Many a potential architecture student has been put off by the mathematical component of the discipline. After all, if math classes give you a headache at school, why would things be any better at university?
If you recognize yourself in this description, you may find the outlook is a little less bleak than you think.
In this article we explain how math is actually used in architecture, and weigh up whether you need to be a math genius to design great buildings. It will introduce some basic mathematical principles, and even show you how math has helped us create some of the world’s most mind-boggling architecture.
Do architects need to be good at math?
The million dollar question! The answer is that architects need to be OK, rather than good, at math. There’s no point pretending it’s not an essential part of the job, but it’s also rare that you’ll need to use anything more difficult than high school math: addition, subtraction, multiplication, division, equations, geometry (working with lines, angles, surfaces and solids), and trigonometry (working with the relationships between the side lengths and angles of triangles).
Architects certainly don’t use things like calculus (working with rates of change) on a day-to-day basis.
Rather than being good at math per se, architects need to be good at mathematical thinking. You need to know how to solve problems with numbers; thankfully, today we have technology that will do the actual solving for us. And using math in the professional world is not the same as using math in a classroom, because in a studio there are practical reasons for making calculations. Will the parts of this building fit together properly? Is there enough space to put another door here? Will that staircase reach the next floor comfortably? These are the kinds of situations when architects need to think mathematically.
What is math used for and why do architects need it?
Primarily, architects use math to make sure their buildings are safe and that resources are not wasted. They also use mathematical principles to make their designs more attractive. Specifically, you might find yourself using math in the following ways:
You could probably do this at the age of nine or ten: convert millimeters to centimeters, centimeters to meters, meters to kilometers, and so on. The only real difficulty with conversion arises when you’re trying to deal with figures on both metric and imperial scales; in places like the United States, the latter (which uses measurements like inches, feet and miles) is more common than the former (which ‘counts in tens’).
Calculating area and volume
You are also likely to have studied this at school: how to calculate the area of a shape with two dimensions, and the volume of an area of three. Area and volume come up all the time in architecture, for example when you’re trying to figure out how many wooden panels you need to cover a floor, or how much cement you’ll need to fill a hole.
Preparing a bill of quantities
Following on from the point above, you may also need math to help you prepare a bill of quantities – an itemized document that shows precisely the materials needed on a construction project. Once you’ve worked out things like area and volume, you can use this data to calculate the number of bricks required for a wall, or the number of tiles for a roof.
Scale describes the size relationship between a drawing and the real world. So if an architect produces a plan using a scale of 1:100, one centimeter on their plan represents 100 centimeters (i.e. a meter) in the real world. A plan at a scale of 1:1000 means that a unit on paper is the same as 1,000 units in the real world. The larger the number after the colon, the larger the area being illustrated; so a house can be shown on a scale of 1:100, but a complex of buildings or a neighborhood might be drawn at 1:1000. Part of an architect’s role is to figure out a suitable scale for their drawings, taking into account the size of the project.
Architects need to understand how math is used within physics. For instance, at the most basic level, we need to know whether a certain amount of one material, arranged in a particular way, will withstand the weight of whatever we want to put on top of it. As part of an architecture degree, you will study quasi-mathematical things like tension, compression, and the physical properties of materials.
Adding beauty to function
Don’t let anyone tell you that architects are ‘engineers who can’t do math’! Architects use math to do something that engineers don’t, intentionally at least: make buildings and structures beautiful. By adjusting proportions – which in math means the ratios between numbers, and in architecture means the relationship between different parts of a building – even a humble block of flats can go from everyday to exquisite.
If you’re interested in art, you may have come across the so-called ‘golden ratio’ (1:1.618), which is found throughout the natural world and which people across all cultures tend to find pleasing. Engaging with principles like this – with ‘math-as-beauty’, or even ‘math-as-magic’ if you like – is nothing like dragging yourself through dry schoolbook exercises! You can see some buildings that use the golden ratio in the section ‘Examples of mathematics in architecture’ below.
How is math related to architecture?
We couldn’t make architecture without math. Sometimes, architects make mathematics visible – think about London’s ‘Gherkin’ building by Foster + Partners, or the CCTV Headquarters by Rem Koolhaas and Ole Scheeren in Beijing – while other times, the calculations that keep a building upright are neither seen nor appreciated.
