Mathematics and architecture

Mathematics and architecture are closely connected because architects use math to create beautiful and strong buildings. Just like some arts, architecture needs mathematics for many reasons. Architects use geometry to design the shape of buildings, making them look balanced and pleasing. From ancient times, people have used math to make buildings harmonious and meaningful.

In places like ancient Egypt, ancient Greece, India, and the Islamic world, buildings such as pyramids, temples, and mausoleums were built with special measurements for religious reasons. Islamic architecture often uses geometric shapes and patterns to decorate buildings. Some Hindu temples have designs that repeat like fractals, showing ideas about infinity. In China, unique round buildings called tulou were made for safety.

During the Renaissance, famous architects like Leon Battista Alberti and Andrea Palladio focused on symmetry and balance in their designs. In more recent times, architects have used advanced math to create exciting new shapes and structures, such as the curved roofs of Denver International Airport and the famous geodesic domes designed by Richard Buckminster Fuller. Math helps architects build not only strong but also beautiful spaces for people to enjoy.

Connected fields

Architects and mathematicians may seem like very different professionals—one makes buildings, the other studies numbers and shapes. But they have been closely linked since ancient times. In old Rome, famous writer Vitruvius said a good architect needed to understand geometry to guide skilled workers. Later, during the Middle Ages and the Renaissance, learning arithmetic, geometry, and other math topics was important for talented builders and thinkers like Leon Battista Alberti and Christopher Wren.

Architects use math for several key reasons. They use geometry to shape the spaces of buildings. They also apply math to create designs that feel beautiful or balanced, following ideas from groups like the Pythagoreans. Sometimes, they decorate buildings with repeating patterns called tessellations. And they use computer models based on math to solve problems, like reducing strong winds around tall buildings.

Secular aesthetics

Ancient Rome

Further information: Vitruvius, Vitruvian module, and De architectura

The ancient Roman architect Vitruvius believed that buildings, like temples, should be designed with two key qualities: proportion and symmetry. Proportion ensures that every part of a building fits well with the other parts. Vitruvius used simple whole number ratios, like 1:2, to create balanced and pleasing designs in structures such as his Basilica at Fano.

Vitruvius also described three important qualities in architecture: firmness, usefulness, and delight. Firmness means the building must be strong and stable. Usefulness involves designing spaces that work well for people. Delight comes from the beauty and harmony created when mathematical principles guide the design.

The Pantheon

Main article: Pantheon (Rome)

The Pantheon in Rome is a great example of classical Roman architecture. Its large dome has a round opening at the top called an oculus that lets in light. The height of the dome to the oculus and the width of the inner space are exactly the same—both are 43.3 metres. This careful use of measurement shows how the Romans used math to create lasting and beautiful buildings.

Renaissance

Further information: Renaissance architecture

During the Renaissance, architects like Leon Battista Alberti and Andrea Palladio used math to design buildings with pleasing proportions. Alberti used shapes like cubes to find harmonious ratios, while Palladio used simple ratios like 3:4 and 4:5 to design rooms that related to the whole building. These mathematical ideas helped spread beautiful and balanced designs across Europe.

Nineteenth century

Hyperboloid structures were introduced near the end of the nineteenth century by Vladimir Shukhov for structures like masts and lighthouses. These shapes are both beautiful and strong, using materials in an efficient way.

Twentieth century

Further information: Modern architecture and Contemporary architecture

In the twentieth century, architects continued to explore new shapes and designs. Some used simple geometric forms, while others tried more complex and free-flowing shapes. Buildings like the Sydney Opera House and the Guggenheim Museum in Bilbao show how architects used math and design to create famous and striking structures.

Religious principles

See also: Golden ratio § Egyptian pyramids

The pyramids of ancient Egypt show the use of mathematical proportions in their design. The exact methods and whether famous rules like the Pythagorean theorem were used are still debated by historians. Some pyramids may have used simple ratios like 3:4:5 triangles, known from ancient texts, to create right angles.

Vaastu Shastra, ancient Indian rules for building, uses symmetrical designs called mandalas. Buildings were planned with complex calculations to match nature and beliefs, using patterns like yantra and directions. Early builders might have found these proportions by simple methods using strings and stakes.

In ancient Greece, the Pythagoreans believed that "all things are numbers." They thought buildings should follow musical ratios. The Parthenon temple in Athens shows these ideas, with careful ratios in its length, width, and height.

Islamic architecture, like the Alhambra in Spain, used mathematical shapes and patterns. The Court of the Lions shows proportions based on square roots. The Selimiye Mosque in Edirne, Turkey, uses an octagon shape inside a square.

Mughal architecture, such as the Taj Mahal, shows symmetry and harmony in its design. The Taj Mahal is laid out on grids with careful ratios.

Christian buildings like the Haghia Sophia in Istanbul used domes and shapes that reflected religious ideas. The Baptistry of Saint John, Florence is octagonal, a shape linked to Christian beliefs. Later churches used circles, stars, and other shapes for symbolic reasons.

Mathematical decoration

Islamic architectural decoration

Main article: Islamic geometric patterns

Islamic buildings often feature beautiful geometric patterns made using special tiles called girih and zellige. These patterns include symmetrical stars with six, eight, or more points, and can be very complex. Famous examples include the decorations in the Sheikh Lotfollah Mosque in Isfahan and the Louvre Abu Dhabi.

Modern architectural decoration

Further information: Ornament (art) and Contemporary architecture

In more recent times, architects have used new mathematical ideas like fractal geometry to create interesting designs on buildings. Modern buildings such as the Harpa Concert and Conference Centre in Reykjavik, the Ravensbourne College in London, the Kanazawa Umimirai Library in Japan, and the Museo Soumaya in México all feature beautiful, decorative patterns made from tiles and glass.

Defence

Further information: Star fort

Fortifications changed from tall walls in medieval times to lower, star-shaped forts that could better resist cannon fire from the fifteenth to the nineteenth centuries. These star shapes were designed so that defenders could fire effectively at attackers from many angles. Famous architects like Michelangelo helped design these strong defenses.

In China, special round buildings called tulou were built for safety. These structures had thick walls and only one door, with roofs sloping both ways to keep out rain. Inside was a open space where people could gather.

Environmental goals

Architects sometimes choose the shape of a building to help the environment. For example, Foster and Partners built 30 St Mary Axe in London, nicknamed "The Gherkin", with a special curved shape to reduce strong winds near the bottom. Even though the building looks round, most of its glass panels are flat and shaped like four-sided figures.

In ancient Persia, people built structures called yakhchal to keep ice cold. These had a dome on top and a space underground where ice was stored. The design helped keep the ice from melting all year.