Mathematics and art have a long historical relationship. Artists have used mathematicssince the 5th century BC when the GreeksculptorPolykleitos wrote his Canon, prescribing proportions based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione(1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, thegraphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led byTheo van Doesberg and Piet Mondrianexplicitly embraced geometrical forms. Mathematics has inspired textile arts such asquilting, knitting, cross-stitch, crochet,embroidery, weaving, Turkish and othercarpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework,Mughal jaali pierced stone screens, and widespread muqarnas vaulting.
Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of symmetry, and mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourfulstellated polyhedra, originally as models for teaching. Mathematical concepts such asrecursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Escher. Computer art often makes use of fractals including the Mandelbrot set, and sometimes explores other mathematical objects such as cellular automata. Controversially, the artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes; the architect Philip Steadman similarly argued that Vermeer used the camera obscura in his distinctively observed paintings.
Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditionalbatiks from different regions of Java have distinct fractal dimensions, and stimuli to mathematics research, especially Filippo Brunelleschi's theory of perspective, which eventually led to Girard Desargues's projective geometry. A persistent view, based ultimately on the Pythagorean notion of harmony in music, holds that everything was arranged by Number, that God is the geometer of the world, and that therefore the world's geometry is sacred, as seen in artworks such as William Blake's The Ancient of Days.
Mathematics is involved in some way in every field of study known to mankind. In fact, it could be argued that mathematics is involved in some way in everything that exists everywhere, or even everything that is imagined to exist in any conceivable reality. Any possible or imagined situation that has any relationship whatsoever to space, time, or thought would also involve mathematics.
Music is a field of study that has an obvious relationship to mathematics. Music is, to many people, a nonverbal form of communication, that reaches past the human intellect directly into the soul. However, music is not really created by mankind, but only discovered, manipulated and reorganized by mankind. In reality, music is first and foremost a phenomena of nature, a result of the principles of physics and mathematics.
I. Music
It is a difficult task to properly define the word "music", since many individuals have quite different opinions. My personal definition, is that music is sound that is organized in a meaningful way with rhythm, melody, and harmony. This is what I consider the three dimensions of music. This definition would exclude such things as "rap music", which has rhythm but has virtually no melody or harmony. I perceive "rap" to be poetry, that is spoken rhythmically with a minimum musical element at best. There are other things that pass as music, such as the works of John Cage, that fail to meet my definition of music. However, many people consider things to be music that I do not. The only definition of music that could be universally agreed upon, then, is that music is any sound, or any combination of sounds, of any kind, that someone, somewhere, enjoys listening to.
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