Sunday, June 26, 2016

Week 1 - Math + Art

Defined by Wikipedia, "In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities." This golden ratio, also known as the golden division, the perfect ratio or the golden mean, is the a mathematical geometry property. It appears in basic constructions of an equilateral triangle, square and pentagon placed inside a circle.

                                                                     Geometry properties of golden ratio                                                                     http://www.goldennumber.net/golden-ratio/
It has math representation

 where the Greek number Phi has value



It seems like that our perception of beauty is tightly wired with Phi, as if it is embedded into our genes (the shape of DNA also follows the golden ratio) that we perceive the beauty in men and women and we judge the appearance of structure by how closely the proportions of facial and body are to Phi. For this reason, Phi is applied in both cosmetics, painting, sculpture and architecture. 

It has been discovered that many artists have found this golden ratio and utilized it to maximize the aesthetic perception produced by the proportion of objects or the division of different parts of the painting. The most famous example is the mixture of art and science, Leonardo da Vinci painted many human figure portrays, and many of them follow the golden ratio as in a pentagon.
        Leonardo da Vinci's human portray.    
In his most famous painting, The Last Supper, we can find Phi everywhere, almost in every sub-structure of the overall structure.: the painting on the arch window on the wall, the position of Jesus and his surrounding men, and the supper table and etc.
                                                             The Last Supper by Leonardo da Vinci                                                       http://www.goldennumber.net/art-composition-design/

Michelangelo's painting on the Sistene Chapel, The Creation of Adam, also is found to follow the golden ratio.
                                                              The Creation of Adam by Michelangelo                                                            http://www.goldennumber.net/golden-ratio/
Moreover, the golden ratio is discovered way before mathematics became a well-defined field. Some ancient architectures and furniture, which are built way before the mathematics burst in 14 century, are discovered to be following Phi as well. It well supports the point that the aesthetic perception of Phi is embedded in our gene and we human kind naturally uses Phi without mathematically defining it.

For example, the Parthenon in Athenian Acropolis, Greece, built by Greek from 443 to 438 BC, has a perfect golden ratio structure.
                                                                  Greek temple, Parthenon.                                                              http://www.goldennumber.net/architecture/
The Egyptians described the golden ratio as "sacred ratio" and utilized it in their design of Pyramid.
                                                          Pyramid follows the golden ratio.                                                http://jwilson.coe.uga.edu/emat6680/parveen/ancient_egypt.htm
Mathematics is widely used in the building of Pyramid. Other than the golden ratio, geometry, calculation of area and other techniques were also learned by Egyptians and utilized in building of the first and oldest Pyramid, the Step Pyramid in Saqqara. Gamwell states that 
Students of the Rhind papyrus were taught the surveyor’s rule for calculating the area of a trapezoid, and scribes have recorded that Egyptian builders knew the related rule for finding the volume of a truncated pyramid.
This integration of mathematics and art gives rise to a more accepted, more appreciated form of aesthetics. It elevates expressions of art and broaden the application of mathematics. Claude summarizes this integration, or the juxtaposition of mathematics, in projective and geometry as 
For at least 32000 years, when they used to ornament the walls of the caves, the artists have painted on plane or curved surfaces using sometimes the rules of perspective in a spontaneous manner. We are indebted to the painters for having founded a rational theory of perspective ... these elements lead the archtect from Lyon, Gerard Desargues, around 1639, to base projective geometry: that is the classical example of the phenomenon of symbiosis between art and mathematics.




Reference:

Gamwell, Lynn, and Neil DeGrasse Tyson. Mathematics Art: A Cultural History. N.p.: n.p., n.d. Print.

Henderson, Linda Dalrymple. “The Fourth Dimension and Non-Euclidean Geometry in Modern Art: Conclusion.” Leonardo. 17.3 (1984): 205-210. Print.

Bruter, Claude Paul. Mathematics and Art: Mathematical Visualization in Art and Education. Berlin: Springer, 2002. Print.

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