Some buildings wear their geometry on their sleeves – the Great Pyramids, or pretty much anything by Frank Gehry for example, but geometry doesn’t usually shout so loudly. It’s the unsung but essential tool that architects use daily, whether they’re designing a bridge or a block of flats, so in this article we’ve decided to shine a light on geometry in architecture.

Read on to find out how architects use proportion and shape, how ideas about geometry have changed over time, and why geometry is so indispensable in our profession. Oh, and we’ve thrown in a list of our top ten international buildings for geometric inspiration, too.

## What does geometry mean in architecture?

Geometry is an ancient branch of mathematics that works with the points, lines, angles and surfaces of 2D and 3D shapes. This is foundational in architecture. Without geometry, we couldn’t be sure that our buildings were safe, and we’d have a much harder time making them look nice. Architectural plans and drawings would communicate very little.

In short, an architect can’t do their job unless they’ve grasped some basic geometrical concepts.

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### What are some examples of geometrical concepts?

You were probably aware of the most basic geometrical concepts – things like right angles and parallel lines – when you were still a child. At high school, geometry may have gotten a little more complicated; maybe you learnt how to use formulae to calculate the area of a 2D shape (e.g. π r² for a circle), the length of their sides (e.g. Pythagoras’ theorem for a triangle), or the volume of a 3D shape (e.g. length x width x height for a cuboid).

Of course, things get more complicated again at college/undergraduate level, but architects don’t wander round with that kind of information in their heads! For advanced geometrical calculations, these days we usually turn to computer software.

### How was geometry historically used in architecture?

In the past, it was thought that adhering to strict geometric rules was the key to ensuring buildings were as beautiful as possible. In 20 BCE, a Roman architect named Marcus Vitruvius wrote *Ten Books on Architecture*, in which he explained the ‘ideal’ proportions that a building should have, based on the proportions of the human body.

Fifteen hundred years later, his writings were rediscovered by Renaissance polymaths such as Leonardo da Vinci (who exemplified them in his drawing ‘Vitruvian man’) and Leon Battista Alberti (who wrote his own version of *Ten Books on Architecture*).

Some of the principles Alberti developed included always beginning the design of religious buildings with a circle, since this is a ‘perfect’ shape that can only have been created by god, and ensuring that ratios between elements were maintained across the whole of a building. These kinds of ideas lasted for centuries, and some of them are rarely questioned even now.

### How do architects use geometry today?

Many historical ‘rules’ about harmonious proportions in architecture still hold good , but they are no longer strictly followed – in fact, subverting them is actively encouraged. (Would Alberti be horrified or delighted by the Guggenheim Bilbao? We’ll never know.) Of course, you have to know the rules before you can break them.

For contemporary architects, geometry is more of a servant than a master. We use it to make sure that our buildings stay standing; that our plans can easily be understood, even by someone on the other side of the world; and that we don’t populate our cities with ugly, imbalanced architecture – in theory, at least.

Another way we see today’s architects use geometry is in the adoption of ‘biomorphic’ geometric forms, which is to say replicating forms found in the natural world. Of course, Vitruvius and his followers were doing this to some extent when they used the proportions of the human body. But modern architects look to animals and plants, too, taking inspiration from the hexagons of honeycomb (as at the Soumaya Museum, Mexico City) or the bubbles on the surface of water (as the National Swimming Centre, Beijing).

Natural geometry includes patterns such as symmetry, spirals, tessellations and fractals.

### How do architects use geometry as a compositional device?

On one hand, geometry limits what architects can actually do – for example, urban buildings tend to be erected perpendicular to the ground, not only because it makes them stable but also because it allows an area to be more densely packed. (When more space is available this rule can be broken, as in most deconstructivist architecture.)

One the other hand, geometry gives architects a toolkit to work with. Shapes can be combined in infinite ways, so architects compose buildings using things like squares and cubes as building blocks. Take the humble triangle, for instance. It is often considered the strongest shape (it has a firm, supporting base and rarely loses its shape) and therefore a good starting point for a design.

It turns up everywhere as the primary compositional element, from the Great Pyramids of Giza to the Eiffel Tower – which is not only triangular in its overall shape, but also made up of smaller iron triangles (see below).

## Why is geometry so important in architecture?

As noted above, architecture without geometry would be impossible. Buildings would collapse (or look awful) if the proper calculations weren’t made. But there’s another reason why 21st century architects are especially interested in this subject.

When we think of geometry, we tend to think of Euclidean geometry – the principles devised by the Greek mathematician Euclid over 2,000 years ago. It is still essential to learn these, but the most innovative contemporary buildings rely on ‘new geometries’ as well.

For example, projective geometry happens in ‘projected’ rather than Euclidian space (see the first image below for a visual representation of this), while fractal geometry is based on hierarchies found in nature such as those of a nautilus shell, or Romanesco broccoli (see second image below).

In the final section of this article, ‘Examples of geometry in architecture’, you can see some real-life buildings that use both old and new geometries.

## Which geometric shapes are mostly used in architecture?

The number of geometric shapes used regularly in architecture is actually very small. Many are quite impractical when it comes to building – with the exception of triangles, squares / rectangles, and circles.

We already mentioned that the triangle is the ‘strongest’ shape, because it can’t be distorted. A rectangle will become a parallelogram if it’s pushed hard enough, but a triangle always holds its form. That’s why it’s so often used in bridges, roofs and electricity pylons, whose collapse would be disastrous. It also tessellates beautifully, of course.

