05/06/2024

Folha: 'many viruses have the shape of regular solids'

Dodecaedro no piso da Basílica de São Marcos – Reprodução

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Proposition 18 of Book 13 of Euclid's "Elements" states that there are five regular solids (polyhedra), that is, whose faces are identical regular polygons with the same number of faces meeting at each vertex. Its proof is attributed to the Athenian Theaetetus (c. 417–369 BC). The five solids are called "Platonic" because they occupy a prominent position in the cosmology of the philosopher Plato (427–348 BC).

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The astronomer Johannes Kepler (1571–1630) also believed that Platonic solids have profound significance in the structure of the universe. In "Mysterium Cosmographicum," published in 1596, he proposed a model of the Solar System, with the six planets known at the time, based on the five solids with inscribed and circumscribed spheres.

However, in 1619, Kepler himself discovered two new regular solids, which were called large and small stellated dodecahedrons. Both were described in 1568 by the German Wenzel Jamnitzer (1507–1585), and one of them is represented in a mosaic in St. Mark's Basilica in Venice, dated 1430 and attributed to the Italian Paolo Uccello (1397–1475). But Kepler was the first to recognize them as regular solids.

The problem is that the proof of Proposition 18 uses a hypothesis that Euclid does not explicitly state: it assumes that the solid is convex, that is, that its surface has no indentations. The new regular solids found by Kepler are not convex.

Almost two centuries later, in 1809, the French mathematician Louis Poinsot (1777–1859) rediscovered Kepler's solids and found two more non-convex regular solids, which we now call the great dodecahedron and the great icosahedron. The saga was completed three years later by his compatriot Augustin-Louis Cauchy (1789–1857). He proved that there are exactly nine regular solids: the five convex Platonic solids and the four non-convex Kepler–Poinsot solids.

The extraordinary beauty of these objects results from their symmetries. For mathematicians, symmetries are transformations that leave the object unchanged.

Our face is (approximately) symmetrical because reflection in a mirror (almost) doesn't change it. The starfish, with its five points, is rotationally symmetrical: if we rotate the creature 72 ^° (360 ^° divided by 5), it remains identical. Regular polyhedra are those that have the maximum possible symmetry.

Symmetry plays a significant role in biology, as it allows the characteristics of a living being to be described by a shorter genetic code: it is no coincidence that several viruses—for example, HIV—have the shape of an icosahedron, octahedron, or dodecahedron.