Visual Art and Mathematics: Common Characteristics of Order

Leonardo. Vol. 14, No. 3. pp. 230-237. 1981. Printed in Great Britain. 0024-094XA I /030230-08$02.00/0 Pergarnon Press Ltd. STATEMENTS ON THE RELATIONSHIPS BETWEEN THE NATURAL SCIENCES AND THE VISUAL FINE ARTS AND, IN PARTICULAR, ON THE MEANING OF ORDER (PART II) This is the second Rroup of contributions 10 rhe discussion o f rhe above subjects obtained through the eflorr.7 of Giorgio Careri. the Italian physicist. and the Founder-Editor o f Leonardo. The conrriburions are .from invited visual artists. scienrisrs and scholars of visual arr. VISUAL ART AND MATHEMATICS: COMMON CHARACTERISTICS OF ORDER In October 1961, M. C. Escher (Holland, 1898-1972) made one of his best known lithographs, entitled ‘Waterfall’, that has endeared him to mathematicians [ I , 21. The picture presents by an ingenious method of perspective (the so-called Japanese perspective) a stream of water that rotates a waterwheel, and then the water impossibly returns to form a closed circuit against gravity. Also in the picture on top of the two towers are stellated polyhedra-much studied during the Renaissance. Furthermore, the vegetation depicted in the garden on the left is not of this world. The picture leads the eyes of viewers to regard it in a circular order. He also used a circular order in some of his other pictures. He also introduced other orders in his pictures, for example, the same depicted thing that becomes progressively smaller, floors with arrays of patterned tiles that cross in different kinds of order, and arrays of angels that are transformed into devils and pigeons into black snakes and vice versa. Another kind of order involves the properties of the Moebius strip [3-51. Visual artists, for example Max Bill (Switzerland), Escher, B. Munari (Italy) and others, have introduced the strip or band into their artworks. The basic property of the Moebius band is that any two points on its surface can be connected by a line without crossing an edge of its surface. Evidently this cannot be done on a simple closed surface, such as a ring, if a point is on one of its two surfaces and another point is on its other surface. Furthermore, on a Moebius band all the points on the edges of its surface can be ordered in a cyclical way. To explain the geometrical properties of the Moebius band, some have applied non-Euclidean geometry and others have spoken in terms of psychic visions. However, it is possible to explain the band’s properties in terms of several definitions of mathematical order. It is known from axiomatic models of contemporary geometry that, starting with the notion of to be in between in a three-way relationship between a ternate of points on a line, one can arrive at the definition of binary order in the intuitive sense of the meaning of order in terms of that which cornes.first and that which comes after. If one makes the statement that a point B is between points A and C, intuitively it follows that one can say A comes first, B comes after A. and C comes after B. But this intuition is correct only when the points A. B and C are on a straight line. If the points A and C are on a circle, then a point B can be placed between them on two different portions of the circle. In such a case it becomes a complex, refined play to define binary order. On a closed line one runs the risk of proving that B comes after B. On the Moebius band it is possible to define binary order, but one encounters the same problem as for a closed line. One encounters other problems of the meaning of order in other situations. For example, one can order mozaic tiles on a plane surface in horizontal and vertical arrays even if the number of tiles is infinite. But in one of Escher’s works in which a depicted thing is repeated and becomes smaller and smaller, there are cyclical orders each of which is closed on itself to become a world in itself, which involves an infinity of an infinite number of tiles...