When we think of architecture, we tend to think of it as an art. But did you know that there are key mathematical principles at play in some of the world’s most iconic buildings?

Let’s start with the golden ratio. You’ve probably heard about this before, but just in case: The “golden ratio” is a number, or proportion, that appears in nature. It’s represented by the Greek letter phi and often denoted as “φ.”

The golden ratio (or at least a close approximation) is found throughout nature: it shows up in flower petals, pine cones, and even our DNA. Some people also believe it’s present in the proportions of the human body and facial features—and many ancient architects believed that using the golden ratio in architecture would create beautiful architectural design.

Another key mathematical principle that influences architecture is symmetry. Symmetry basically means that two halves of something are mirror images of each other—and it can also be used to describe how patterns repeat themselves again and again on different scales. Symmetry shows up all over the place in architecture: you’ll find it in everything from rooflines to doorway designs to entire buildings themselves.

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importance of mathematics in architecture

Mathematics is a technical science that can be applied to architectural design, both artistically and practically. It is crucial for generating a design proposal. Mathematics has two functions: first, it serves as economic factors relevant to the proposed design solution. Through this method, one can decide the budget for the construction and maintenance with the help of floor area, heights, materials, and developments. This helps in determining costs at each stage of development of a building.

A construction project is a major undertaking, and it takes a lot of time, effort, and money. A lot can go wrong if you don’t plan ahead, but if you do have a plan, you can avoid the worst of it (or at least mitigate the damage). The key to success is a great plan.

Just like in any other field, mathematics plays an equally vital role in construction projects. Whether the design is for a renovation or new construction or an extension to the existing building, math helps create the atmosphere required through size and scale. It also helps monitor construction maintenance to achieve a cost-effective design. You know that construction can be expensive—but math can help you keep costs down so that your project doesn’t break the bank.

Dramatically, it can be stated that mathematics, an essential factor in the architectural design process, remains behind the scene many times. And comes to notice when a piece of bad news or wrong solutions are highlighted when they face some issues. Wise and faithful use of mathematics in the process from the beginning will foster the project for potential success.

Mathematics, as much as is beneficial in ensuring design practicality, also adds an intangible element to the design in the form of beauty. The architectural design provides a basis, and mathematics supplements a design with beauty, life, and imagination. There have been numerous architects and designers who keep this thought close to their hearts. Despite the complexities involved in mathematics, its ultimate intention to create visual simplicity and satisfaction cannot be stressed enough. 

Architects and design theorists of the world, rejoice! We have found the secret to true architectural beauty.

Mathematician Christopher Alexander has analyzed thousands of architectural structures and discovered that a building’s beauty is directly correlated to 15 specific mathematical properties. Here are just a few of the amazing things you can accomplish by incorporating these into your work:

-A beautiful space lets people feel like themselves, giving them the confidence to truly be their best self.

-A beautiful space allows people to accomplish their goals without running into obstacles or being held back.

-A beautiful space makes us capable of understanding the world around us and our relationship to it.

-A beautiful space is flexible enough that we can create new worlds with it.

-A beautiful space fosters happiness and satisfaction in life, allowing people to be at ease and free from worry.

1) Level of scale

2) Strong centers

3) Boundaries

4) Alternating repetition

5) Positive space

6) Good shape

7) Local symmetries

8) Deep interlock and ambiguity

9) Contrast

10) Gradients

11) Roughness

12) Echoes

13) The void

14) Simplicity and inner calm

15) Not-separateness 

Mathematical principles in architecture

Every building you spend time in––schools, libraries, houses, apartment complexes, movie theaters, and even your favorite ice cream shop––is the product of mathematical principles applied to design and construction. Have you ever wondered how building professionals incorporate math to create the common structures you walk in and out of every day?

Before construction workers can build a habitable structure, an architect has to design it. Geometry, algebra, and trigonometry all play a crucial role in architectural design. Architects apply these math forms to plan their blueprints or initial sketch designs. They also calculate the probability of issues the construction team could run into as they bring the design vision to life in three dimensions.

