*Opera House in Sydney, Australia*

From the first pyramids to today’s intricate building designs, architecture and mathematics have been inextricably linked. It is essentially impossible to understand architectural design without considering the mathematics behind it..

Using mathematics we can try to understand the aesthetics of architecture way beyond simple measurements. Aspects such as proportion and symmetry can be explained through mathematical language. Architects today can create groundbreaking forms through complex mathematics and modern technology.

Though we all know architecture would not exist without the use of mathematics, we rarely consider the complexity and abstract mathematical equations that play a part in some of the world’s most unique buildings.

One of the most notable examples is the iconic Sydney Opera House. Unlike anything else of its time, Danish architect Jørn Utzon’s imaginative design took six years of searching for a workable combination of mathematics, materials, and construction methods to even begin building.

One would assume that an architect could figure out how to make a roof that looks like open sails in less than six years, yet it took that long for Utzon to come up with a workable formula that would give birth to the roof he envisioned.

The size and complexity of the roof design required a mathematical model to successfully cover all functional aspects of the structure. Like many unconventional designers today, Utzon said “I don’t care what it costs, I don’t care what scandal it causes, I don’t care how long it takes, but that’s what I want.”

A spherical solution of curving triangles on a sphere was the geometrical model the roof was finally based on. Due to the time it took to find a method, plus budget overruns and construction delays, the opera house was completed 16 years after Utzon was commissioned to design it.

Iconic buildings all over the world have been modeled after mathematics boasting genius designs. Some notable examples include London’s 30 St Mary Axe (The Gherkin), the Cube Village in the Dutch city of Rotterdam, and The Sagrada Familia cathedral and Endesa Pavillion in Barcelona. These buildings all show hints of mathematical genius.

The Gherkin used parametric modeling to minimise whirlwinds around the base. Using half the energy of other buildings the same size, the tapered top and bulging centre maximises ventilation.

Rotterdam’s Cube Village by Dutch architect Piet Blom sits on top of a pedestrian bridge. Every façade of the three split levels has windows to make them feel like separate structures, built using elaborate formulas to create the peculiar design.

The Endesa Pavillion is based on solar inclination and used mathematical algorithms to change the geometry of the building. The algorithm was used for planning the optimal form for the building’s location and the best form to maximise natural light. Genius mathematical formulas were used to study the sun’s movement to find the primal positioning.

Gaudí’s Sagrada Familia cathedral features hyperbolic paraboloid structures throughout. Much more than basic measurements and calculations, the cathedral’s Magic Square has an arrangement of numbers that equals 33 in every column, row, and diagonal.

We can never underestimate the ability of advanced mathematical formulas and equations behind the buildings we admire. Abstract and convoluted mathematics has been used in several iconic buildings and is increasingly being used by architects to create some of today’s modern buildings.

We must consider that there has previously been a gap in information on the link between architecture and modern mathematics. As the field of mathematics evolves and changes, so too does architecture. Modern architects that are up to date on modern mathematics would hopefully have the ability to find a solution to a problem more easily.

Traditional architecture prominently uses the application of ‘number’ which has somewhat disappeared in the depths of modern software which can lead to a void in the language of architecture. A book by Jane and Mark Burry called *The New Mathematics of Architecture *attempts to rewrite the language and reinvent the numbers hiding behind the complex software.

An instant flash of mathematical wizardry in the head of an architect can result in a solution to an ongoing problem, or a more rational and logical way to design a structure. But sometimes they are so complex it can take years to overcome, as it did with Utzon’s design. Perhaps by delving further into modern mathematics, architects will save themselves some frustration.

Lionel March, UCLA professor of design and computation says “The interface between mathematics and architecture is vital to the future of architectural dialect.”