Kepler, Snowflakes, and Pomegranates
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HISTORICAL NOTE
Kepler, Snowflakes, and Pomegranates In late December of 1610, the German astronomer and mathematician Johannes Kepler walked the streets of Prague, returning home after a visit to his “master and benefactor,” the “illustrious Counsellor at the Court of His Sacred Imperial Majesty,” Johannes Matthäus Wacker von Wackerfels. As Kepler crossed the Karlsbrucke Bridge, he noted his embarrassment at not having a New Year’s gift for his benefactor. “Just then by a happy chance watervapor was condensed by the cold into snow, and specks of down fell here and there on my coat, all with six corners and feathered radii,” Kepler wrote later in a short treatise. “’Pon my word, here was something smaller than any drop, yet with a pattern; here was the ideal New Year’s gift… since it comes down from heaven and looks like a star.” The experience hardly seems like one of scientific significance. However, Kepler turned it into one. Reaching home, he wrote down his ruminations on the hexagonal shape of snowflakes in an essay titled A New Year’s Gift, or On the Six-Cornered Snowflake, which he presented to Wacker shortly thereafter. Full of wit and contemplation of the mystical “formative faculty” (a property of the Earth embodying a universal spirit that shapes everything) that Kepler employed to account for the genesis of forms, it also contained a complete discussion of the orderly arrangement of spheres when packed together. As the mathematical physicist and philosopher of science Lancelot Law Whyte wrote in the introduction to a 1966 English translation of the work, “Kepler’s essay provides the first published evidence, in diagrams as well as text, of the ideas of regular arrangements and close-packing which have proved fundamental to crystallography.” Whyte noted that the Englishman Thomas Harriot wrote a manuscript in 1600 dealing with the same issues, but it was never published. How did a winter’s walk and a few stray snowflakes lead to such valuable scientific deductions? Kepler allowed his mind to wander along with his legs, leading him to consider honeycombs and pomegranates in addition to snowflakes. In the end, it was his investigation of the pomegranate, with its inner chambers, or “loculi,” as Kepler called them, which yielded the most important results from a materials science perspective. Initially, he assigned the cause of the shape of the loculi to the plant’s soul, or “life principle.” However, he noted that the life principle “is not a sufficient cause
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of its shape: for it is not from its formal properties that it induces this shape in its fruit, but it is assisted by material necessity [italics added].” The loculi are very small and round to begin with, he said, encased in a soft rind. But as the loculi grow and the rind hardens, they become squeezed into the available space. The most efficient packing of spheres is thus a major consideration.
“How did a winter’s walk and a few stray snowflakes lead to such valuable scientific deductions?” “In general equal pellets when collected in any ves
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