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       From left to right: the tetrahedron, the hexahedron, the octahedron, the dodecahedron and the icosahedron.

      To this day, the polyhedra are usually named according to the number of their faces, written in Ancient Greek, followed by the suffix -hedron. For example, in geometry, the cube with its six square faces is referred to as the hexahedron. The tetrahedron, the octahedron, the dodecahedron and the icosahedron have four, eight, twelve and twenty faces, respectively. These five polyhedra subsequently became known as the Platonic solids.

      Platonic? Why are they not associated in name with Theaetetus? History is sometimes unfair, and the discoverers are not always those who receive the honours of posterity. The Athenian philosopher Plato had nothing to do with the discovery of the five solids, but he made them famous through a theory that associated them with the elements of the cosmos: fire is associated with the tetrahedron, earth with the hexahedron, air with the octahedron and water with the icosahedron. As for the dodecahedron, with its pentagonal faces, Plato claimed that it was the shape of the Universe. This theory was abandoned by science a long time ago, and yet these five regular polyhedra are still conventionally associated with Plato.

      As it happens, Theaetetus himself was not actually the first to discover these five solids. There exist considerably older sculpted models and written descriptions of them. For example, a collection of small sculpted stone balls reproducing the shapes of the Platonic solids came to light in Scotland and is said to date from one thousand years before Theaetetus. These pieces are currently conserved in the Ashmolean Museum in Oxford.

      So is Theaetetus no more worthy than Plato? Is he also an impostor? He is not a total impostor, for while the five figures were known before his time, he was the first to demonstrate clearly that the list was complete. There is no point in looking any further, Theaetetus tells us, no one will ever find any more. This assertion has something reassuring about it. It removes a dreadful doubt. Phew – they’re all present and correct!

      This step is emblematic of the way in which Greek mathematicians came to approach mathematics. For them, it was no longer a matter of simply finding solutions that worked. They wanted to exhaust the problem. They wanted to make sure that nothing escaped them. And for that reason, they developed the art of mathematical exploration to its peak.

      Let us now return to our Geode. Theaetetus’ proof is final: it is impossible for a polyhedron with several hundred faces to be perfectly regular. But then what do you do when you are an architect and you want to create a building that resembles a regular sphere as closely as possible? It is technically difficult to design the building in a single piece. No, there’s nothing you can do about that, and you have to assemble a multitude of small faces. But how do you create such a structure?

      We can imagine various solutions. One of these involves taking one of the Platonic solids and modifying it. Consider the icosahedron, for example. With its twenty triangular faces, it seems the roundest of the five. To smooth it out, each of its faces can be subdivided into several smaller faces. The polyhedron obtained in this way can then be deformed, as though one were inflating it by blowing into it, in order to come closer to a sphere.

      Here, for example, is what happens when you subdivide each face of the icosahedron into four smaller triangles.

       The icosahedron

       Inflated icosahedron with subdivided faces

       Icosahedron with faces subdivided into four

      In geometry, such a polyhedron is called a geode. Etymologically, this means a figure that is shaped like the Earth – that is to say, it resembles a sphere. There is nothing very complicated about this principle. It is precisely this construction that was used for the Geode of la Villette! The subdivision of the faces is, however, very much finer: the basic triangles of the icosahedron are here subdivided into 400 smaller triangles, making a total of eight thousand triangular facets.

      In reality, the Geode has rather fewer than 8,000 facets – only 6,433 – since it is not complete. Its base, which rests on the ground, is truncated and some triangles are missing. Nevertheless, this structure allows us to explain the presence of the twelve irregularities. The latter correspond simply to the twelve vertices of the basic icosahedron. In other words, these are the points where the original large triangles met in fives to form the points of the icosahedron. These vertices, which were originally pointed, have been flattened with the increase in the number of faces to the point of becoming almost invisible. However, their presence remains firmly fixed in the layout of the triangles, and the twelve irregularities are there to remind attentive passers-by of this.

      Theaetetus probably didn’t imagine that his studies would one day permit the construction of buildings such as the Geode. And the great power of mathematics, as in the kind that the scholars of ancient Greece

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