Squares and rectangles are also practical shapes for architecture. They are easy to construct out of common building materials like metal and concrete; their right angles provide sturdiness and reliability; and they fit together in a straightforward way, reducing cost and waste. It’s also easy to scale a square or rectangle up or down, which is not the case with more complex polygons.

As for circles, you might think they’re rarely used in buildings but that’s not true in reality. Sure, you don’t *see* them as much, but they’re doing a lot of the legwork in things like domes and arches. They work well in sustainable architecture for a number of reasons (e.g. less material is needed overall; they allow wind to flow more easily around them), and they are nearly as strong as triangles because stress is distributed evenly across the whole shape. However, circles are much harder to work with (imagine cutting a square of wood, then imagine cutting a circle).

### A note on ‘geometric’ and ‘natural’ shapes

Traditionally, geometric shapes were defined as those that could be constructed using Euclidean geometry: squares, triangles, cones, prisms and so on. These were contrasted with organic forms, which were thought to be complex and irregular. But the new geometries began to make sense of nature’s apparent chaos; we now understand that while a tree is very different from a rectangle, it has its own pattern – and one that has been copied in architecture for a long time. You may find older textbooks that maintain a distinction between geometry and nature, but in many ways they rely on one another.

Some people even believe in the idea of ‘sacred geometry’ – that everything in nature, from a leaf to a galaxy, has its own geometric pattern. The problem is just that humans are too dim to understand them all, so we stick to manageable things like rectangles!

## Examples of geometry in architecture

The following are some buildings in which the beauty of geometry is revealed; you can find a lot more examples in our articles Math in Architecture and Physics in Architecture.

### 1. Virupaksha Temple, Hampi (~C14)

If you’re interested in the idea of fractal architecture, take a closer look at this Hindu temple in the Indian state of Karnataka. The patterns on the *gopuram *(entrance tower) divide and repeat as they ascend – just as in they do towards the edges of a snowflake or a leaf.

### 2. Palazzo Rucellai, Florence by Leon Battista Alberti (1446-51)

This Italian palace was designed by our friend Alberti, who wrote a treatise on how geometry should be used in architecture. The building shows that Alberti practiced what he preached – he designed the façade using a grid, with pilasters and entablatures dividing it into harmonious proportions.

### 3. Rietveld Schröder House, Utrecht by Gerrit Rietveld (1924)

This Dutch house is one of the only examples of De Stijl architecture. It made no attempt to integrate with its environment, and aimed to shock by eliminating walls and rooms altogether, and existing as an open, changeable space. Rietveld designed it using the golden ratio (1:1.618) which occurs frequently in the natural world, and is a key idea within sacred geometry.

### 4. Bauhaus School, Dessau by Walter Gropius (1925)

Gropius’ design for the first Bauhaus school in Germany demonstrates the principles that were taught there. The form of a building was to follow its function (which is to say that decoration was out, and clean lines were in), and all man-made products were to be designed around the square, triangle and circle.

### 5. Melnikov House, Moscow by Konstantin Melnikov (1927-29)

This house by Soviet architect and artist Melnikov, who refused to build in the style demanded by Stalin, is one of the few cylindrical buildings that has stood the test of time. It actually comprises two cylinders, made of bricks in a honeycomb lattice formation, with asymmetrically-organized hexagonal windows. If this sounds a little eccentric, Melnikov’s original plan was to make a house in the shape of an egg.

### 6. Headquarters of the United Nations, New York by Harrison & Abramovitz (1951)

The UN building is a slab block that was designed using Le Corbusier’s *Modulor *scale of proportions. The *Modulor *was based on the height of a man with his arm held above his head, and was meant to bridge the gap between the metric and imperial systems. UNHQ has always divided opinion, but its distinctive form is iconic.

### 7. Philips Pavilion, Brussels by Iannis Xanakis (1958)

Le Corbusier was commissioned to design a pavilion for Expo ‘58 but found himself too caught up in his planned city at Chandigarh, India. He handed over the reins for the pavilion to Xanakis, who created an improbable collection of nine hyperbolic paraboloids in concrete. Sadly, Xanakis’ monument to technological progress was demolished the year after it was built.

### 8. Opera House, Sydney by Jørn Utzon (1973)

The famous ‘sails’ of Utzon’s Opera House are in fact slices of spheres, cast in concrete. Each slice is coated with triangular and diamond-shaped tiles, arranged in a chevron pattern, that are only visible when standing close to the building. Though today curved buildings are more commonplace, the Opera House was bold and startling when it was first unveiled.

### 9. CN Tower, Toronto by multiple architects (1976)

The CN Tower was the world’s tallest tower for over three decades (Guangzhou’s Canton Tower swiped its crown in 2009), a title you don’t win without the help of geometry. The ratio of the observation deck to the tower’s total height is 0.618, taking into account the principle of the golden ratio.

### 10. Sagrada Família, Barcelona by Antonio Gaudí (begun 1882, construction ongoing)

Where to begin with this one? Gaudí’s basilica looks like it shouldn’t exist outside the pages of a fairy tale, and yet it does – mostly. The original plan includes three grand facades, 18 spires, and an interior in which no surface is flat, and some of these are now on their way to completion. Thanks to advances in CAD, it is now estimated that the Sagrada Família could be finished by 2026.

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## Summary

Where would we be without geometry? Under a pile of rubble, probably! At best, we’d be living and working in buildings that would barely warrant a second thought, never mind pondering in open-mouthed wonder. Understanding geometry helps architects to make towns and cities beautiful, though thankfully today you needn’t be a slave to ‘the rules’. All you really need is a square, a triangle, a circle and a dash of imagination – then the world is your oyster!