A certain ratio: Theodore Cook thought The Laughing Cavalier was based on mathematics.

Divine proportion and the Holy Grail

Everyone loves a conspiracy. Or so we read in Dan Brown's massive bestseller The Da Vinci Code. The conspiracy in this case involves the Priory of Sion, the secret society of which Leonardo was the reputed grand master between 1510 and 1519. The visual secrets include Leonardo's portrayal of St John in the Last Supper — young and somnambulant, as was customary — who is re-identified as Mary Magdalene, pregnant with Jesus's child. Mary, as divine mother, is herself the elusive Holy Grail. The linked properties of the ‘divine proportion’ and the Fibonacci series, beloved of mathematical mystics, provide clues in the decoding of the historical secrets.

Leonardo was certainly aware of the divine proportion (or golden section), illustrating the regular and semi-regular polyhedra in Luca Pacioli's manuscript of the same name in 1498. This section arises when a line is cut so that the ratio of the larger part to the smaller is equivalent to the ratio between the whole and the larger, approximately 1:1.618. The ratio between successive numbers in the Fibonacci series — 1,1,2,3,5,8,13,21,34... — approaches this ratio. There is no obvious problem in thinking that Leonardo knew the work of his namesake, the thirteenth-century mathematician Leonardo Fibonacci da Pisa.

The problem lies in granting a privileged significance to the ratio in the Renaissance, or in any period before the later nineteenth century. Incommensurable ratios were regarded as special, but, in Renaissance design, the ratio 1:√(2), based on Pythagoras' theorem, was used far more widely, not least because it was a simple construction based on the diagonal of a square. It features in architectural planning and occasionally to determine the dimensions of paintings.

An ardent advocate of the ubiquity of the golden section in art and nature was the English writer and Olympic swordsman Theodore Andrea Cook. In his Spirals in Nature and Art (1903) and The Curves of Life (1914), he adduced instances of the ‘phi’ ratio, which he named after the legendary Greek sculptor Phidias, in such natural configurations as shells, spiral horns and phyllotaxis. He recognized comparable formations in art and artefacts from across the centuries and around the globe. Such was his faith in the ratio that he used it to disclose the aesthetic secrets of some unlikely candidates, including The Laughing Cavalier by Frans Hals, reproduced here from Cook's 1914 book.

Some artists in the early twentieth century did design their pictures in this way, in particular the Spanish Cubist Juan Gris. But there is no evidence that Renaissance and Baroque artists used such surface geometry in constructing their paintings. Among the thousands of preparatory drawings and the growing number of underdrawings detected in paintings, not one reveals constructional methods based upon the ‘secret’ division of their surfaces.

There is quite a fashion for the drawing of fat lines on thin reproductions of Renaissance paintings. Draw enough lines on a small image — equilateral triangles, pentagrams and so on — and it is hard not to hit some ‘significant’ feature. The latest manifestation of this strategy is Bulent Atalay's surprisingly ahistorical Math and the Mona Lisa (Smithsonian, 2004), which is determined to credit Leonardo with Cook-like obsessions.

In the service of fiction, the adducing of mathematical secrets in the quest for the Holy Grail is fine if it works as story-telling. As serious history of image-making in the Renaissance it is nonsense. The problems with Brown's code are not the fantasies and anachronisms, but that his invented truth has been taken seriously by those who cannot recognize fiction.