Mathematics and architecture
Encyclopedia
Mathematics and architecture have always been close, not only because architecture depends on developments in mathematics, but also their shared search for order and beauty, the former in nature and the latter in construction. Mathematics is indispensable to the understanding of structural concepts and calculations. It is also employed as visual ordering element or as a means to achieve harmony with the universe. Here geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 becomes the guiding principle.

In Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...

 architecture, the Golden mean, (also known as the Golden rectangle, Golden Section, and Golden Ratio) served as a canon for planning. Knowledge of the golden mean goes back at least as far as 300BC, when Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

 described the method of geometric construction in Book 6, Proposition 30 of his book the Elements. It corresponds to a proportion of 1: 1.618, considered in Western
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...

 architectural theory to be very pleasing. This number is also known as Phi
Phi
Phi may refer to:In language:*Phi, the Greek letter Φ,φ, the symbol for voiceless bilabial fricativeIn mathematics:*The Golden ratio*Euler's totient function*A statistical measure of association reported with the chi-squared test...

. Jay Hambidge
Jay Hambidge
Jay Hambidge was an American artist, born in Canada. He was a pupil at the Art Students' League in New York and of William Chase, and a thorough student of classical art. He conceived the idea that the study of arithmetic with the aid of geometrical designs was the foundation of the proportion and...

 believed that the golden mean was the ratio used by Attic Greek architects in the design of the Parthenon
Parthenon
The Parthenon is a temple on the Athenian Acropolis, Greece, dedicated to the Greek goddess Athena, whom the people of Athens considered their virgin patron. Its construction began in 447 BC when the Athenian Empire was at the height of its power. It was completed in 438 BC, although...

 and many other ancient Greek buildings, as well as sculptures, paintings, and vases.

In Islamic architecture
Islamic architecture
Islamic architecture encompasses a wide range of both secular and religious styles from the foundation of Islam to the present day, influencing the design and construction of buildings and structures in Islamic culture....

, a proportion of 1: √2 was often used—the plan would be a square and the elevation would be obtained by projecting from the diagonal of the plan. The dimensions of the various horizontal components of the elevation such as mouldings and cornices too were obtained from the diagonals of the various projections and recesses in plan.
Ancient architecture such as that of the Egypt
Egypt
Egypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...

ians and India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...

ns 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 mandala
Mandala
Maṇḍala is a Sanskrit word that means "circle". In the Buddhist and Hindu religious traditions their sacred art often takes a mandala form. The basic form of most Hindu and Buddhist mandalas is a square with four gates containing a circle with a center point...

s. Extremely complex calculations are used to arrive at the dimensions of a building and its components. Some of these calculations form part of astrology
Astrology
Astrology consists of a number of belief systems which hold that there is a relationship between astronomical phenomena and events in the human world...

 and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 whereas others are based on considerations of aesthetics
Aesthetics
Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...

 such as rhythm.

Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...

 architecture used symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 as a guiding principle. The works of Andrea Palladio
Andrea Palladio
Andrea Palladio was an architect active in the Republic of Venice. Palladio, influenced by Roman and Greek architecture, primarily by Vitruvius, is widely considered the most influential individual in the history of Western architecture...

 serve as good examples. Later High Renaissance
High Renaissance
The expression High Renaissance, in art history, is a periodizing convention used to denote the apogee of the visual arts in the Italian Renaissance...

 or Baroque
Baroque
The Baroque is a period and the style that used exaggerated motion and clear, easily interpreted detail to produce drama, tension, exuberance, and grandeur in sculpture, painting, literature, dance, and music...

 used curved and dramatically twisted shapes in as varied contexts such as rooms, columns, staircases and squares. St. Peter's Square in Rome
Rome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...

, fronting the St. Peter's Basilica
St. Peter's Basilica
The Papal Basilica of Saint Peter , officially known in Italian as ' and commonly known as Saint Peter's Basilica, is a Late Renaissance church located within the Vatican City. Saint Peter's Basilica has the largest interior of any Christian church in the world...

, is an approximately keyhole-shaped (albeit with non-parallel sides) exterior space bounded by columns giving a very dynamic visual experience.

