Mathematicians tend to see beauty in mathematical results which establish connections between two areas of mathematics which at first sight appear to be totally independent. A good example of this is Euler's identity, called "the most remarkable formula in mathematics" by Feynman, for relating all the critical constants (0, 1, e, i, pi) and operations (equality, addition/subtraction, multiplication/division, and exponentation/roots) in a neat formula. This is widely recognized to have a property that mathematics professors often describe as beautiful, elegant or important.
Such results may be obtained by clever, unconventional or innovative means, but these three terms are not used interchangeably, and not every mathematical result is described in these terms. Just as often, results or calculations are called ugly, clumsy, pedestrian, conversely, such results are usually correct, but are obtained by laborious, conventional, or even kludgy means.
The statement that a specific mathematical result is beautiful is rarely seen in published mathematical research, but this is a result of the modern emphasis on impersonality. Classic mathematical works from as late as the 19th century are often written in a very personal style and contain grandiose statements which the modern reader would classify as anything from quaint to preposterous, or in any case more appropriate to writing popularizations. This is true of most scientific writing. It is also true of most bad writing. Whether this style is reasonably a form of literature or theology is quite difficult to say.
In ancient ages famous mathematicians such as Pythagoras (and his entire philosophical school) believed in the literal reality of numbers. Indeed, the discovery of irrational numbers was a great shock for them - they considered the existence of numbers not expressable as the ratio of two natural numbers to be a terrible flaw in nature. Indeed, from the modern perspective Pythagoras was as much a numerologist as a mathematician, and his treatment of numbers was more like that of a religious believer than of a modern mathematician - including his dealing with heresy: he reputedly drowned a student in a barrel for revealing irrational numbers.
Another example is Archimedes, who was so impressed by one of his own theorems that he asked that a representation of it be used as his epitaph. He is also reputed to have been killed by a soldier for ignoring him to concentrate on diagrams he was drawing with a stick in the sand.
About the relationship between art and mathematics, one may say that ancient Egyptians and Greeks knew the golden ratio, regarded as an aesthetically pleasing ratio, often used when building monuments (e.g., the Parthenon). The pentagram so popular among the Pythagoreans also contains the golden mean. Recent studies showed that the Golden ratio plays a role in human perception of beauty, as in body shapes and faces. Johannes Kepler believed that the regular solids (tetrahedron, cube, dodecahedron) were important to the organizing of the Solar System before finding the even-more-elegant solution of the elliptical orbit. So it is hard to say that beliefs of this sort play no role in science or should not be in philosophy of science.
Ludwig Boltzmann and Carl Friedrich Gauss also requested that their most famous theorems be used as an epitaph, in the forms of Boltzmann's equation and a 17-sided polygon, respectively. This is more clearly like a religion.
The use of mathematics to describe or derive other art is now common:
The work of M. C. Escher is plenty of impossible constructions, made using geometrical objects that cannot exist but are pleasant to the human sight. Relationship between the work of Goedel (mathematician), Escher (painter) and Bach (music) is explained in Goedel, Escher, Bach, a Pulitzer Prize-winning book.
In the modern industry of computer animation, fractals play a key role in modelling mountains, fire, trees and a other natural objetcs. See fractal art for examples about the use of this mathematical objects with only aesthetical motivations. See low-complexity art for Juergen Schmidhuber's minimal art form explicitly based on short computer programs.
A recent study published in Scientific American (December 2002) shows that an interesting property in Jackson Pollock's art is that his works have a fractal dimension, which make them different from purely random strokes.
The association of mathematics and architecture is particularly notable as the latter employs the former in its search for beauty, truth and the absolute, however elusive these qualities may be.
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