Since ancient times, architects have used geometric principles to plan the shapes and spatial forms of buildings. In 300 B.C., the Greek mathematician Euclid defined a mathematical law of nature called the Golden Ratio. For more than two thousand years, architects have used this formula to design proportions in buildings that look pleasing to the human eye and feel balanced. It is also known as the Golden Constant because it manifests literally everywhere.

The Golden Ratio still serves as a basic geometric principle in architecture. You could even call it a timeless archetype, as it evokes in human beings a universal sense of harmony when they see or stand in a building designed with this principle. And perhaps not surprisingly, we see the Golden Ratio demonstrated throughout “architectures” of the natural world. Read here to learn more!

Calculating ratio is essential, as well, when it’s time to construct a building from the architectural blueprints. For example, it’s important to get the proportions right between the height and length of a roof. To do that, building professionals divide the length by the height to get the correct ratio.

The Pythagorean theorem, formulated in the 6th century B.C., has also come into play for centuries to calculate the size and shape of a structure. This theorem enables builders to accurately measure right angles. It states that in a triangle the square of the hypotenuse (the long side opposite the right angle) is equal to the sum of the squares of the other two sides. Read here to find out more about how builders use the Pythagorean theorem to make roofs!

The most remarkable ancient architecture of all may be the pyramids of Egypt, constructed between 2700 B.C. and 1700 B.C. Most of them were built and scaled at about a 51-degree angle. The Egyptians clearly and mysteriously possessed knowledge of geometry, as evidenced by the accuracy of pyramid construction. Just in case you’re curious about the geometry and triangle mathematics that ancient Egyptians applied to build their pyramids, read here.

In the modern world, builders use math every day to do their work. Construction workers add, subtract, divide, multiply, and work with fractions. They measure the area, volume, length, and width. How much steel do they need for an office building? How much weight in books and furniture will the library floors need to bear? Even building a small single-family home calls for careful calculations of square footage, wall angles, roofs, and room sizes. How many square yards of carpet? How much water do you need to fill up a swimming pool?

Chances are you’re in a building right now. Look around at the walls and windows. Math is everyplace you walk into––work, school, home, or pet store. Imagine that you are an architect. How would you apply math to build a small dwelling?

Calculus in architecture

The age-long complaint of bored math students is, of course, that what they are learning won’t have any real-world value. If such a student wants to become an architect, he should sit up and pay attention in calculus class. Architecture blends several subjects together, including art, physics, geometry and calculus. Since calculus is used for examining forces over time, it is the main reason buildings don’t topple over in hurricanes and heating systems don’t overload in the winter.

What Calculus Is Used For
Calculus was developed in the 17th century by scientists, including Isaac Newton, as a means of describing the physical universe. In a nutshell, you could say calculus is the mathematical study of how things change in the physical universe. It provides a means of modeling any system that is subject to change and for predicting the effects of change on that system. Taking a class in calculus itself doesn’t teach you how to design buildings, but it does give architects a language to use when designing structures and for understanding how forces like gravity and wind will affect them.

The Math Behind Architecture
In designing structures, an architect must visualize the solutions to both functional and aesthetic problems — balancing art with science and math. Calculus is important for an architect to understand the forces acting on the structures she designs. It also gives her a means of calculating factors such as heat loss over time. This understanding must be mathematically precise if the structures architects design are to be stable and safe for use.

Arches and Domes: Beyond Hyperbole
The familiar hyperbolic curves math students learn in calculus class are just one example of how this branch of mathematics is important in architecture. Famous landmarks, such as the Gateway Arch in St. Louis and the dome of St Paul’s Cathedral in London, both incorporate hyperbolic curves called catenaries into their design. You will recognize this curve if you hold a slack chain between two hands. A catenary arch can support the weight of a structure with a minimum amount of material.

Specific Course Requirements
If you look at the course requirements for architecture schools, you will see calculus is either a part of or prerequisite for the program. The prerequisites for the University of Massachusetts’ three-year Master of Architecture program, for example, are calculus, physics and an introduction to architecture history. Cornell University’s Bachelor of Architecture program does not specifically require calculus at the high school level for admission. Plane geometry, intermediate algebra and trigonometry are mandatory, while calculus is recommended.

