Symmetry in the Arts

In a painting or a sculpture, a high degree of symmetry suggests formality and is difficult to combine with the suggestion of movement. While it was typical of, for example, statuary in ancient Egyptian art or medieval European religious painting, it is rarely found in Western art after the Renaissance. Symmetry, however, features largely in crafts. The forms and decorations of vases, utensils, and other useful objects, frequently exploit symmetric patterns. Symmetry is central to much architecture as well. Royal palaces, Gothic cathedrals, country houses, and large civic buildings were normally designed symmetrically, though since the 20th century asymmetry has been widely used in both large and small buildings.

There are countless areas in which the principles of symmetry play an important role. The different rhyme schemes in poetry represent different types of symmetry. For example, the rhyme scheme a b b a shows a kind of reflective symmetry in time, the rhymes of the last two lines being a mirror image of those of the first two. There are many types of rhyme scheme more complex than this, but symmetrical patterns can often be discovered in them. Musical forms reveal similar types of symmetry, too. On the shortest timescale, musical phrases in most Western music usually come in symmetrical pairs, usually two or four bars long, in which the second phrase seems to answer or complete the first. On a larger scale, one type of rondo has the form ABACABA, which again shows a time symmetry. Such musical palindromes are employed by composers ranging from Guillaume de Machaut, in his rondeau Ma Fin est Mon Commencement (My End is My Beginning), to Johann Sebastian Bach (for instance, in the mirror fugue of the gigue from the English Suite in D minor).
         One can also look at the symmetry of a single motif or ornament. Examples of such motifs (illustrated from patterns used in batiks) are shown below. Such ornaments typically have rotational symmetry and/or reflection symmetry.

A more complex pattern such as the one below can be built up from simple motifs. Such patterns have translational symmetry in one direction. Designs or patterns of this kind are known as strip, band, or frieze patterns.

The motifs used to make such frieze patterns may be isolated from one another or coalesce into a "continuous" geometric design along the strip. If a pattern has translations in two directions, then the pattern is often referred to as a wallpaper pattern.

Whereas an artist may choose to create a pattern with absolute and strict adherence in all details to have symmetry in the pattern, this is not all that common for "tribal" artists or artisans. Thus, if one looks carefully at a rug which at first view looks very symmetrical, it is common to see that at a more detailed level it is not quite totally symmetric either in the use of the design or of the colors used in different parts of the design. One can see the small liberties that are taken either because of the difficulty of making patterns exact by hand or because the artist wants consciously to make such small variations. In analyzing the symmetry of such a pattern it probably makes sense to idealize what the artist has done before applying some mathematical classification of the symmetry involved.

In the patterns shown above no color appears. We have a black design on a white background. However, in discussing the symmetry of a pattern one can study the symmetry involved if color is disregarded or by taking color into account. If you look at the batik below from a symmetry point of view you must idealize (model) what is going on to use mathematics. This batik is not infinite in either one or two directions. You must decide what colors have been used and what is the background color.

Many find it interesting to use mathematics to decide what symmetry pattern is involved for various interpretations of the whole or parts of a design. E. Fedorov (1859-1919) enumerated the seventeen 2-dimensional patterns in 1891 in a paper which did not receive wide attention because it was in Russian. P. Niggli (1888-1953) and G. Polya (1887-1985) developed the seven 1-dimensional and the seventeen 2-dimensional patterns in the 1920's; it was through this work that a mathematical approach to the analysis of symmetry patterns became more widely known. One extension of this work to color symmetry was accomplished by H. Woods in the 1930's. It turns out that there are 46 two-color types of patterns. Subsequently much work has been done with regard to studying symmetry in higher-dimensional spaces and using many colors. Recently Branko Grünbaum and Geoffrey Shephard, in a long series of joint papers and in their seminal book Tilings and Patterns, explored many extensions and facets of pattern, tilings, and their symmetries. In particular, Grünbaum and Shephard explored the interaction between symmetry and the use of a motif. This enabled them, for example, to develop a "finer" classification of the seven frieze patterns and seventeen wallpaper patterns. Unfortunately, this work is not as widely known as it should be.

Many people have been instrumental in disseminating mathematical knowledge of symmetry and pattern to scholars outside of mathematics as well as to the general public. One of the most influential and early books of this kind was Hermann Weyl (1885-1955)'s book Symmetry. Also noteworthy among these popularizers are Doris Schattschneider, Branko Grünbaum, Geoffrey Shephard, Marjorie Senechal, Michele Emmer, H. S. M. Coxeter, Dorothy Washburn (an anthropologist), Donald Crowe and Kim Williams. These individuals called attention to the use of symmetry as a tool for insight into various aspects of fabrics, ethnic designs and culture, architecture, and art, as well as to artists such as Escher whose work tantalizes people with a mathematical bent. 

Symmetry exists in architecture all around the world.  The best known example of this is the Taj Mahal.

Exeter Cathedral is built to a highly symmetrical plan. The west front has obvious left-right reflection symmetry, and so, less obviously, does the ground plan. The tower seen here, for example, is balanced by another that is not visible in this view.

 

There is an unusually high degree of symmetry in this 15th-century fresco by Perugino, Christ Giving the Keys to St Peter. The positions of the figures closely follow an imaginary grid with reflection symmetry, the left-hand and right-hand halves of the grid being mirror-images of each other.

SYMMETY IN UKRANIAN EGGS

ADVANCED DESIGNS

COMPARISON DESIGNS		GOOSE EGG DESIGNS