**Of an unusual geometric structure in a XIII century fresco re-discovered in Ferrara**

A fifteenth century manuscript was exhibited at the Medici Riccardi building on 30 January 2007. It had been discovered there in 2005 by the English historian James Banker, and contained a treatise on Archimedes related to spirals and included written text and geometric designs. The inestimable value of the work lies in the fact that the entire text was acknowledged as having been written by Piero della Francesca, the great artist and mathematician. I began to examine the mathematical structure of some of the curves on the copies that are now available. These curves recur in all of his most famous works. In most cases the lines were identified as exponential in accordance with the interpolation equation; but there were also Gaussian and spiral lines found too. I should make it quite clear that I do not intend to assume that Piero knew the formulas to these equations; I just used this data for its "objective and quantitative character" like "a signature of the artist". He also makes frequent use of rectangles, often to frame figures that he considered important. These are the famous golden rectangles, the most famous discovery of the Pythagorean school of Metaponto. The ratio between the longer side and shorter side of these figures is 1.61808, known as the Greek letter phi, one of the most interesting numbers in mathematics. The clearest example of the golden figure is the scroll that the minor prophet holds in his hand in a fresco painted by Piero della Francesca in the basilica of San Francesco in Arezzo. The June 2008 edition of "Ferrara. Voci di una città" published the story of a fresco which had originated in the old Santa Caterina church in Ferrara, dating back to the XIII century. The fresco was of a Crucifixion and had been detached in 1935-1936. Twenty years later it was acquired by the Casa Romei. A large piece was missing form the original fresco, showing Our Lady fainting as she was held by two holy women. This missing fragment now forms part of a private collection. A copy of the rediscovered fragment was added to the original, so you can now see the entire painting at Casa Romei. A careful mathematical examination of the fresco clearly revealed that the unknown "artist of the re-discovered fresco" had managed to design the three haloes of the holy women in accordance with a geometric-mathematical layout imitating the golden ratio. The number 2 also features, a number favoured by Pythagoras. This type of arrangement would definitely not have occurred by chance. We will never know if the artist applied the phi principles of his own accord, or if he executed the fresco in accordance with the instructions of another(s). The other geometric designs confirm the mathematical precision of the arrangement as they were designed on the basis of the lengths of the sides and angles of the triangle. In addition, the fact that the ratio of two of the triangle sides are so close to the golden ratio is significant; this value was often used in the works of Piero della Francesca; however it was only present in rectangular figures in accordance with my previous research. The ratio between the side and diagonal of this polygon equals phi, as Piero della Francesca writes in his two mathematical works, the Liber Abaci and the Libellus de Quinque corporibus regularibus. In conclusion, if we believe that the re-discovered fresco had been created in accordance with a golden ratio inspired project, it would mean that this was the first reappearance of a geometrical figure based on phi since the classical period. This reappearance would have been made in the XIII century, a good two centuries before the frescos of Piero della Francesca in the basilica of San Francesco in Arezzo.