Religious architecture has relied on math since ancient times, often for symbolic reasons; for example, many Hindu temples in India have symbolic, fractal-like structures, in which the component parts have a similar form to the whole.
In secular design, too, architectural aesthetics have long been supported by mathematics. The first book on architecture ever to be published, Alberti’s On the Art of Building (1450), explained linear perspective and gave the author’s thoughts on shapes to be embraced and avoided.
Mathematical principles for architecture
Mathematical principles are fundamentals of math that we apply every time we do a particular task. You might not think you’ve encountered them before, but if you were ever taught the order in which to tackle parts of an equation – i.e. brackets, indices, multiplication, division, addition, subtraction – then you understand what a mathematical principle is. We use the same rule to solve every equation, which is why you were told to repeat ‘BIMDAS’ 50 times before that test!
In the same vein, certain mathematical principles occur over and over again in architecture. The following are the most common of these:
Proportion – the golden ratio and Fibonacci sequence
We saw above that using the golden ratio (1:1.618) leads to visually harmonious proportions in architecture. Similarly, the Fibonacci sequence (1, 2, 3, 5, 8, 13 and so on – each number is the total of the previous two) can be seen in many of the world’s most beautiful buildings. The golden ratio and Fibonacci sequence are intimately related; after 13 in the Fibonacci sequence, each number divides into the next 1.61 times!
Perpendicular lines are lines that meet at right angles. As an architect, you will learn how to draw these perfectly with a ruler, pencil and compass.
Parallel lines are also essential, not only in architectural drawings but in buildings themselves. Beneath even Gaudi’s most organic forms are structures that rely on parallel lines to stay standing.
How does math influence buildings?
In short, buildings are influenced by what is mathematically possible. For example, weight is generally concentrated at the bottom of a building to avoid collapse, though there are exceptions to this rule such as the Slovak Radio Building or London’s Peckham Library which seem to defy gravity! math also helps us to produce architecture that is more environmentally-friendly.
Consider London City Hall, which has been compared amongst other things to a headlight and a testicle because of its uneven but roughly spherical shape. City Hall was designed this way because spheres have a low surface area compared to their volume, meaning the building costs less to heat in winter; it also costs less to cool in summer, since upper-floor windows on the south side provide shade to the floors below.
One (relatively) recent innovation that has given us a better understanding of what is mathematically possible in architecture is parametric modelling. Now a common feature of CAD software, parametric modelling tools allow a user to change one aspect of a building’s design while keeping others constant. For example, if you have a fixed-size plot on which to build, you can lock those dimensions and play with the length of a potential cantilever until it reaches – quite literally – tipping point.
Examples of mathematics in architecture
If you’re looking for inspiration – or just a little evidence that math creates beauty as well as tension headaches – check out our top 10 mathematically-inspired buildings.
1. The Great Pyramid of Giza, Cairo, Egypt (2580-2560 BCE)
All kinds of mathematical secrets have been uncovered in this ancient pyramid, which used to be the tallest man-made structure in the world. If you walk around the base with a trundle wheel, you’ll find the perimeter is 365.24 cubits to correspond with the number of days in a year (a cubit was a unit of length used by the Egyptians, measuring just under half a meter). What’s more, if you divide the perimeter by twice the height of the pyramid, the answer is 3.1416, or pi.
2. The Parthenon, Athens, Greece (447-432 BCE)
Some scholars have argued that the Parthenon was based on the golden ratio; whether or not this is accurate, it has a width to length ratio of 4:9 which reflects harmonious proportions according to classical taste. There is also clear evidence at the Parthenon that the Greeks understood how optical illusions work. The façade is six centimeters higher in the center than at the sides, and the columns swell in the middle – both of which ‘correct’ the human eye’s distorted perception, making the building appear straight!
3. Chichen Itza, Yucatan, Mexico (~700-1100)
It is hardly surprising that the civilization responsible for the concept of zero also produced one of the architectural wonders of the world. Chichen Itza was a city built on mathematical principles: for example the Temple of Kukulcan, in the center, has 52 panels on each side (one for each year of the Mayan cycle); 18 tiers (one for each month of the Mayan year); and 365 steps inside its mirror (one for each day of the solar year).
4. Villa Capra, Vicenze, Italy (1567-1592)
Andrea Palladio was influenced by the style of ancient Greece and Rome, and expanded on Renaissance interpretations of classical ideals in his treatise, The Four Books of Architecture. All of Palladio’s buildings show a balance and harmony that feels almost extreme, but his Villa Capra is completely symmetrical with a square plan, four projecting porticoes and a central circular hall, topped with a dome.
