Mathematics is an underlying factor constantly used in the design of products, structures, and spaces. A crucial aspect indeed, its extent can be seen applied in various proposals both practical and conceptual serving its functions of engineering and geometry. Where engineering is used for the construction to build a habitable structure, an architect has to design it initially. 

Geometry, algebra, and trigonometry all have a significant role in the design of size, proportions, and area integrated for the structure. Architects apply these math forms to lay out the blueprints or initial designs and also to calculate the prospects of the construction issues faced by the team as they bring the design vision to life.

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The Parthenon in Ancient Greece has obvious geometric shapes and patterns ©upload.wikimedia.org

History

In ancient Rome, Vitruvius described an architect as a multi-skilled worker, well versed in a range of many disciplines such as geometry, and mathematics to enable him to oversee the skilled artisans in all areas, comprising masons and carpenters. In the middle ages, the same was applied where graduates were taught geometry, arithmetic, and aesthetics in their curriculum to craft a master builder, at the top of his profession. The Renaissance period, however, saw the quad of geometry, music, arithmetic, and astronomy become an extra syllabus expected of the Renaissance man. Noted examples are those of Leon Battista Alberti and Sir Christopher Wren in England.

The Pythagorean Theorem, developed by Pythagoras, has been in use for centuries to calculate the size and shape of a structure. This theory allows builders to correctly measure right angles. The Pythagoreans argued that buildings too should be designed with harmonious ratios to create forms considered symmetrical and melodious according to mathematical, aesthetic, and sometimes religious principles. In ancient India, the temples and town planning were laid out with symmetrical drawings called the Mandalas. The building dimensions arrived from complex calculations and designs that integrated ancient beliefs and Hindu cosmology. 

One of the most remarkable of all ancient architecture may be the pyramids of Egypt, constructed by the Pharaohs to bury their bodies. Most of them were built with mathematical proportions and scaled at about a 51-degree angle. The Egyptians mysteriously possessed an immense knowledge of geometry, as demonstrated by the accuracy of pyramid construction.

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The Vitruvian Man exhibiting The Golden Ratio ©rouleauc.blogspot.com

Since time immemorial, architecture has been inlaid with geometric principles to plan the shapes and spatial forms of various structures. The Greek mathematician, Euclid, developed the principle of Golden Ratio, a mathematical law of nature in 300 B.C, which has been used for more than two thousand years now by architects all around the globe to design ratios and proportions in buildings that appear pleasing to the human eye and with a sense of balance. 

The Golden Ratio is still used as a fundamental geometric principle in architecture, inducing a universal sense of harmony when designed with this principle. Thus it is not surprising to see the Golden Ratio demonstrated throughout the natural architecture of the world.

Islamic mosques are usually seen adorned with multiple geometric patterns that make use of several mathematical tessellations, formed of ceramic tiles that may themselves be plain or decorated with stripes. Symmetric symbols such as stars with multiples of six or eight points are used in Islamic patterns. Islamic patterns use many of the 17 possible wallpaper groups of architecture and decorative art. 

Antoni Gaudí used a wide variety of geometric designs in many of his structures. In the Sagrada Família, Barcelona, some surfaces are designed with tessellations, catenary arches, hyperbolic parabolas, helicoids, and lined surfaces. This varied mix of geometries is creatively combined in different ways around the church exploiting natural patterns such as columns inspired by shapes of trees. 

Mathematics in the 20th and 21st Century

The elimination of ornament in the 20th century removed all ordered structural differentiations from the smaller scales and since patterns cannot be defined on those scales in the Modernist design standard, it removed mathematical information from the built environment. 

Most 20th century buildings do not possess many visual patterns but architects disguised patterns on human scales that are preordained because of the building activities. These can be seen in materials, as a consequence of structural stability and weathering and the design is only at the largest scale, without any hierarchy in scales, and can be seen in the 20th-century buildings such as the Guggenheim Museum in Bilbao, Denver International Airport, and the Sydney Opera House. 

The end of the 20th century saw mathematical concepts of fractal geometry and aperiodic tiling introduced by architects to provide interesting and attractive coverings for buildings. In the 21st century, architects have designed ornamentation that is extremely diverse. With the computer software interface providing the opportunity to experiment with mathematical relations as a generative tool, architects have reintroduced both curvature and parametric into their designs. 

With parametric and algorithmic design methods and the use of digital fabrication, new aptitudes are required from architects for the design of details which are evident in structures such as Shenzhen Bao’an International Airport and Japanese Zen gardens. The newer concepts of Dynamic Architecture where buildings are designed after calculating the sun path and moving to the wind make modern architecture design more efficient and environmentally friendly. They keep modifying their shape and are evident in the works of Zaha Hadid and Rem Koolhaas.

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The Louvre Museum, Abu Dhabi, designed by Foster and Partners has a dome built up of layers of stars made of octagons, triangles, and squares ©www.architecturaldigest.com

Usage of mathematics in architecture designs is not just restricted to the engineering of the buildings. Geometry is vastly exploited because it defines the spatial forms of a building. Mathematics is used to design forms that are considered beautiful and harmonious. Decoration of buildings also uses mathematical objects such as tessellations. Mathematics is also employed nowadays in the form of computer modeling to achieve efficiency and environmental goals, such as minimizing air currents at the base of tall buildings. 

Architects, since the eras of ancient Greece, ancient Rome, the Renaissance, and the Islamic periods have calculated the proportions of their buildings to the built environment mathematically to arrive at aesthetic and religious principles. Mathematics has been making an art out of structural purity, using simple geometric forms for aesthetic as well as functional purposes endlessly.

Author

An architecture graduate, Merina is a strong believer of the "Less is More" idealogy, a principle which is not only evident in her designs, but one that bleeds into other facets of her life. A passionate writer with an insatiable curiosity for all things design, she is ever ready for soaking in some Vitamin D and a conversation over some freshly brewed chai.

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