Descartes ( rediscovered by Euler ) stated that for any polyhedron the following identy holds $$F-E+V=2.$$ Where $F$ is the number of faces, $E$ is the number of edges and $V$ is the number of vertices. A tetrahedron, for example has $4$ vertices, $4$ faces and $6$ edges: $4 - 6 + 4 = 2$. One of the most beautiful and fascinating mathematical theorems I know states that this formula implies that there are only five regular polyhedra. A theorem that doesn't involve any deep topology or geometry, given $F-E+V=2$ all it takes is some number theory, i.e. solving some Diophantine equations and modular congruences.
Du Sautoy demonstrates the 2D version of the formula above in this episode. He shows that in 2D there can only be three regular lattices created from a regular polyhedron. He visits a beehive and shows how bees create lattices based on perfect pentagons. He then calculates the amount of required wax for the three possibilities and concludes that the pentagon is the best solution. "Nature is lazy", he says and is obviously part of The Code. The fascinating fact here is that nature, in the form of the bee, "knows" this, and for thousands of years. The knowledge about the pentagon and the skill to create pentagons seems encoded in the bee lifeform. Then, what is nature? And what exactly is the role of mathematics? That seem to be the questions he is trying to answer.
From there on Du Sautoy shows us more regular polyhedra in nature. Like a virus in the shape of an icosahedron.
He visits a salt mine with perfectly cube shaped crystals and using a model of the molecule explains the creation of the cube.
If you look closer to the shapes however the creations are not mathematically perfect. Are we using mathematics merely to understand nature? Or is nature driven by mathematics? Fractals are introduced to explain tree like shapes. But although close mathematically perfect fractal-type-of trees don't exist either.
Next week episode 3.
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