Mathematics and architecture

Minimal surfaces have been exploited in tent-like roof coverings as at Denver International Airport, while Richard Buckminster Fuller pioneered the use of the strong thin-shell structures known as geodesic domes.

In ancient Rome, Vitruvius described an architect as a man who knew enough of a range of other disciplines, primarily geometry, to enable him to oversee skilled artisans in all the other necessary areas, such as masons and carpenters.

The same applied in the Middle Ages, where graduates learnt arithmetic, geometry and aesthetics alongside the basic syllabus of grammar, logic, and rhetoric (the trivium) in elegant halls made by master builders who had guided many craftsmen.

[13][14] Fourthly, they may use mathematics in the form of computer modelling to meet environmental goals, such as to minimise whirling air currents at the base of tall buildings.

[20][21] Alberti also documented Filippo Brunelleschi's discovery of linear perspective, developed to enable the design of buildings which would look beautifully proportioned when viewed from a convenient distance.

This widely printed book was largely responsible for spreading the ideas of the Italian Renaissance throughout Europe, assisted by proponents like the English diplomat Henry Wotton with his 1624 The Elements of Architecture.

[30][31][32] The early twentieth century movement Modern architecture, pioneered[d] by Russian Constructivism,[33] used rectilinear Euclidean (also called Cartesian) geometry.

[35] In 1938, the Bauhaus painter László Moholy-Nagy adopted Raoul Heinrich Francé's seven biotechnical elements, namely the crystal, the sphere, the cone, the plane, the (cuboidal) strip, the (cylindrical) rod, and the spiral, as the supposed basic building blocks of architecture inspired by nature.

[53] The proportions of some pyramids may have also been based on the 3:4:5 triangle (face angle 53°8'), known from the Rhind Mathematical Papyrus (c. 1650–1550 BC); this was first conjectured by historian Moritz Cantor in 1882.

[54] It is known that right angles were laid out accurately in ancient Egypt using knotted cords for measurement,[54] that Plutarch recorded in Isis and Osiris (c. 100 AD) that the Egyptians admired the 3:4:5 triangle,[54] and that a scroll from before 1700 BC demonstrated basic square formulas.

Cooke concludes that Cantor's conjecture remains uncertain; he guesses that the ancient Egyptians probably knew the Pythagorean theorem, but "there is no evidence that they used it to construct right angles.

The designs are intended to integrate architecture with nature, the relative functions of various parts of the structure, and ancient beliefs utilizing geometric patterns (yantra), symmetry and directional alignments.

[12][58] 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.

'mountain') about the tallest, central, tower which represents the holy Mount Kailash, abode of Lord Shiva, depicts the endless repetition of universes in Hindu cosmology.

The Greek word symmetria originally denoted the harmony of architectural shapes in precise ratios from a building's smallest details right up to its entire design.

[69] The historian of Islamic art Antonio Fernandez-Puertas suggests that the Alhambra, like the Great Mosque of Cordoba,[70] was designed using the Hispano-Muslim foot or codo of about 0.62 metres (2.0 ft).

[73] Mughal architecture, as seen in the abandoned imperial city of Fatehpur Sikri and the Taj Mahal complex, has a distinctive mathematical order and a strong aesthetic based on symmetry and harmony.

The white marble mausoleum, decorated with pietra dura, the great gate (Darwaza-i rauza), other buildings, the gardens and paths together form a unified hierarchical design.

The historians of architecture Koch and Barraud agree with the traditional accounts that give the width of the complex as 374 Mughal yards or gaz,[g] the main area being three 374-gaz squares.

[77] The Christian patriarchal basilica of Haghia Sophia in Byzantium (now Istanbul), first constructed in 537 (and twice rebuilt), was for a thousand years[i] the largest cathedral ever built.

[82] Saint Ambrose wrote that fonts and baptistries were octagonal "because on the eighth day,[j] by rising, Christ loosens the bondage of death and receives the dead from their graves.

[86] The number five is used "exuberantly"[87] in the 1721 Pilgrimage Church of St John of Nepomuk at Zelená hora, near Žďár nad Sázavou in the Czech republic, designed by Jan Blažej Santini Aichel.

For example, in the Passion Façade of Sagrada Família, Gaudí assembled stone "branches" in the form of hyperbolic paraboloids, which overlap at their tops (directrices) without, therefore, meeting at a point.

[100] Towards the end of the 20th century, novel mathematical constructs such as fractal geometry and aperiodic tiling were seized upon by architects to provide interesting and attractive coverings for buildings.

[101] Foreign Office Architects' 2010 Ravensbourne College, London is tessellated decoratively with 28,000 anodised aluminium tiles in red, white and brown, interlinking circular windows of differing sizes.

[102][103][l] Kazumi Kudo's Kanazawa Umimirai Library creates a decorative grid made of small circular blocks of glass set into plain concrete walls.

[101] The architecture of fortifications evolved from medieval fortresses, which had high masonry walls, to low, symmetrical star forts able to resist artillery bombardment between the mid-fifteenth and nineteenth centuries.

Well-known architects who designed such defences include Michelangelo, Baldassare Peruzzi, Vincenzo Scamozzi and Sébastien Le Prestre de Vauban.

"[106] In Chinese architecture, the tulou of Fujian province are circular, communal defensive structures with mainly blank walls and a single iron-plated wooden door, some dating back to the sixteenth century.

[87] For example, Foster and Partners' 30 St Mary Axe, London, known as "The Gherkin" for its cucumber-like shape, is a solid of revolution designed using parametric modelling.