The term Cartesian
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 planning given to the planning of cities using a grid plan
Grid plan
The grid plan, grid street plan or gridiron plan is a type of city plan in which streets run at right angles to each other, forming a grid...

, shows the close association between architecture and geometry. Ancient Greek cities such as Olynthus
Olynthus
Olynthus was an ancient city of Chalcidice, built mostly on two flat-topped hills 30–40m in height, in a fertile plain at the head of the Gulf of Torone, near the neck of the peninsula of Pallene, about 2.5 kilometers from the sea, and about 60 stadia Olynthus was an ancient city of...

 had such a pattern superimposed on rugged terrain giving rise to dramatic visual qualities, though proving difficult to negotiate heights. Modern town planning used the grid pattern extensively, and according to some, resulting in monotony and traffic problems.

The beginning of the twentieth century saw the heightened use of Euclidean
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

 or Cartesian
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

 rectilinear geometry in Modern Architecture
Modern architecture
Modern architecture is generally characterized by simplification of form and creation of ornament from the structure and theme of the building. It is a term applied to an overarching movement, with its exact definition and scope varying widely...

. In the De Stijl
De Stijl
De Stijl , propagating the group's theories. Next to van Doesburg, the group's principal members were the painters Piet Mondrian , Vilmos Huszár , and Bart van der Leck , and the architects Gerrit Rietveld , Robert van 't Hoff , and J.J.P. Oud...

 movement specifically, the horizontal and the vertical were seen as constituting the universal. The architectural form therefore is constituted from the juxtaposition of these two directional tendencies, employing elements such as roof planes, wall planes and balconies, either sliding past or intersecting each other. The Rietveld Schröder House
Rietveld Schröder House
The Rietveld Schröder House in Utrecht was built in 1924 by Dutch architect Gerrit Rietveld for Mrs. Truus Schröder-Schräder and her three children. She commissioned the house to be designed preferably without walls. Rietveld worked side by side with Schröder-Schräder to create the house...

 by Gerrit Rietveld
Gerrit Rietveld
Gerrit Thomas Rietveld was a Dutch furniture designer and architect. One of the principal members of the Dutch artistic movement called De Stijl, Rietveld is famous for his Red and Blue Chair and for the Rietveld Schröder House, which is a UNESCO World Heritage Site.-Biography:Rietveld was born in...

 is a good example of this approach.

The most recent movement-Deconstructivism
Deconstructivism
Deconstructivism is a development of postmodern architecture that began in the late 1980s. It is characterized by ideas of fragmentation, an interest in manipulating ideas of a structure's surface or skin, non-rectilinear shapes which serve to distort and dislocate some of the elements of...

-employs non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

 to achieve its complex objectives resulting in a chaotic order. Non-parallel walls, superimposed grids and complex 2-D surfaces are some external manifestations of this approach which is exemplified by the works of Peter Eisenman
Peter Eisenman
Peter Eisenman is an American architect. Eisenman's professional work is often referred to as formalist, deconstructive, late avant-garde, late or high modernist, etc...

, Zaha Hadid
Zaha Hadid
Zaha Hadid, CBE is an Iraqi-British architect.-Life and career:Hadid was born in 1950 in Baghdad, Iraq. She received a degree in mathematics from the American University of Beirut before moving to study at the Architectural Association School of Architecture in London.After graduating she worked...

, and Frank Gehry
Frank Gehry
Frank Owen Gehry, is a Canadian American Pritzker Prize-winning architect based in Los Angeles, California.His buildings, including his private residence, have become tourist attractions...

. Topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 has been a fascinating influence.

In recent times, the concept of fractals has been used to analyse many historical or interesting buildings and demonstrate that such buildings have universal appeal and are visually satisfying because they are able to provide the viewer a sense of scale at different levels/ distances of viewing. Fractals have been used to study Hindu
Hindu
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent. As used in the Constitution of India, the word "Hindu" is also attributed to all persons professing any Indian religion...

temples where the part and the whole have the same character.

As is apparent, architecture has always tried to achieve ends that not only relate to function, but also to aesthetics, philosophy and meaning. And in many a case, the means has been the beauty and structure of mathematics.

External links

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