Engineers are often math enthusiasts who got bored with the abstract. Even though number crunching is significant to engineers’ work, math is no more than a convenient means to arrive at a physical end. The type of math an engineer uses will depend on the type of engineer she is and the type of project in which she’s involved.

Basic Arithmetic
All math is based on the idea that 1 plus 1 equals 2, and 1 minus 1 equals 0. Multiplication and division –2 times 2 and 4 divided by 2 – are variations used to avoid multiple iterations of either subtraction or addition. One example of an engineer’s use of basic arithmetic is the civil engineer’s calculations for describing water flow across an open basin. The flow is reckoned in cubic feet per second, or Q, where Q equals the runoff coefficient times the intensity of the rain for a specified period, times the area of the basin. If the runoff coefficient is 2, the intensity, in inches of rain, is 4 and the basin – a specified area of land – is 1/2 acre, the engineer’s formula resembles this: (2×4)/(.5×43,560), or 8/21,780. The result, 0.0003673, is the volume of water, in cubic feet per second, flowing across the land.

Algebra and Geometry
When several of the factors of a problem are known and one or more are unknown, engineers use algebra, including differential equations in cases when there are several unknowns. Because engineers work to arrive at a solution to a physical problem, geometry – with its planes, circles and angles – determines such diverse things as the torque used to turn a wheel, and reduces the design of a roadway’s curve to an accurate engineering or construction drawing.

Trigonometry
Trigonometry is the science of measuring triangles. Engineers may use plane trigonometry to determine the size of an irregularly shaped parcel of land. It may also be used or to determine the height of an object based solely on the distance to the object and the angle, up or down, from the observer. Spherical trigonometry is used by naval engineers in ship design and by mechanical engineers working on such arcane projects as the design of mechanical hand for an underwater robot.

Statistics
We all love statistics. They tell us where we stand in the world, among our peers and even in our family. They tell us who’s winning. The engineer uses them for the same reasons – by statistical analysis of the design, the engineer can tell what percentage of a design will need armor or reinforcement or where any likely failures will occur. For the civil engineer, statistics appear as the concentration of rainfall, wind loads and bridge design. In many locations, engineers designing drainage systems must design for a 50- or 100-year storm in their calculations, a significant change from the normal rain concentration.

Calculus
Calculus is used by engineers to determine rates of change or rates by which factors, such as acceleration or weight, change. It might tell NASA scientists at what point the change in a satellite’s orbit will cause the satellite to strike an object in space. A more mundane task for calculus might be determining how large a box must be to accommodate a specific number of things. An engineer who designs packaging, for example, might know that a product of a certain weight must be packaged in groups of no more than 10 because of their weight. Using calculus, he can calculate both the optimum number of objects per box, plus the optimum size of the box.

Mechanical Engineering Required Courses
By
Will Charpentier

Mechanical engineers are instrumental in the design of hydroelectric dams.
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Does a Mechanical Engineer Require a Lot of Math?
While not all mechanical engineering curricula include all possible courses, the courses in any curriculum have one common characteristic: all study motion and control in dynamic systems. As the nature of materials and processes changes, mechanical engineering itself evolves from the single focus on fundamental engineering to one of both fundamental and systems engineering.

What Its Worth
Part of the value of an education as a mechanical engineer is financial. According to the U.S. Bureau of Labor Statistics, mechanical engineers earn a median salary of $79,230 per year, or $38.09 per hour. The average annual salary, though, is somewhat higher, at $83,550, an hourly wage of $40.17.

Material Science
A mechanical engineering curriculum routinely includes courses in material science. Material science covers the uses and limitations of various materials. The student learns about the characteristics of the materials, from steel to ceramics, used to create products. The student also learns the methods used to turn these materials into useful products.

