A journalist asked me this week if there was a ‘Eureka!’ moment while I was researching Double Entry. I said no, there wasn’t a Eureka moment, there were many Eureka moments. And several led me down sidetracks so far from my main story that they either didn’t make it into the finished book at all or have become asides.
One of these asides is the story behind the portrait of Luca Pacioli, which I discovered in ‘The Portrait of Fra Luca Pacioli‘, an article by English mathematician Nick Mackinnon published in 1993. Here Mackinnon writes about using Pacioli’s mathematics and the mathematics of the portrait in an attempt to unravel the portrait’s many mysteries. Mackinnon’s investigation was also filled with Eureka moments. He writes: ‘I have tried in this article to preserve some of the excitement of the chase, at the cost, perhaps, of a systematic presentation.’
In the portrait above, painted after he’d become a celebrity following the success of his Summa de arithmetica, geometria, proportione et proportionalita, Fra Luca Pacioli is depicted in grey Franciscan robes demonstrating a mathematical problem to a young man. Pacioli points to a diagram drawn in chalk on a slate, the edge of which is marked ‘EVCLIDES’. To his left is a large book, probably the Summa, which supports a wooden dodecahedron, one of the five Platonic solids which were revered in the Renaissance. Hanging at the top left of the painting is a crystal rhombicuboctahedron (as it was later named by Kepler), one of the 13 semi-regular solids discovered by Archimedes. All its faces (18 squares and 8 equilateral triangles) are simultaneously visible and some reflect buildings, probably the ducal palace of Urbino.
Who painted the portrait, the exquisite precision of the rhombicuboctahedron (especially compared to the crude representation of the much simpler-to-paint dodecahedron) and the identity of the young man beside Pacioli have long been the subject of debate by historians. The portrait, now in Naples’ Galleria Nazionale di Capodimonte, was dedicated to Pacioli’s patron and probable student Guidobaldo da Montefeltro, Duke of Urbino, and so it’s always been assumed that the young man in the portrait was Guidobaldo.
But as Mackinnnon points out, no amount of documentation can make ‘the handsome ambitious red-head of the Pacioli portrait into the gifted but sickly Guidobaldo’. Mackinnon was led to the ‘astonishing answer’ to the question of the young man’s identity by a remark made in Pacioli’s next book, De divina proportione (‘The divine proportion’). This led him to Pacioli’s unpublished manuscript on magic and mathematical games, De viribus quantitatis (‘On the powers of numbers’), which has languished in the Bologna University library for 500 years. De viribus includes a section on magic squares and astrology – and magic squares made Mackinnon think immediately of Albrecht Durer’s famous 1514 engraving Melencolia I, which also contains a magic square, the first 4 x 4 magic square to appear in print in Europe. (Magic squares were new on the scene and Durer’s magic square was only the third to appear in Europe, the other two being Pacioli’s and one in a manuscript now in Cracow, Jagiellonian MS 753.)
And so speculating that Durer might somehow be connected to Pacioli and therefore possibly be the young man in the portrait, Mackinnon compared a black and white portrait of Durer with the man in Pacioli’s portrait. He found they shared a similarly confident manner. He then spent another week ‘on tenterhooks’ trying to find out the colour of Durer’s hair. He discovered that Durer’s hair was ‘ginger’, like the young man’s in the portrait. And so, Mackinnon’s Eureka moment.
Mackinnon then set off in pursuit of Durer. He discovered that Durer was in Venice in 1495, the year given on Pacioli’s portrait (which is signed JACO. BAR. VIGENNIS. P. 1495), and that Durer knew the artist who probably painted it, Jacopo de’ Barbari.
And so Mackinnon supports the majority view that it was Barbari who painted the portrait – but then makes a brand new claim for the nature of its subject. Mackinnon believes the portrait shows Pacioli teaching a real maths lesson. That here the Franciscan monk is teaching the young German painter, Durer, the geometry of ancient Greece; he is expositing Book XIV.8 of Euclid’s Elements.
Mackinnon thus argues that Pacioli’s portrait records ‘the most important meeting of the Renaissance’: the meeting of Luca Pacioli and Albrecht Durer. The young German artist would go on to become ‘one of the most original geometers in Europe’ and one of the greatest artist-mathematicians of his age.
And so Mackinnon claims that this ‘under-regarded’ Portrait of Fra Luca Pacioli captures ‘one of the greatest moments of the Renaissance, the transmission to Albrecht Durer, and hence to the world north of the Alps, of the geometry of Ancient Greece and the basis of the new art of Italy.’
Needless to say I was incredibly excited to discover this possible link between Luca Pacioli and Albrecht Durer, because it meant that Pacioli, the central character of Double Entry, was directly connected (through his knowledge of the mathematics of the new perspective painting) to all three of the great Renaissance artist-mathematicians and perspective painters: Albrecht Durer, Piero della Francesca and Leonardo da Vinci. And therefore that Pacioli held a pivotal position not just in commercial history and the history of capitalism, but also in the history of European art.
The other mystery that Mackinnon attempts to solve is the riddle of the exquisitely drawn glass rhombicuboctahedron, which seems to have been painted by a different hand from the one which painted the awkward dodecahedron.
Mackinnon argues that the two polyhedra were indeed painted by different hands. He believes there was ‘only one man in Europe capable of painting this object, and he is not the author of the rest of the painting!’ According to Mackinnon, in the masterly execution of the rhombicuboctahedron – which is half-filled with water and yet all its refractions and reflections are precisely captured – ‘we surely see the ineffable left hand of Leonardo da Vinci’.
Mackinnon’s argument is persuasive – especially as Leonardo would soon collaborate with Pacioli on his next project, for which Leonardo would draw a series of extraordinary illustrations of polyhedra (which will be the subject of my next Double Entry post).
For lovers of arcane Renaissance symbolism, Mackinnon also suggests a connection between the water-filled rhombicuboctahedron in Pacioli’s portrait and the egg suspended over the Virgin Mary in Piero della Francesca’s famous Montefeltro altarpiece now in the Brera in Milan. The altarpiece was commissioned by Guidobaldo’s father Federico da Montefeltro, Duke of Urbino, to celebrate the birth of his son. (By chance it’s said to include a portrait of Luca Pacioli as Saint Peter Martyr, standing second from the right with St Peter Martyr’s distinctive head wound.) Mackinnon applies Kenneth Clark’s interpretation of the double significance of the egg – as a symbol not only of rebirth and the resurrection, but in medieval iconography as a symbol of the four elements – to the rhombicuboctahedron in the portrait.
His argument stems from the chemical theory of Plato’s Timaeus, the only Platonic dialogue known in medieval Europe, which sets out Plato’s cosmogony based on Pythagoras. According to Plato, the elements fire, earth, air and water are modelled respectively by the tetrahedron, cube, octahedron and icosahedron. The rhombicuboctahedron, with its square and triangular faces, therefore represents all four polyhedra and evokes all four elements.
According to Mackinnon, by filling it with water Leonardo (assuming he painted it) made the symbol more concrete: ‘for now the elements are represented physically too: earth by the glass, water and air contained in the glass, and fire by the bright reflections’. The fifth solid is the dodecahedron, which for Plato represented the universe. So not only does Pacioli’s portrait refer to the elements of geometry (the Platonic solids), but also to the four elements of nature. And therefore, says Mackinnon, by painting a water-filled rhombicuboctahedron the artist has ‘contrived a brilliant double reference to the Timaeus‘.