the construction firm as a chainman (because he had just failed his engineering exams at the University of NSW.

Peter Rice had moved to Sydney to be assistant engineer to Ian MacKenzie. After one month MacKenzie fell ill and was hospitalised, leaving Rice in total charge at the age of 28. On site his geometrical knowledge enabled him to write a computer program to locate the segments of the shells in space correctly.The surveying techniques were highly complex and required the use of backsights on known positions across the harbor and around the site. 

The laborious trigonometrical calculations were initially done by hand, but in the interests of speed, the whole procedure was dramatically updated and computerized.  In what were then even pre-faximile days, Peter Rice’s mathematical coordinates — converted by the young mathematical wiz from Hornibrook’s surveying team into computer code — were taxied across town to the AGE Company’s computer centre in York Street.  There they were processed into Cartesian and modified Cylindrical polar coordinates, and taxied back to the site office at Bennelong Point. 

The process, in the days of punchcards and room sized computers, usually took about one and a half hours.  If the sight readings were sent late in the evening, they would be ready for use next morning.  That this could be done so quickly was one of the marvels of computer applications to architectural and engineering works at that time.  That it could be done at all was due to Utzon’s geometry for the roofs.  Peter Bergin was able to carry most of the critical figures of the spherical geometry in his head, and could tell the errors revealed by the computer’s figures almost always by mental arithmetic.

Whenever necessary, these usually small errors could be rectified by the positioning of the next precast element.  Cumulative elements could be rectified by lengthening or shortening of elements, by a simple adjustment of the formwork.

After working on the Sydney Opera House for seven years, Peter Rice went on to work on the Pompidou Centre with Piano and Rogers, and later became Piano’s partner in an early incarnation of Piano’s Building Workshop where he remained until his death in 1992, at the age of 57.  History does not to my knowledge record what happened to Peter Bergin.

Utzon was working with similar geometries for the hall interiors when he resigned in 1966."

Acknowledgement:

The Peter Rice/Hornibrook story is mostly from John Yeoman's 1968  The Other Taj Mahal