29/05/2024

Folha: 'the discovery of Platonic solids'

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

As a teenager, one day I found a book at home, " Euclidean Geometry ," from my mother's time at teacher training college. Dona Isaura reacted without enthusiasm: "I hated it. It was just 'necessary condition' this, 'sufficient condition' that, I didn't understand anything!" She wasn't lucky enough to have good teachers, a pity.

Because Euclidean geometry is a gem, the best way to understand that combination of intuition, discovery, and rigor that only mathematics provides. But in my own experience as a student, I've seen geometry increasingly "forgotten" in the classroom, a trend that's difficult to reverse.

Among my readings in "Euclidean Geometry," the most memorable was the theorem that states that there are exactly five regular solids (polyhedra). I think I learned the concept at that very moment: a polyhedron is regular if all its faces are regular polygons – all sides and angles are equal to each other – identical, and the same number of faces always meet at each vertex.

I knew two or three: the cube (or hexahedron), which has six square faces, the tetrahedron, with four triangular faces, and perhaps the octahedron, with eight triangular faces. But I had never heard of the icosahedron, with 30 triangular faces, much less the spectacular dodecahedron, with its 12 pentagonal faces.

What impressed me most was learning that it's possible to determine, rigorously and definitively, how many of these marvelous objects exist.

The argument is as follows. At each vertex, there are three or more faces, whose internal angles can be 60º (triangles), 90º (squares), 108º (pentagons), etc. The sum of these angles must be less than 360º, a full turn. This only leaves the possibility of having three, four, or five triangular faces (tetrahedron, octahedron, icosahedron), three square faces (cube), or three pentagonal faces (dodecahedron) at each vertex.

Later, I learned that this theorem is Proposition 18 from Book 13 of Euclid's "Elements," and that the proof is by the Athenian mathematician Theaetetus (417 BC-369 BC).

In the philosophy of his contemporary Plato (427 BC-348 BC), the five solids – which we now call Platonic solids – occupy a prominent role: the tetrahedron, cube, octahedron, and icosahedron were responsible for generating the four classical elements: fire, earth, air, and water, while the dodecahedron "was used by the demiurge to organize the constellations in the firmament."

To read the full text, visit the newspaper's website.