Kepler sought patterns and symmetry in the laws of nature. In 1611 he wrote a booklet, De Niva Sexangular (The Six-Cornered Snowflake), in which he attempted to explain the structure of familiar symmetrical objects. Almost 300 years before the existence of atoms was definitively established, he concluded that the symmetrical shape of crystals is due to the regular arrangement of the atoms of which they are formed. He also investigated the structure of geometrical objects such as the Platonic solids and the regular stellated polyhedra, known today as the Kepler–Poinsot polyhedra. Like Kepler, today’s theoretical physicists are seeking patterns and symmetries that explain the universe. According to string theorists, the universe includes six extra hidden spatial dimensions, forming a shape known as a Calabi–Yau manifold. No-one knows whether string theory will revolutionize physics like Kepler’s brilliant insights, or whether it will turn out to be a red herring.
Samelson products in p-regular exceptional Lie groups