On the Application of the Golden Ratio in the Visual Arts

Leonardo, Vol. 14, pp. 31-32 Pergamon Press, 1981. Printed in Great Britain. ON THE APPLICATION OF THE GOLDEN RATIO IN THE VISUAL ARTS Roger Fischler* The Golden Ratio or Section or Mean for a line divided into a shorter length a and a longer length b is determined by alb = bl(a + b). The value ofthe Ratio is the irrational number (v' 5-1)/2 = 0.618.... A Golden Rectangle is one in which the Ratio applies to its sides of length a and b. The ratios of the successive Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34, ... oscillate rapidly to the limiting value 0.618, the Golden Ratio [1]. There are two aspects of the Golden Ratio that are of interest to contemporary artists: (1) whether it should be used as a theoretical basis for their own works of art and (2) whether works from another era were designed with the Golden Ratio taken as a theoretical basis. In this Note I examine these two aspects and give some examples. In his Entretiens sur l'architecture (1863), Violet-leDue explained why a visual artist or a design architect might divide a line in the ratio 5/8 or 0.625, which is a rational number and which is close to the Golden Ratio [2]. Despite his book being well-known in the Occident , a '5/8-ratio' school has not developed among either visual artists or design architects. In 1921, R. Carpenter [3] compared two analyses of the proportions of an ancient Greek lecythus vase. One analysis involved J. Hambidge's 'dynamic symmetry' system [4], which includes, as a special case, the Golden Ratio (and, therefore, the irrational-number division of aline). The other analysis was 'static', that is, it involved rational-number divisions of a line. Carpenter concluded that 'practically speaking, this lecythus ["based on a static system"] would be indistinguishable from that constructed dynamically ... and if it were drawn on paper and subjected to the same analysis of squares and diagonals, all the geometry would be the same' [3, p. 33]. In 1921, E. Monod-Herzen pointed out that the ratio 2/rr = 0.637 ..., an irrational number, is also close to the Golden Ratio and that furthermore n: is used, for example, in mathematical analyses of wave motion [5]. Still, a '2/rr - ratio' school has not appeared among visual artists and design architects. At least eight different hypotheses have been proposed to explain the form of the Great Pyramid of Cheops in Egypt [6]. Two of these involve the Golden Ratio irrational number. One of the two gives an excellent agreement with actual measurements, but it is based on a quotation that does not exist [7]. There is, however, a simple hypothesis based on rational numbers that gives just as good an agreement and, furthermore , this hypothesis is supported by archeological and textual evidence [6, 7]. •Mathematician, Dept. of Mathematics, Carlton University , Ottawa K1S 5B6, Canada. (Received 30 Aug. 1979) 31 It is often stated that Paccioli in his Divine Proportione (1509) advocated the use of the Golden Ratio. In fact, while in the Divina Proportione proper, he praised highly the mathematical properties of the Golden Ratio, in the accompanying Architectura, which deals with design and proportions, he advocated a classical Vitruvian system, that is a system based on simple proportions [8]. On the basis of measurements, it has been stated that Seurat used Golden Ratio divisions as a basis for his paintings. However, a detailed analysis of his writings, sketches and paintings shows that this was not the case [9]. An analysis was made of the works of the cubist Juan Gris using the diagonal of a Golden Rectangle and the fit seemed to be rather close. However, there is still in existence a letter written by Gris in which he categorically states that he did not use the Golden Ratio to proportion his paintings [10]. Le Corbusier used the Golden Ratio in his Modulor system and several authors have stated that he had used it in paintings in his early 'Purist' period [11]. But, Le Corbusier's (or rather Jeanneret, as he was known at that time) own writings...