Controls and Instrumentation
Another part of the curriculum includes the basics of controls and instrumentation. Without a set of controls, whether mechanical or electromechanical, the movement of a part within a machine is unlimited and may not serve its intended purpose. Instrumentation is essential as well, as the instruments apprise the machine operator of the machine’s situation at any time. Instrument design is, in itself, a specialty in mechanical engineering. Instruments may include something as familiar as a water meter or as esoteric as a device to measure the salinity of seawater.

Fluid Dynamics
Fluid dynamics finds a place in the mechanical engineering curriculum because the behavior of fluids in motion affects power systems. The behavior of water as it flows through the piping system of a hydroelectric dam or the behavior of gasoline as it flows into a cylinder in the engine of your family car influences the power each can provide.

Thermal and Fluid Engineering
Hydraulic cylinders and systems, such as those used to raise and lower the booms and buckets on construction machinery, are examples of the objects of study in fluid engineering or fluid mechanics, another course, or set of courses, that appears in a mechanical engineering curriculum. Thermal engineering — the heating and cooling processes that involve heat transfer – is related to fluid mechanics because of the heat-transfer capacity of liquids, such as those in a car’s radiator, household radiant heat systems or the cooling towers of a nuclear reactor.

Design and Manufacturing
Design and manufacturing are part of the common curriculum of mechanical engineering because all modern manufacturing systems involve machines that move. Manufacturing facilities include systems to form, join and assemble parts into a finished product. A mechanical engineer not only designs each portion of a machine, but a series of machines to perform the required functions, one step at a time. Not only must the engineer understand design, but the manufacturing process.

Numerical Computation
Engineering mathematics courses introduce students to basic computer programming. They also introduce linear algebra, approximation and integrations. Students learn to solve linear and nonlinear equations, as well as ordinary differential equations. The student also learns about deterministic and probabilistic approaches to engineering.

mathematical names for buildings

Math and architecture are more closely linked that you might think, so read on to discover what’s behind it all…

It doesn’t matter whether you’re a mathematics student or an architecture student; the chances are that your career could be influenced by the other. That’s why we’re going to show you the link between these two disciplines, and how you could benefit from learning about both.

Math is used in architecture for a number of reasons, but perhaps most obviously to calculate the strength of materials. Structural engineers use math to measure the stresses that will impact on a building, allowing them to design structures which can handle them.

But math also allows architects and designers to express their creativity without compromising on safety. Math allows us to create new shapes and designs which would be impossible using traditional techniques. This has opened up a world of possibilities for both architects and their clients, who can now have buildings designed around their specific needs rather than working around existing limitations.

1) The Great Pyramid of Giza, Cairo, Egypt

The superlatives that describe the Great Pyramid of Giza speaks for itself: its the largest and oldest of the three pyramids and was the tallest man-made structure in the world for 3,800 years, but there’s also plenty of math behind one of the Seven Wonders of the Ancient World.

Did you know that in cubits (the first recorded unit of length), the pyramid’s perimeter is 365.24 – the number of days in the year? That the pyramid’s perimeter divided by twice its height is equal to pi (3.1416)? Or that the King’s Chamber measurements are based on a Pythagorean triangle (3, 4, 5)?

2) Taj Mahal, Agra, India

Sitting firmly at the top of many traveler’s wish lists, the Taj Mahal in India is a delight for tourists, with many waiting to get that iconic photo in front of this beautiful building. But look closer and you’ll find a great example of line symmetry – with two lines, one vertical down the middle of the Taj, and one along the waterline, showing the reflection of the prayer towers in the water…

3) The Eden Project, Cornwall, UK

The Eden Project, in South West England, opened in 2001 and now ranks as one of the UK’s most popular tourist attractions. Although visitors come to check out what’s inside, the greenhouses – geodesic domes made up of hexagonal and pentagonal cells – are pretty neat too.

‘The Core’ was added to the site in 2005, an education center that shows the relationship between plants and people. It’s little surprise that the building has taken its inspiration from plants, using Fibonacci numbers to reflect the nature featured within the site.

