Term Paper: Art of Math

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Mathematics and Art

Mathematics is often treated as a distant and very different discipline from the arts, but in fact the arts make use of mathematics in a number of ways. The relationship between mathematics and music should be evident, while the relationship between mathematics and the graphic arts may be less apparent. Paintings, drawings, and designs can be analyzed according to mathematical principles to see ways in which the artist balances different shapes and forms according to mathematical principles or draws on mathematical theory for inspiration. The art of different periods may reflect different mathematical ideas and give more or less emphasis to these ideas, but in some degree, art is always based on mathematics in that math explains relationships, identifies what would be considered ideal in different forms, and also explains the relationships seen in nature. Composition involves recreating these relationships in the medium chosen, and just as the eye sees and absorbs patterns and the brain categorizes and makes sense of them unconsciously, so does the viewer of art make many of the same adjustments and so is affected by the mathematics of art.

The classical era was one in which mathematics was used quite consciously in developing artistic styles, and some of these styles have even been named with mathematical references. The artworks of a given era reflect the formalist, social, and economic realities of the period, exemplifying the prevailing artistic styles and the social and economic structures which influence the arts.

In Greek art, the Geometric period was an era which produced a good deal of pottery and other geometrically regular works. The Geometric krater from the Dipylon cemetery from the eighth century B.C. (De La Croix, Tansey, and Kirkpatrick 130) exemplifies the style of the period. The Geometric period is the name given to the era between the end of the Mycenaean age and the beginning of the Classic age. Greek society was marked then by tribal hereditary power and a growing land-owning aristocracy. The worship of particular gods in certain sacred places united Greeks of different tribes and cities through common sacrifices and common competitive games. The Geometric style reached its apex about the time of this krater, and the largest and most characteristic vases came from the area of the Dipylon Gate. These kraters served as sacrificial vessels and as tomb-monuments (Kjellberg and Saflund 53-55). These vases are marked by their decorative patterns of squares, rhomboids, triangles, and zigzags. On later Geometric vases, such as the one under discussion, representations of figures also appear, in a two-dimensional, analytical style. The aspect of the Greek character seen in these works is an analytical clarity and order and a desire for rhythmic regularity, the sort satisfied by mathematics (Kjellberg and Saflund 56).

The Early Classical period can be seen in a work such as the Charioteer of Delphi from the Sanctuary of Apollo at Delphi, c. 470 B.C., representing the king's driver from a grouping that included chariot and horses (De La Croix, Tansey, and Kirkpatrick 150). The charioteer is an example of the Severe Style. The figure is three dimensional but contained:

The bearing of the entire figure conveys the solemnity of the event commemorated, for chariot races and similar contests at that time were competitions for divine favor, not sporting events in the modern sense (Janson 104).

The size of the figure and of the group of which it was a part shows an economic change, with more community-based artwork on a much grander scale than was seen in the Geometric period. The Greeks had become a more powerful state, recognized as such in the world after their defeat of the Persians. Athens was now the cultural and economic center of all of Greece, and artists from all parts of the Greek world were drawn to Athens (Kjellberg and Saflund 105). This also contributed to the Severe Style as the people of Greece manifested a sort of austere grandeur in their art of the period, with stern simplification of outline and surface, fixed pose, firm stance, and immobility of expression (De La Croix, Tansey, and Kirkpatrick 149).

The Renaissance was a period in which classical learning was revived, and this also meant a return to classical ideas of design. Those ideas were based on mathematical principles, for the Greeks understood the use of mathematics in developing a sense of balance and composition in art and architecture. Brunelleschi was an early Renaissance architect who sought a new way to make visual records of architecture on a flat surface, and he accomplished this using a style that made it possible to measure precisely the depth of the foreshortened flanks of buildings. This was a geometric procedure of some complexity, and it utilized the central feature of the vanishing point, the point toward which parallel lines converge when an image is drawn on a flat surface, reproducing what is seen by the eye when looking at distant objects. Brunelleschi's discovery of this vanishing point and of the fact that the point at which lines perpendicular to the picture plane disappeared was on the horizon exactly corresponding in position to the eye of the viewer would become very influential on painters who followed him and was also useful to sculptors (Chilvers, Osborne, and Farr 77).

Masaccio was the first to take up this approach in painting, though it was already well established in architecture and sculpture, and to use it in his Holy Trinity, produced early in the fifteenth century. This picture was a fresco that showed this startling new style to good advantage. From Donatello Masaccio took the idea of the "clothed nude," meaning that the figures are anatomically correct and have been drawn realistically in the way the clothing fits a real body. Masaccio's picture shows a rational development of space in which the architecture shows a new realism along with the figures. Brunelleschi had also wanted to rationalize architectural design, and this was carried over into the other arts through his discovery of perspective and with a new sense of the relation ships among proportions (Chilvers, Osborne, and Farr 316). Donatello took up the new approach in works like the Feast of Herod, which reflected Brunelleschi's system of linear perspective that now made it possible to represent three-dimensional space on a flat surface in a new way.

In the nineteenth century, the prevailing artistic style for the first part of the century was romanticism, an art based on a form of "disorder," but a disorder seen as the emblem of the unfettered processes of the imagination:

In historical terms, fully developed Romanticism is the successor to the cults of nature and of feeling which sprang up in the course of the eighteenth century... Romanticism took pride in its own contradictions: it embraced free thought on the one hand, and religious mysticism on the other (Lucie-Smith 373).

Romanticism was the heir to the spirit of the French Revolution, a spirit of freedom and self-determination manifested artistically as freedom of expression. It contrasts sharply with the controlled and ordered world of classicism in the Renaissance period, but it bears a relation to the mode of thought that created humanism and an emphasis on individual thought. Yet, mathematics still played a part in composition, in the balance of different elements in painting so that no one element would overpower all others.

Architecture in the nineteenth century was not stylistically unified. There was a nostalgia for the past which resulted in a revival of a number of styles, another link with the Renaissance period. Karl Freidrich Schinkel was the most important early Romantic architect. There was a Gothic revival in this era that was seen by critic John Ruskin as opposed to the evils of capitalism and as symbolic of the freedom allowed to medieval artisans (Lucie-Smith 375). Here again, mathematical principles remained a necessary element in architecture for both practical and aesthetic reasons.

In each era, aesthetic principles will differ, with some ideas discarded and others newly accepted. There are rarely major breaks with the past, and even those shifts that seem to be major breaks depend on the existing aesthetic even if only as a counter for comparison. New aesthetics indicate not only a change in tastes but also in economics, social relations, politics, and other societal forces. There have been many such shifts in aesthetic viewpoint in this century, each redefining art in terms of the existing social relations and prevalent aesthetic ideas. Proponents of these new aesthetics often claim to have discovered what art really is or should be, but in truth art is always in the act of becoming, leaving behind examples of each style and aesthetic on the continuum. Art involves notions of the self, and differing perspectives on the self are seen in works by an abstract expressionist such as Jackson Pollock and a post-modernist like Andy Warhol.

More recently, artists have started using mathematics more consciously as they find ways to utilize computer technology in their designs:

Computers are… [END OF PREVIEW]

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