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Sydney Harbour Bridge
Two parabolas with a fine view of Sydney

Parabolas in art and architecture

The parabola is a beautiful and elegant curve. For this reason, not just mathematicians and physicists like it: but architects and engineers too. Not surprisingly, we find that it has been used in many man-made structures.

In this step we look at some of the occurrences of the parabola in architecture and design.

A pair of parabolas spanning Sydney harbour

The image to this step shows one of Australia’s great icons: the Sydney Harbour bridge. Built in the 1930’s and spanning one of the world’s greatest harbours, and with a view of one of the world’s most beautiful cities, it shows off two parabolas in action: but of slightly different shapes and sizes.

For a detailed discussion of the mathematics behind the Sydney Harbour bridge, check out the interesting post

Passy’s World of Mathematics and the Sydney Harbour Bridge.

A popular tourist activity to do in Sydney is walking up one of those parabolic arches; evidently the view is quite remarkable.

The shape of a suspended cable

When a cable is hung between two points, it assumes a shape that looks much like that of a parabola near the vertex called a catenary. In such a case the cable is only supporting its own weight. The mathematics of a catenary is somewhat more advanced, and requires understanding of a hyperbolic function: \(\normalsize cosh(x)\).

Graph of the catenary y=cosh x

But when the suspension cables are used to uniformly support a bridge, especially a heavy bridge, as in the Golden Gate bridge in San Francisco, then the shape is a parabola. This is another classic engineering masterpiece.

Golden gate bridge © “Golden Gate Bridge” by Bernard Gagnon/Wikimedia Commons CC BY SA 3.0

Parabolic arches and domes

Arches and domes in the shape of parabolas have been known and used since antiquity. They have some particularly pleasant load-bearing properties. And they still feature prominently as aspects of modern architectural design.

Picture of building with dramatic parabolas over its windows © “L’Oceanografic (Valencia, Spain)” by Felipe Gabaldón/Wikimedia Commons CC BY 2.0

Surfaces created from parabolas

In our three-dimensional world, we are often interested in surfaces more than curves. Mathematicians have of course also intensively studied degree-two surfaces, called quadrics: such as the sphere, or ellipsoid, or the hyperboloids of one and two sheets, and many more complicated, higher degree surfaces too.

One quadric that is particularly subtle yet intriguing is the hyperbolic paraboloid: which pleasantly combines parabolas opening up in opposite directions! Sometimes also called a saddle shape, for obvious reasons, this is a favourite surface for mathematically minded architects:

Plot of a saddle shaped surface Saddle point” Public Domain/Wikimedia Commons

One of the intriguing aspects of the hyperbolic paraboloid is that although it is a completely smooth surface, it is also a ruled surface — in fact it is a doubly-ruled surface, with two families of lines that lie completely on it.

You can make such a surface yourself if you join corresponding points on two parallel pencils with rubber bands, stretch the bands, and then twist the pencils so they are no longer parallel. Or for something more permanent, just join up corresponding points on two fixed skew line segments in space with thread.

Because a hyperbolic parabola can be constructed with straight beams, it is particularly amenable to construction for buildings.

Here is the Warszawa Ochota Railway station.

Warszawa Ochota Railway station © “Warszawa Ochota Railway station” by Panek/Wikimedia Commons CC BY SA 4.0

Another example is the SaddleDome in Calgary, Canada.

SaddleDome in Calgary, Canada © “Pengrowth Saddledome” by Abdallahh/Wikimedia Commons CC BY 2.0

Questions for Discussion

Do you know of any pleasant or interesting examples of parabolas used in architecture or design? We have to ask the famous question: are MacDonalds’ arches parabolas? What do you think?

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Maths for Humans: Linear and Quadratic Relations

UNSW Sydney

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