Mathematical Excursions To Architecture

[quote]On the one side is mathematics, and its capacity to enhance the understanding of architecture, both aesthetic aspects such as symmetry and proportion, and structural aspects such as loads, thrusts, and reactions. On the other side is architecture, as an attractive setting that allows basic abstract and abstruse mathematics to become visible and more transparent.[/quote]

Mathematical Excursions To Architecture

One of the world’s great buildings, the Sydney Opera House provides an illuminating example of a challenging architectural — and mathematical — problem. // photo by Pamela L. Reynolds

[quote]One of the world’s great buildings, the Sydney Opera House, provides an illuminating example. In January 1957, the young Danish architect Jørn Utzon won a competition to design an opera and concert hall complex on a dramatic piece of land that juts into Sydney Harbor. Utzon’s design featured an arrangement of soaring vaulted roofs that looked like a cluster of sailboats under full sail. However, the path from Utzon’s imaginative design of the vaults to their realization would prove to be a formidable challenge. It would take from 1957 to 1963 — years of exploration, analysis, disagreements, and hard work — to determine a workable combination of geometry, materials, and methods of construction.[/quote]

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[twocol_one_last]Roof structures of this size and complexity cannot be built without an explicit geometry that can be expressed mathematically. Without such a mathematical model, it is not possible to calculate the loads, stresses, and rotational forces to which the vaults will be subjected and to estimate the impact of wind and temperature changes on their stability. Parabolas and ellipses were Utzon’s first choices for the profiles of the vaults, but neither provided a buildable option.[/twocol_one_last]

Ove Arup and the engineers of his London-based construction firm had been chosen to partner with Utzon in the execution of the structure. It had been their initial thought to build the vaults as thin, concrete, egg-like shells. Such roofs were coming into use at the time. In fact, Arup’s firm had executed such roofs with concrete shells only 3 inches thick. However, Utzon’s pointed, steeply rising vaults were not compatible with such a design. Utzon and Arup had to find a different solution.

Arup became convinced that each of the sail-like roof structures could only be built as a sequence of curving ribs — narrow at the bottom and increasingly wide as they rise — that would spring from a common point and fan outward and upward from there. Each roof vault would consists of two such curving fan-like structures — one the mirror image of the other — soaring upward from opposite sides to meet at a circular ridge at the top. Utzon endorsed this concept enthusiastically.

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