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Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define spatial forms; from the Pythagoreans onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings. In ancient Egypt, ancient Greece, India, and the Islamic world buildings including pyramids, temples, mosques, palaces and mausoleums were laid out with specific proportions for religious reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about Hindu cosmology. In the 21st century, mathematical ornamentation is again being used to cover public buildings. In Renaissance architecture, symmetry and proportion were deliberately emphasized by secular architects such as Leon Battista Alberti, Sebastiano Serlio and Andrea Palladio, influenced by Vitruvius's ''De architectura'' from Ancient Rome and the arithmetic of the Pythagoreans from Ancient Greece. At the end of the nineteenth century, Vladimir Shukhov in Russia and Antoni Gaudí in Barcelona pioneered the use of hyperboloid structures; in the Sagrada Família, Gaudí also incorporated hyperbolic paraboloids, tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. In the twentieth century, styles such as modern architecture and Deconstructivism explored different geometries to achieve desired effects. 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. ==Connected fields== Michael Ostwald and Kim Williams, considering the relationships between architecture and mathematics, note that the fields as commonly understood might seem to be only weakly connected, architecture being a profession concerned with the practical matter of making buildings, mathematics being the pure study of number and other abstract objects. But, they argue, the two are strongly connected, and have been since antiquity. 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. A master builder at the top of his profession was given the title of architect or engineer. In the Renaissance, the extra syllabus became the quadrivium of arithmetic, geometry, music and astronomy, studies expected of the Renaissance man such as Leon Battista Alberti. Similarly in England, Sir Christopher Wren, known today as an architect, was firstly a noted astronomer. Williams and Ostwald, further overviewing the interaction of mathematics and architecture since 1500 according to the approach of the German sociologist Theodor Adorno, identify three tendencies among architects, namely to be revolutionary, introducing wholly new ideas; reactionary, failing to introduce change; or revivalist, actually going backwards. They argue that architects have avoided looking to mathematics for inspiration in revivalist times. This would explain why in revivalist periods, such as the Gothic Revival in 19th century England, architecture had little connection to mathematics. Equally, they note that in reactionary times such as the Italian Mannerism of about 1520 to 1580, or the 17th century Baroque and Palladian movements, mathematics was barely consulted. In contrast, the revolutionary early 20th century movements such as Futurism and Constructivism actively rejected old ideas, embracing mathematics and leading to Modernist architecture. Towards the end of the 20th century, too, fractal geometry was quickly seized upon by architects, as was aperiodic tiling, to provide interesting and attractive coverings for buildings. Architects use mathematics for several reasons, leaving aside the necessary use of mathematics in the engineering of buildings.〔(【引用サイトリンク】url=http://www.careercornerstone.org/pdf/archeng/archeng.pdf )〕 Firstly, they use geometry because it defines spatial forms. Secondly, they use mathematics to design forms that are considered beautiful or harmonious. From the time of the Pythagoreans with their religious philosophy of number, architects in Ancient Greece, Ancient Rome, the Islamic world and the Italian Renaissance have chosen the proportions of the built environment – buildings and their designed surroundings – according to mathematical as well as aesthetic and sometimes religious principles.〔〔〔〔 Thirdly, they may use mathematical objects such as tessellations to decorate buildings. 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.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mathematics and architecture」の詳細全文を読む スポンサード リンク
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