Leonardo

INTRODUCTION VISUAL MATHEMATICS: ANEW ERA? On 12 October 1492, after a long trip, Christopher Columbus landed on an island of the Americas. On that same day, Italian Renaissance artist Piero della Francesca died. Giorgio Vasari (1511-1574) wrote the following about della Francesca in The Lives of theArtist:"He was regarded as a great master of the problems of regular bodies, both arithmetical and geometrical, but he was prevented by the blindness that overtook him in his old age, and then by death, from making known his brilliant researches and the many books he had written, which are still preserved in Borgo San Sepolcro, his native town" [1]. George Bull, editor of the English edition ofVasari's book, adds in a note: "Piero based the relationships in his painting on the laws of Euclidean geometry. The regular bodies were theoretically perfect forms, which, it was thought, could provide the artist with certain , or precisely measurable, relationships through which art could reveal and reproduce the order of nature" [2]. Three treatises by della Francesca have come down to us: Trattatod'abaco [3]; Libellusde quinque corporibus regularibus [4]; and De prospectiva pingendi [5]. As Margaret Davis wrote in 1977, even though it is universally agreed that geometry is a central element of della Francesca's painting, his mathematical treatises are known only superficially, if at all, to art historians. 'There exists no study of their contents, sources and significance that places them within the context ofPiero's own art and mathematics, as well as in the context of the art and mathematics ofPiero's time" [6]. Perhaps the situation will change in the near future; celebrations for the anniversary of della Francesca's death will include an exhibition on the mathematical aspects of his works and perhaps the first publication of the Latin version ofDeprospectiva pingendi. Morris Kline, in Mathematics in Western Culture, describes della Francesca as one of the greatest mathematicians of the fifteenth century: The artist who perfected the science of perspective was Piero della Francesca. This highly intellectual painter had a passion for geometry and planned all his works mathematically to the last detail. The placement ofeach figure was calculated so as to be correct in relation to other figures and to the organization of the painting as a whole. He even used geometry forms for parts of the body and objects of dress and he loved smooth curved surfaces and solidity. Piero's TheFlagellation [7] is a masterpiece ofperspective. The choice ofprincipal vanishing point and the accurate use of the principles of the focused system tie the characters in the rear of the courtyard to those in front, while the objects are all accommodated to the clearly delimited space. The diminuition of the black inlays on the marble floor is also precisely calculated. A drawing in Piero's book on perspective shows the immense labor which went into this painting. Here, as well as in other paintings, Piero used aerial perspective to enhance the impression ofdepth. The whole painting is so carefully planned that movement is sacrificed to unity ofdesign [8]. This fascinating description ofdella Francesca's painting is written by a mathematician ! TheFlagellation is one of the paintings I most love. Every summer I spend 2 weeks on the Adriatic Sea, in a small town called Senigallia, founded by the Senonian Gauls in circa 400 B.C. (One of the most famous of della Francesca's paintings is entitled The Madonna of Senigallia.) There I take advantage of the opportunity to take a short trip to the nearby town ofUrbino, birthplace ofRaphael, 19921SAST Pergamon Press ltd. Printed inGreat Britain. 0024-094X192 $5.00+0.00 LEONARDO, Vol. 25, No. 3/4, pp. 235-237, 1992 235 and look at The Flagellation inside the Galleria Nazionale delle Marche, located in the rooms of the Palazzo Ducale. It is a nearly universal opinion that all mathematical study by Renaissance artists deserves to be viewed from the vantage point of artists and mathematicians studying and learning together [9]. "Although the mathematicians often undeniably furnished the lead, this background is as revealing for artists who developed their mathematical skills independently as for those who were trained and encouraged to formulate mathematical...