Periodic arrangements of closely packed spheres

Abstract The symmetry aspects of close packing of hexagonal layers of equal spheres are usually presented with reference to the two basic patterns: AB… ( P 6 3 / mmc ) and ABC… ( Fm 3 m ), of which the latter is the fundament of the Kepler conjecture, published in 1611, stating that no arrangement of equal spheres can achieve higher density. In fact, however, there are infinitely many arrangements of N > 1 layers that, when repeated periodically, will give the same closest packing density of spheres in 3D. One such possibility for N = 4 was discovered by Barlow in 1883, and other authors gave this issue particular attention. The present paper systematically analyzes the symmetry of periodic patterns of hexagonal layers with growing N . It also presents the colorful history of this subject and introduces an ingenious system to describe the symmetry of close stacking of hexagonal layers of spheres.