From cubism to icosahedrism

Among the many scientific breakthroughs which have brought solid state physics to be considered as a distinct field of physics, with its own concepts and methods, the most important one was probably the discovery, by Max Von Laue, Walter Friedrich and Paul Knipping in 1912, of the diffraction of X-rays by a crystal. It proved that crystals were made of a periodic array of atoms or molecules, an idea which was already supported by the work of abbé René-Juste Haüy at the end of the eighteenth century. Indeed, the latter proposed a periodic microscopic structure for crystals, based on the observation of the regular facets of crystal grains at a macroscopic level.

The mathematicians of the nineteenth century contributed to this story by inventing a very important tool, the concept of a transformation group, which would prove useful in almost every field of physics. As far as the groups of space symmetry operations are concerned, the complete classification of the spatial groups was fulfilled by the end of the nineteenth century for the three-dimensional Euclidean space. A very important result which follows this classification is the crucial restriction on the compatibility between rotations and translations: only rotations of order 2, 3, 4, 6 can let a crystal be invariant. An important ingredient in the description of an ordered structure is its point group, which enumerates the symmetry operations which leave a point fixed.