There’s even more math to be found in the building structure, which is derived from phyllotaxis, the mathematical basis for most plant growth (opposing spirals are found in many plants, from pine cones to sunflower heads).

4) Parthenon, Athens, Greece

Constructed in 430 or 440 BC the Parthenon was built on the Ancient Greek ideals of harmony, demonstrated by the building’s perfect proportions. The width to height ratio of 9:4 governs the vertical and horizontal proportions of the temple as well as other relationships of the building, for example the spacing between the columns.

It’s also been suggested that the Parthenon’s proportions are based on the Golden Ratio (found in a rectangle whose sides are 1: 1.618).

The Ancient Greeks were resourceful in their quest for beauty – they knew that if they made their columns completely straight, an optical illusion would make them seem thinner in the middle, so they compensated for this by making their columns slightly thicker in the middle.

5) The Gherkin, London, UK

The Gherkin’s unusual design features – the round building, bulge in the middle, the narrow taper at the top and spiraling design – create an impact in more ways than you might think. The cylindrical shape minimizes whirlwinds that can form at the base of large buildings, something that can be predicted by computer modeling using the math of turbulence.

What’s more, the bulging middle and tapered top give the illusion of a shorter building that doesn’t block out sunlight, helping to maximise natural ventilation and saving on air conditioning, as well as lighting and heating bills. Built with the help of CAD (Computer Aided Design) and parametric modeling, the Gherkin is now a distinctive feature in London’s city skyline.

6) Chichen Itza, Mexico

Chichen Itza was built by the Maya Civilization, who were known as fantastic mathematicians, credited with the inventing ‘zero’ within their counting system. At 78 feet tall, the structure of El Castillo (or ‘castle’) within Chichen Itza is based on the astrological system.

Some fast facts: the fifty two panels on each side of the pyramid represent the number of years in the Mayan cycle, the stairways dividing the eighteen tiers correspond to the Mayan calendar of eighteen months and the steps within El Castillo mirror the solar year, with a total of 365 steps, one step for each day of the year.

7) Sagrada Familia, Barcelona, Spain

Designed by Antoni Gaudi, the Sagrada Familia is one of Spain’s top tourist destinations. There’s plenty of math to get your teeth into too. Gaudi used hyperbolic paraboloid structures (a quadric surface, in this case a saddle-shaped doubly-ruled surface, that can be represented by the equation z = x2/a2 – y2/b2), which can be seen within particular façades.

The Sagrada Familia also features a Magic Square within the Passion façade – an arrangement where the numbers in all columns, rows and diagonals add up to the same sum: in this case, 33. The Magic Constant, or M is the constant sum in every row, column and diagonal and can be represented by the following formula M = n (n^2 +1)/2.

8 ) Guggenheim Museum, Bilbao, Spain

Bilbao may not be the first place you’d think to travel to in Spain, but the Guggenheim Museum certainly gives you a good excuse to pay this northern port city a visit. Since opening to the public in 1997, the Guggenheim Museum Bilbao has been celebrated as one of the most important buildings of the 20th century and it’s not hard to see why.

Intended to mimic a ship, the titanium panels, which look like fish scales, were designed to appear random but actually relied on Computer Aided Three Dimensional Interactive Application (CATIA). In fact, computer simulation made it possible to build the sorts of shapes that architects from earlier years could have only imagined.

9) Philips Pavilion, Brussels, Belgium

Known for great beer, delicious waffles and moules frites, if you visited Belgium’s capital Brussels in 1958, you would have most likely come across the Philips Pavilion instead. Commissioned by electronics company Philips, the Pavilion was a mind-boggling collection of asymmetric hyperbolic paraboloids and steel tension cables, intended to be used as a venue to showcase technological progress after the Second World War.

In this article, we’ve examined the ways in which the fields of architecture and mathematics interact with one another and have found many similarities. Both fields require a keen awareness of structure, form, and relationships to other elements. It is easy to see how math is necessary for architecture, but it is also fascinating to see how certain mathematical principles are reflected in the world that we build around us.

We hope you’ve enjoyed reading about these interactions as much as we enjoyed writing about them!