A ti, maravillosa disciplina,
media, extrema razón de la hermosura,
que claramente acata la clausura
viva en la malla de tu ley divina.
A ti, cárcel feliz de la retina,
áurea sección, celeste cuadratura,
misteriosa fontana de mesura
que el Universo armónico origina.
A ti, mar de los sueños, angulares,
flor de las cinco formas regulares,
dodecaedro azul, arco sonoro.
Luces por alas un compás ardiente.
Tu canto es una esfera transparente.
A ti, divina proporción de oro.
A la divina proporción, Rafael Alberti
O God, I could be bounded in a nut shell and count myself a king of infinite space, were it not that I have bad dreams. (act II, scene II)
William Shakespeare, Hamlet
In one of his most beautiful and perceptive conferences, Alain Badiou reminds us of the poet Fernando Pessoa when he says that Newton’s Binomial theorem is as beautiful as the Venus of Milo, the problem being that very few people realize this. This suggests that there is somebody who does. That person is the philosopher, destined therefore, to explore the relation between mathematics and art.
Arthur Koestler also established that timely encounter between objective knowledge and the creative act: Einstein’s space is no closer to reality than Van Gogh’s sky. The glory of science is not in a truth more absolute than the truth of Bach or Tolstoy, but in the act of creation itself. The scientist’s discoveries impose his own order on chaos, as the composer or painter imposes his; an order that always refers to limited aspects of reality, and is based on the observer’s frame of reference, which differs from period to period as a Rembrant nude differs from a nude by Manet. (The Act of Creation, London, 1970, p. 253)
And we could include Hardy when he says that: The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. (Cambridge University Press, 1994)
The first ones to appreciate and cultivate this encounter between rigour and beauty were the Pythagoreans. Approximately two thousand four hundred years ago they had already discovered the existence of the only five convex regular polyhedra: the tetrahedron (four triangular faces), the cube or hexahedron (six square faces), the octahedron (eight triangular faces), the dodecahedron (twelve pentagonal faces) and the icosahedron (twenty triangular faces).
In search of a rationale for the creation of the world, they, the Pythagoreans, had allotted each one of these polyhedra with an element of nature, thus finding harmony between the geometric bodies and the cosmic order. It was Plato who gave shape to this concept in one of his dialogues, Timaeus , considered to have had resounding influence throughout classical antiquity right up to the renaissance. In Raphael’s fresco, the “School of Athens”, Plato is depicted holding the Timaeus in his hands.
Three principles, which have been and are of great importance in the development of science, are expounded in this book: the union of science and beauty, the application of objects and mathematical rules known to entities or unknown processes (association of the polyhedra with cosmic or microscopic entities) and the construction of complexity starting from simple elements. Let’s remember the Platonic style …We must imagine all these things to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen. And the ratios of their numbers, motions and other properties, everywhere God, as far as necessity allowed or gave consent has exactly perfected, and harmonized in due proportion. (Timaeus, translated from Greek by Benjamin Jowett)
In accordance to this necessary order, the existence of four elements stems from the following reasoning: Things must have fire, because they can be seen, and earth, because they are material, and two things have need of a third to be joined together. If the universe were flat a third element would be enough, but as it has depth it needs a fourth one to complete that union. Thus, in order to unite fire and earth there is a need for two other elements: air and water. Analysing the properties of the elements and the proportion in which they must be in nature, Plato comes to the conclusion that the atoms of fire are tetrahedra, those of earth are cubes, those of air are octahedra and those of water are icosahedra. Only one combination remains, the dodecahedron, which Plato reserves for the intangible of the cosmos.
This concept whereby the sensitive world consists of numbers and geometric figures, followed a long, strange path. It started with the Pythagoreans, it reached the Renaissance and then slowly petered out during Modern times. But surprisingly it has returned to centre stage in face of two contemporary scientific events: the use of Boolean logic in the field of technology especially in the ever-growing field of digital technology - practically all the images that house our lives are nothing other than zeros and ones, and the discovery of the electronic microscope which made it possible for us to observe molecular structures which proved to be analogous to the various platonic solids such as the tetrahedron, the cube and the icosahedron: Nature takes on the shapes of regular polyhedra.
This peculiar “revenge” of Pythagorean thought has, however, a limit, a tension that needs to be pointed out. If there is a relation between science and art, it is not one of symmetry, complement or opposition but of supplementation. For Alain Badiou, in accordance to his developments in terms of the being and the event:
…mathematics is the transparent model of the relation between thought with what is. On the other hand, for any new event, for anything that arises, that appears, a new language is always needed. By definition an occurrence cannot be received in the existing language, a new form has to be created, which is like a welcome of what is to come. And this time, it is in the realm of art that we will find the creation of new forms and consequently, what we could call new names for that which is to come.
This edition of Aesthethika proposes to explore the scenario of what is and of what is to come, by means of articles that are central to this double movement. Its paradigmatic form may possibly be found in Andrés Jacob’s discoveries. Studying regular polyhedra, the art of stained glass leads him to build objects that end up surprising him when chance recovers, twenty-five centuries later, the epiphany of virtual icosahedra.
The two epigraphs that open this editorial account for that dialect between what is and what is to come: the divine proportion of the Pythagoreans, supplemented by Hamlet’s nightmares. Two forms of beauty: the one that proceeds from perfection and the one that emerges from fault.
As the collage we have prepared to head this edition, which begins with the Neolithic stones carved in the shape of rudimentary polyhedra, followed by the prodigal drawings of Leonardo Da Vinci and Kepler’s designs, to finally lead us to the astonishing visions of the microscope –from elemental molecules, the fullerene, the icosahedral bull, the braids of DNA and the virus rings to the Lacanian Borromean knots that illustrate Maria Elena Domínguez’s article.
The divine proportion of a theorem and the trauma that carves in marble the body of a woman, are indeed two forms of beauty. So Newton’s binomial theorem can be as beautiful as the Venus of Milo
