The mysterious powers of mathematics and its application to art were favourite subjects in the court of Milan when Luca Pacioli and Leonardo da Vinci were there in the late 1490s – and they were the subject of Luca Pacioli’s next book. Called De divina proportione (The Divine Proportion), it explores the mathematics of the divine proportion (or golden ratio) and its relation to the heavenly spheres.
Published in Venice in 1509, De divina proportione was Pacioli’s most successful book. It was also his most mystical. The ‘divine proportion’ of the title – which clearly indicates Pacioli’s supernatural interests – is better known today as phi. This ratio results when a line is divided so that the short portion is to the longer portion as the longer portion is to the whole, and is expressed numerically as 1.61830339887…
Pacioli calls the golden ratio ‘essential, ineffable, awesome and invaluable’. In De divinahe explores its relation to the five regular (or Platonic) solids: the cube, tetrahedron and octahedron (used by the Pythagoreans and probably learnt from the Egyptians), and the icosahedron and dodecahedron (developed by the Pythagoreans). The Pythagoreans had assigned each figure to an element: the cube to earth; the tetrahedron to fire; the octahedron to air; the icosahedron to water; and the dodecahedron to heavenly ether. These solids had been revered by philosophers and mathematicians since the ancient Greeks, and in the Renaissance were seen as the ‘supreme expression of the majesty of geometry’.
On the first page of De divina, Pacioli declares his desire to reveal to artists the secret of harmonic forms through the use of the divine proportion, calling his book:
‘A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.’
Pacioli says he’s included in his book all the material forms of geometric bodies which have ‘hitherto been unknown to the living’. This is true. These geometric bodies – a collection of regular and semi-regular solids – had never before been visually represented. But Pacioli was fortunate to have the ‘ineffable left hand’ of Leonardo da Vinci at his disposal – and Leonardo made a set of 60 3-dimensional illustrations which appear in two surviving manuscripts of De divina and in the printed edition of 1509.
The drawings show Leonardo’s extraordinary spatial imagination. To represent the 3-D figures in 2-D space, Leonardo devised a way of drawing them in perspective, systematically shaded as if they’re real objects, rather than geometrical diagrams, and invented a method of showing their spatial configurations in skeletal form.
Pacioli acknowledges Leonardo’s contribution with the following praise: ‘the most excellent painter in perspective, architect, musician, the man endowed with all virtues, Leonardo da Vinci who deduced and elaborated a series of diagrams of regular solids’.
De divina contains three texts: ‘Compendio de divina proportione’ (Compendium of divine proportion), ‘Tractato del’ architectura’ (Treatise on architecture) and ‘Libellus in tres partiales tractatus divisus quinque corporum regularium et dependentium’ (Treatise on the five regular bodies).
The first volume includes a detailed summary of the properties of the golden ratio and a study of the Platonic solids and other polyhedra. In the fifth chapter Pacioli discusses the divinity of numbers and explains why he’s called his book ‘divine proportion’ (the golden ratio had previously been known as ‘extreme and mean ratio’ or ‘proportion having a mean and two extremes’). Why does Pacioli call this proportion ‘divine’? Because ‘we expect in this most useful discourse God himself to come’. He gives five reasons for why he’s chosen to rename the extreme and mean ratio the divine proportion:
1. ‘That it is one and only one and not more’. That is, there’s only one value for the divine proportion and only one Christian God.
2. The geometric expression of divine proportion involves three lengths and God also contains three (the Holy Trinity of Father, Son and Holy Ghost).
3. Just as God can’t be properly defined nor understood through words, so the divine proportion can’t be designated by any intelligible number nor by any rational quantity, ‘but always remains concealed and secret, and is called irrational by mathematicians’.
4. The omnipresence and invariability of God is like the self-similarity associated with the divine proportion: its value is always the same and does not depend on the length of the line being divided or the size of the pentagon in which ratios of lengths are calculated.
5. Pacioli proposes a fifth, esoteric, quality shared by the divine proportion and the Christian God, which he derives from Plato: ‘As God has conferred being to heavenly virtue as a fifth substance, and by means of this fifth substance has extended being to the other four simple bodies or four elements (earth, water, air and fire) and through these to every other thing in nature, so in our divine proportion, following the ancient Plato in hisTimaeus, we give formal being to Heaven itself by creating for it the body called the dodecahedron or the body of twelve pentagons.’
By the Middle Ages the five regular solids were relatively well known among educated Europeans because of their metaphysical significance. Euclid had demonstrated how these five solids could be constructed geometrically. Although the resulting figures were highly abstract, they did introduce into the western mathematical tradition the challenge of representing 3-dimensional figures. The work of Fibonacci continued this tradition and was picked up by Pacioli’s confrere, the painter and mathematician Piero della Francesca.
By representing the solids in three dimensions in De divina, Leonardo transformed the problem into a 3-dimensional one, an approach later taken by Durer. As Kim H. Veltman says: ‘By transforming the treatment of the regular and semi-regular polyhedrons from a two-dimensional construction problem to a three-dimensional perspectival challenge, Leonardo initiated a programme for translating the whole of Euclidean geometry into three-dimensional terms.’
And for the first time, in De divina, his colleague Luca Pacioli demonstrated the significance of these polyhedra beyond their metaphysical and geometric importance: he showed them to be the building blocks of the everyday world. For example, he discusses their application to architecture: the 26-sided rhombicuboctahedron and its stellated truncated form with 72 sides were used in the construction of the Pantheon in Rome and the Santa Maria delle Grazie in Milan.
Scientists would later discover that the presence of these forms in the everyday world goes even deeper than their use in art and architecture, to the structure of matter itself, eg the tetrahedron form of silicates and other polyhedral forms of fluorides, garnets, cuprite, etc.