5. Taj Mahal, Agra, India (1632-53)
The Taj Mahal was designed on grids within grids, with every element precisely arranged. The angles, weight and size of the minarets were calculated to protect them from earthquakes, using geometrical knowledge way ahead of its time, and the building is frequently hailed as a monument to symmetry. Inside, tessellations and other patterning on floors and walls show how highly the Mughals valued visual harmony.
6. Geodesic domes, various locations (1926-)
Geodesic domes appear to be covered in triangles, as they are made from intersecting geodesics – the shortest lines between two points on a curved surface. They are often assumed to be the brainchild of American architect Buckminster Fuller, but in fact it was a German, Walther Bauersfeld, who first had the idea. In the UK, you can see geodesic domes at the Eden Project in Cornwall.
7. Cube houses, Rotterdam/Helmand, Netherlands (1977)
Piet Blom’s houses are designed to look like trees. Hexagonal pylons represent the trunks, while the treetops are cubes turned to an angle of over 45 degrees. The improbable-looking homes have fantastic views, but have been criticized for a lack of available space within. If you find yourself in Rotterdam, one resident has opened their cube to the public.
8. Guggenheim Museum, Bilbao, Spain (1997)
Frank Gehry’s Guggenheim has been credited with rejuvenating Bilbao’s economy, and it’s easy to understand why visitors make international journeys just to see it. The building – a diverse cluster of forms, which appear to be covered in shimmering fish scales – was unlike anything that had come before, thanks to developments in modelling software in the late 90s. Performing the necessary calculations to construct the Guggenheim would be beyond most humans, but technology is now creating seemingly impossible architecture in the museum’s image.
9. International Culture and Art Centre, Changsha, China (2019)
Two decades after the Guggenheim opened, Zaha Hadid Architects took its principles one step further and created this organic, three-part arts center which looks a bit like toothpaste coming out of a tube. Nowhere has Hadid’s nickname, Queen of the Curve, been more apparent, nor has the incredible power of today’s CAD.
10. ‘Mobius Temple’, Taicang, China (yet to be built)
A Mobius strip, with no beginning or end, no inside or outside, is the inspiration for a new Buddhist temple in eastern China. The infinite form of the strip symbolises Buddhists’ belief in the eternal oneness of the universe, and you can see the proposed design (by Miliy Architects) here.
Can architecture be done without math?
In sum: no. Buildings without sound mathematical underpinnings are a recipe for disaster, which is why math is not an optional extra for architect students. The good news, however, is that you don’t need to be a total math whizz; high-school level competence will do, since computer software does most of the legwork these days. You just have to give it the correct instructions on what to calculate.
Is math in architecture hard?
In general, the math required for architecture is not that difficult. You’ll need to do things like addition and multiplication, as well as constructing and solving equations; you won’t need to pass an advanced calculus exam to work in the profession. Unless you really have trouble with basic calculations and logic, with a bit of hard work you’ll be fine.
If I’m terrible at math, should I give up my dream of becoming an architect?
Doing math is hardly the mainstay of being an architect, so no. If you’re creative, with a passion for buildings, having not-so-great math skills is an obstacle you can overcome. If you’re applying for an architecture course at university, highlight the things you can do and show how you’re working to improve other areas.
So how can I get actually better at math?
The answer may not be what you want to hear: to get better at math you have to keep doing math. Try out a few online courses (for example, courses in trigonometry are available for free at Khan Academy and for a minimal fee at Udemy) until you find one that suits your learning style.
Get hold of some workbooks from your local library, and dedicate a small amount of time each day to practicing the skills you struggle with.
The world’s most awe-inspiring buildings have relied on math for their impact, but that doesn’t mean all architects are in love with math. It’s ok just to be average at math, even to hate it – as long as you’re prepared to keep practicing. In your day-to-day work not much is likely to be asked of you mathematically, so once you’ve mastered the basics you can set your number-related nightmares aside.
One thing is for sure: lukewarm feelings about math shouldn’t put you off a career in architecture. There are so many skills required for architecture you’re bound to be stronger in some areas and weaker in others. Instead of losing sleep over long division, try thinking about the buildings you love most in mathematical terms – and who knows, you might even come to enjoy it.