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Mathematical Treasure: Kepler and Sphere-packing

Frank J. Swetz (The Pennsylvania State University)

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

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Kepler and Sphere-packing," Convergence (June 2017)