Mathematics and Art
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.
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.
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