Mathematics and architecture

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Deconstructivist architecture of the Disney Concert Hall, Los Angeles, uses disorder to achieve its effect.

Mathematics and architecture are related. Architects intentionally or accidentally use mathematical proportions to shape buildings.

In ancient Greece, the golden ratio may have been used to lay out some buildings. In Islamic architecture, geometrical shapes and tiling patterns are used. The pyramids of ancient Egypt have mathematical proportions including the golden ratio, for whatever reason. Hindu temples may have been laid out using the mathematics of astrology; they also have a fractal-like structure where parts resemble the whole.

In Renaissance architecture, symmetry and mathematical proportion were deliberately emphasized.

In the twentieth century, styles such as modern architecture and Deconstructivism explored different geometries to achieve desired effects.

Ancient times[edit]

The Parthenon has been claimed to follow the proportions of the golden rectangle
The complex geometry and tilings of the Lotf Allah mosque, Isfahan
The ancient Egyptian pyramids at Giza have mathematical proportions, either by accident or by design.
The Virupaksha temple at Hampi has a fractal-like structure where the parts resemble the whole.
Further information: Golden Ratio

Ancient Greece[edit]

In Greek architecture, the golden rectangle (whose sides are in the ratio known as Phi, the golden section, golden ratio, or golden mean), served as a canon for planning architectural designs. Knowledge of the golden mean goes back at least as far as 300BC, when Euclid described the method of geometric construction.[1] The golden rectangle's short and long sides are in the ratio 1: 1.618, considered pleasing in Western architectural theory.

Jay Hambidge believed that the golden rectangle was the ratio used by Attic Greek architects in the design of the Parthenon and other ancient Greek buildings, as well as sculptures, paintings, and vases.[2]

More recent authors, however, question nineteenth-century assumptions about the Golden Mean in Classical buildings such as the Parthenon,[3] where experiments by George Markowsky failed to find any preference for the golden rectangle.

The Cartesian grid plan of cities such as ancient Greek Olynthus shows another association between architecture and geometry.

Islamic architecture[edit]

Main article: Islamic architecture

Islamic buildings are often decorated with tiling patterns which typically make use of several mathematical tessellations.[4] A variety of symmetries such as stars with six, eight, or multiples of eight points are used in Islamic patterns. Some of these are based on the 'Khatem Sulemani' or Solomon's seal motif, which is an eight-pointed star made of two squares, one rotated 45 degrees from the other on the same centre. Islamic patterns exploit many of the 17 possible wallpaper groups; as early as 1944, Edith Müller showed that the Alhambra made use of 11 wallpaper groups in its decorations, while in 1986 Branko Grünbaum claimed to have found 13 wallpaper groups in the Alhambra, asserting controversially that the remaining 4 groups are not found anywhere in Islamic ornament.[5] Jason Elliot has suggested that the golden ratio was used by the designers of the Naqsh-e Jahan Square and the adjacent Lotfollah mosque.[6]

Other civilizations[edit]

Ancient architectures in Egypt and India employed planning principles and proportions that rooted the buildings to the cosmos, considering the movements of sun, stars, and other heavenly bodies. Vaastu Shastra, the ancient Indian canons of architecture and town planning employs mathematical drawings called mandalas. Complex calculations are used to arrive at the dimensions of a building and its components. Some of these calculations were astrological, while others were based on aesthetics such as rhythm.

However, early builders may have come upon mathematical proportions by accident. Georges Ifrah notes that simple "tricks" with string and stakes can be used to lay out geometric shapes, such as ellipses and right angles.[4][7]

The mathematics of fractals has been used to show that the reason why existing buildings have universal appeal and are visually satisfying is because they provide the viewer with a sense of scale at different viewing distances. For example, in Hindu temples such as the Virupaksha temple at Hampi, the parts and the whole have the same character.


Renaissance architecture used symmetry as a guiding principle. The works of Andrea Palladio serve as good examples. Later High Renaissance or Baroque used curved and dramatically twisted shapes in as varied contexts such as rooms, columns, staircases and squares.[citation needed]

Twentieth century[edit]

The Rietveld Schröder House shows the De Stijl architecture of horizontal and vertical planes sliding past each other.
Sydney Opera House is composed of shells all of uniform curvature in both directions

The early twentieth century movement Modern Architecture used rectilinear Euclidean (also called Cartesian) geometry. In the De Stijl movement, the horizontal and the vertical were seen as constituting the universal. The architectural form consists of putting these two directional tendencies together, using roof planes, wall planes and balconies, which either slide past or intersect each other, as in the Rietveld Schröder House by Gerrit Rietveld.[8]

The late twentieth century movement Deconstructivism creates deliberate disorder with what Nikos Salingaros in his A Theory of Architecture calls random forms[9] of high complexity[10] by using non-parallel walls, superimposed grids and complex 2-D surfaces, as in Frank Gehry's Disney Concert Hall[11][12] and the works of Peter Eisenman and Zaha Hadid.

See also[edit]


  1. ^ Euclid. Elements. Book 6, Proposition 30.
  2. ^ Archibald, R. C. Notes on the Logarithmic Spiral, Golden Section and the Fibonacci Series
  3. ^ Applications of the Golden Mean to Architecture
  4. ^ a b O'Connor, John; Robertson, Edmund F (July 2012). "Mathematics and Architecture". University of St Andrews. Retrieved December 11, 2012. 
  5. ^ Rønning, Frode. "Islamic Patterns And Symmetry Groups" (PDF). University of Exeter. Retrieved 18 April 2014. 
  6. ^ Elliot, Jason (2006). Mirrors of the Unseen: Journeys in Iran. Macmillan. pp. 277, 284. ISBN 978-0-312-30191-0. 
  7. ^ Ifrah, 1998.
  8. ^ "Rietveld Schröderhuis (Rietveld Schröder House)". World Heritage Centre. UNESCO. Retrieved 13 December 2012. 
  9. ^ Salingaros, 2006. pp139–141
  10. ^ Salingaros, 2006. pp124–125
  11. ^ Gehry, Frank O.; Mudford, Grant; Koshalek, Richard (2009). Symphony: Frank Gehry's Walt Disney Concert Hall. Five Ties. 
  12. ^ Garcetti, Gil (2004). Iron: Erecting the Walt Disney Concert Hall. Princeton Architectural Press. 


  • Ifrah, Georges (1998). A Universal History of Numbers. Penguin. 
  • Salingaros, Nikos (2006). A Theory of Architecture. Umbau. 

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