Mathematics Used in Arts and Architechture

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Mathematics and art have a long historical relationship. The ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid,[1] theParthenon, the Colosseum. There are many examples of artists who have been inspired by mathematics and studied mathematics as a means of complementing their works. The Greek sculptor Polykleitos prescribed a series of mathematical proportions for carving the ideal male nude. Renaissance painters turned to mathematics and many, including Piero della Francesca, became accomplished mathematicians themselves. Contents  [hide]  * 1 Overview * 2 Ancient times * 2.1 The Golden Ratio * 2.1.1 Pyramids * 2.1.2 Parthenon * 2.1.3 Great Mosque of Kairouan * 2.2 Polykleitos * 3 Renaissance * 3.1 Paolo Uccello * 3.2 Piero della Francesca * 3.3 Notre Dame * 3.4 Albrecht Dürer * 3.5 De Divina Proportione * 3.6 Da Vinci * 4 Industrial and modern times * 4.1 Penrose tiles * 4.2 Eden Project * 4.3 California Polytechnic State University * 4.4 M.C. Escher * 4.5 Salvador Dalí * 4.6 Pablo Palazuelo * 4.7 John Robinson * 4.8 The Eightfold Way * 4.9 Fractal art * 4.10 Platonic solids in art * 4.11 Bridges conference * 5 See also * 6 References * 7 External links| -------------------------------------------------

Galileo Galilei in his Il Saggiatore wrote that “[The universe] is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.”[2] Artists who strive and seek to study nature must therefore first fully understand mathematics. On the other hand, mathematicians have sought to interpret and analyse art through the lens of geometry and rationality. -------------------------------------------------

[edit]Ancient times
[edit]The Golden Ratio
The Golden Ratio, roughly equal to 1.618, was first formally introduced in text by Greek mathematician Pythagoras and later by Euclidin the 5th century BC. In the fourth century BC, Aristotle noted its aesthetic properties.[3] Aside from interesting mathematical properties, geometric shapes derived from the golden ratio, such as the golden rectangle, the golden triangle, and Kepler’s triangle, were believed to be aesthetically pleasing. As such, many works of ancient art exhibit and incorporate the golden ratio in their design. Various authors can discern the presence of the golden ratio in Egyptian, Summerian and Greek vases, Chinese pottery, Olmec sculptures, and Cretan and Mycenaean products from as early as the late Bronze Age.[4] The prevalence of this special number in art and architecture even before its formal discovery by Pythagoras is perhaps evidence of an instinctive and primal human cognitive preference for the golden ratio.[5] [edit]Pyramids

Pyramid of Khufu
Evidence of mathematical influences in art is present in the Great Pyramids, built byEgyptian Pharaoh Khufu and completed in 2560BC. Pyramidologists since the nineteenth century have noted the presence of the golden ratio in the design of the ancient monuments. They note that the length of the base edges range from 755–756 feet while the height of the structure is 481.4 feet. Working out the math, the perpendicular bisector of the side of the pyramid comes out to 612 feet.[6] If we divide the slant height of the pyramid by half its base length, we get a ratio of 1.619, less than 1% from the golden ratio. This would also indicate that half the cross-section of the Khufu’s pyramid is in fact a Kepler’s triangle. Debate has broken out between prominent pyramidologists, including Temple Bell, Michael Rice, andJohn Taylor, over whether the presence of the golden ratio in the pyramids is due to design or chance. Of note, Rice contends that experts of Egyptian architecture have argued that...
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