The Cosmology of the Divine Comedy

Divina Commedia: Online at Columbia University.

If you think poetry and maths are poles apart, think again. Around the sixth century, Indian poet and mathematician Virahanka codified the structure of Sanskrit poetry, formulating rules for the patterns of long and short syllables. In this process, a sequence emerged in which each term is the sum of the preceding two. This is precisely the sequence studied centuries later by Leonardo Bonacci of Pisa, which we now call the Fibonacci sequence [TM241 or search for “thatsmaths” at].

Just as for music and maths, structure is the factor linking mathematics and poetry. One of the most fascinating examples of this interplay of ideas is Dante Alighieri’s Divina Commedia. The Divine Comedy is divided into three parts, or cantiche, evolving from the consequences of sin (Inferno) to the Christian life (Purgatorio) and the soul’s ascent to God (Paradiso). Each cantica comprises 33 cantos and an introductory canto brings the total number to 100.

The cantos in The Divine Comedy average 142 lines, so the poem is about 14,200 lines in length. The lines are divided into tercets, or groups of three, each line having eleven syllables, so that the number of syllables in each tercet is 33, the same as the number of cantos in each cantica. The tercets have an intriguing rhyming pattern, aba, bcb, cdc, ded, … forming an interlocking chain in which each triplet is linked rhythmically to those immediately before and after it.

Dante’s Cosmology

The Empyrean, from the illustrations to The Divine Comedy by Gustave Doré, 1868.

But the most amazing mathematical aspect of The Divine Comedy is Dante’s conception of the universe, what we might now call Dante’s cosmology. He imagines the material universe following the model of Aristotle, with nine spheres. The Earth is central, surrounded by spheres for the Moon, Sun, planets and fixed stars. The spheres rotate about the Earth at different rates. Surrounding them is the sphere of the Prime Mover. The entire model of Aristotle can be contained in one vast sphere. But what lies outside it? Dante also conceived a model of the spiritual universe, with nine choirs of angels in spheres surrounding God. But he  then asks how can the two universes “go together”? How can they be one?

In canto 28 of Paradiso, his guide Beatrice provides a rational answer to the question. Beatrice, representing divine revelation, guides Dante from Purgatory into Paradise. As they ascent through the nine concentric spheres to the outermost limits of the material universe, Dante and Beatrice gaze upward to another vast realm, with the nine angelic spheres surrounding a glorious light. From their vantage point, they see that the material and spiritual domains are, in fact, a single universe.

Schematic diagram of Dante’s Universe. The material domain (left) is linked directly to the spiritual realm (right) [Image from Peterson (1979)].

The 3-Sphere

Dante showed his extraordinary genius in imagining this merging of two spheres into what we now call a 3-sphere. On Earth, we know that if we travel in one unchanging direction we arrive back at the same point. We can visualise the curved surface as two flat maps, one for each hemisphere, joined smoothly at the equator. In a similar manner, we can (with some effort) visualise the 3-sphere by considering two solid globes and identifying corresponding points on the surface of each. Passing through the surface of one, we enter the other in a seamless manner.

In a 1917 paper of Albert Einstein, the 3-sphere formed the basis of a cosmological model which was finite but unbounded. Travelling without change of direction, we ultimately arrive back at our starting point. We still do not know the shape, or topology, of the universe, but the cosmology first imagined by Dante may yet turn out to be scientifically relevant.




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