In March 2016, Ukrainian mathematician Maryna Viazovska provided the final solution for the Sphere Packing Problem in dimension 8 and later on – in collaboration – in dimension 24. Both papers Sphere Packing Problem in dimension 8, and 24, put an end to almost four centuries of studies of Kepler’s Conjecture.
As Dr. Ulrich H. Kurzweg, Professor Emeritus of Mechanical and Aerospace Engineering at the University of Florida explains on his website:
Johannes Kepler was the first to conjecture that the tightest packing of constant radius spheres occurs for a face centered cubic configuration. That is a configuration where there is an arrangement of parallel rows of spheres such that each sphere just touches its neighboring spheres as shown in the top view of a single layer of this packing.
A second identical layer is laid on top after being shifted so that its component spheres rest in the depressions shown in red. The height of the second layer center of spheres lies at z=2sqrt(2/3) above the x-y plane as is readily determined by treating the triangle ABC shown as the bottom of a regular tetrahedron.
After Kepler conjectured the densest packing density for sphere packing in a three-dimensional Euclidian space in 1611, it took nearly two hundred years until German mathematician Gauss to exhibited a proof for Kepler’s Conjecture in 1831.
In 1953, Hungarian mathematician László Fejes Tóth was the first to suggest that the Sphere Packing problem could be reduced to a finite case analysis and, ultimately, solved by a computer. In the end of the 20th century, it was Thomas Hales who spent almost twenty years on research with computers – a proof by exhaustion – in order to formulate a final solution for the problem in three-dimensional Euclidean space.
Fifty years later, Henry Cohn and Noam Elkies found a new approach and published New Upper Bounds On Sphere Packings I, introducing the notion that the sphere packing problem in dimensions 8 and 24 could be resolved by adopting a language of free analysis. All that remained was to find the function with certain properties which would be capable of sorting every packing problem.
After this new breakthrough, Maryna Viazovska, number theorist and researcher at the Humboldt University of Berlin, ventured to find this special function. The researcher teamed up with a colleague but after some missteps her peer left to focus on another project, leaving her alone in the quest to solve the sphere packing problem in dimension 8.
Fortunately, the setback did not discourage Viazovska. After a few years of working on this problem she found a way to find this magic function that led to the solution for Sphere Packing Problem in Dimension 8, at first, and later on with collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko, in Dimension 24. Read PaperHive’s Conversation with Dr. Viazovska to find more about her groundbreaking work.