T5.JPGThe sphere packing problem asks how to arrange congruent balls as densely as possible without overlap. The density is the fraction of space covered by the balls, and the problem is to find the maximal possible density. This problem plays an important role in geometry, number theory, and information theory.  Aside from the trivial case of one dimension, in 1892 the optimal density was previously known in two dimensions.  Then in 2005 for three dimensions by Thomas C. Hales and then in 2015 by the team consisting of Thomas Hales(Thomas Hales, Mark Adams, Gertrud Bauer, Dat Tat Dang, John Harrison, Truong Le Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Thang Tat Nguyen, Truong Quang Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, An Hoai Thi Ta, Trung Nam Tran, Diep Thi Trieu, Josef Urban, Ky Khac Vu, Roland Zumkeller).  Lattices are far more algebraically constrained, and it is widely believed that they do not achieve the optimal density in most dimensions.  By contrast, periodic packings at least come arbitrarily close to the optimal sphere packing density.  A lattice is a discrete subgroup of Rn of rank n, and a lattice packing uses spheres centered at the points of a lattice, while periodic packings are unions of finitely many translates of lattices.  In this connection, Henry Cohn and Abhinav Kumar in 2009 recognized Optimality and uniqueness of the Leech lattice among lattices.  Now, Maryna Viazovska proved, based on linear programming bounds, that no packing of unit balls in Euclidean space R8(8-dimension) has density greater than that of the E8-lattice packing.  Subsequently a team consisting of Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska proved that the Leech lattice is the densest packing of congruent spheres in 24 dimensions, and that it is the unique optimal periodic packing.