Mathematical Treasure: Kepler and Sphere-packing

Johannes Kepler (1571-1630) is perhaps best remembered as an astronomer, but his curiosity extended far beyond just astronomy. Kepler was a true natural scientist and was interested in many aspects of the natural world. In 1611, he published a brief treatise, Strena seu de Nive Sexangula (New Year’s Gift concerning Six-cornered Snow), as a gift to his friends. In this work, he explored the topic of sphere-packing and related it to the six-fold symmetry of the snowflake. Kepler discovered that, in the plane, a hexagon provides the most efficient polygon for a packing or tessellation. The six-fold symmetry of the hexagon imparts a six-fold symmetry to water droplets it confines: thus the form of a snowflake.

Various configurations for sphere-packing are discussed and explored.

These images are obtained through the cooperation of the University of Oklahoma History of Science website.

Index to Mathematical Treasures