Visual Art and Mathematics: Comments on the Meaning of Order

Leonardo, Vol. 15, No.1, pp. 65-67, 1982 Printed in Great Britain 0024-094X/82/010065-03$03.00/0 Pergamon Press Ltd. STATEMENTS ON THE RELATIONSHIPS BETWEEN THE NATURAL SCIENCES AND THE VISUAL FINE ARTS AND, IN PARTICULAR, ON THE MEANING OF ORDER (PART IV) This is the fourth group ofcontributions to the discussion ofthe above subjects obtained through the efforts of Giorgio Careri, the Italian physicist. and the Founder-Editor of Leonardo. The contributions are from invited visual artists. scientists and scholars of visual art. VISUAL ART AND MATHEMATICS: COMMENTS ON THE MEANING OF ORDER In a recent manuscript [I] I wrote: 'It can be said that for thousands of years there has been an interest in geometrical plane shapes and solid forms, in particular those possessing regular features of proportion and of syn try'. An important meaning of order in the visual arts and in mathematics is regularity, in particular geometric regularity. As G. Careri states: 'In the visual arts the idea of order is sometimes applied to stress certain compositional geometrical ratios and some color relationships ' [2]. There is a precise definition of order in mathematics: A set P is said to be totally ordered by the binary relation ~ if: (I) every pair a, b ofP satisfies either a ~ b or b ~a; (2) a ~ band b ~ c implies a ~ c; (3)a ~a for every a in P and (4) a ~ band b ~ a implies a = b. (For a discussion of a different meaning of order in mathematics see, for example, Refs. 3 and 4.) Many geometrical surfaces are interesting from the viewpoint of mathematical order, for example the Moebius band, which has the property of being a nonorientable surface [5-7]. I realize that this property is not of interest to some artists who have depicted the band. As I have pointed out, Max Bill made his first sculpture of this type without being aware of the work of Moebius [6, p. 108]. Bill was attracted to this type of surface because he said 'it demonstrates the possibility of developing surfaces that lead to forms proclaiming the existence of aesthetic reality' [6, p. 109]. The mathematician Herman Weyl said: 'Symmetry is a vast subject, important to art and to nature. Mathematics is its root, and it would be difficult to find a better field in which to demonstrate the working of the mathematical intellect [8]. The arrangement of symmetric patterns (for example in mosaics) has long been practised. The discovery in 1928 of the 17 bidimensional groups of symmetry is credited to G. Polya [9, 10]. Ignorance ofthe number of these groups did not inhibit, for example, the 65 surprising results contained in the mosaics of The Alhambra mosque in Granada, Spain [II]. In 1978 Max Bill said: 'I am convinced it is possible to evolve a new form of art in which the artist's works could be founded to quite a substantial degree on a mathematical line of approach to its content .... Just as mathematics provides us with a primary method of cognition, ... , so, too, some of its basic elements will furnish us with laws to appraise the interactions of separate objects ... , since it is only a natural step from having perceived them to desiring to portrait them' [12] (in this regard, see also Ref. [13]). One should also be aware of the possible indirect influence of mathematical ideas, such as non-Euclidean geometry or the so-called geometry of the fourth dimension, on Cubism and Futurism [14, 15]. Some efforts have been made to apply mathematics to the aesthetics of visual fine art [16, 17]. For example, George D. Birkhoff [17] suggested the following for the quantification of beauty: 'The typical aesthetic experience can be regarded as compounded of three successive phases: (1) A preliminary effort of attention, which is necessary for the act of perception, and which increases in proportion to what we shall call the complexity (C) of the object; (2) the feeling of value or aesthetic measure (M) which rewards this effort; and finally (3) a realization that the object is characterized by a certain harmony, symmetry, or order (0), more or less concealed, which seems to...