By George G. Szpiro
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But with falling temperatures, they calm down and start to bond. 17 Viewed from the top, this arrangement appears as a hexagon. And this is whence six-corneredness arises. When ice crystals start to grow around the nucleus, they initially retain the hexagonal shape of their molecular structure. As more water molecules travel through the vapor-filled air, looking for a good place to land, the swirling hexagons offer themselves as ideal airports. And since the corners of the hexagons stick out farther into space than the edges, this is where the molecules like to dock.
It may be—and often is—much harder to find the solution to the nonlattice variety of the problem, since it allows more possibilities. Who was the first person to put his thoughts about the densest arrangement of circles to print? Generally it is assumed that it was Kepler in the treatise The Six-Cornered Snowflake (see chapter 2). But while researching this book I found otherwise. In Zürich I was invited to the office of Caspar Schwabe, a designer of geometrical objects. Sitting among the strange mathematical objects that Schwabe had either produced himself or col2 If we consider that the hoses must reach a building’s upper floors, we are faced with a three-dimensional covering problem.
Throughout his professional life, Dürer would put this new technology to good use. Albrecht was number three in a line of children that ran to the number eighteen. His father was a jeweler who had immigrated from Hungary to the German city of Nüremberg. He was a religious man who managed to instill a healthy fear of God in his children. Dürer described him as “gentle and patient . . ” Albrecht Dürer (self portrait) 38 KEPLER’S CONJECTURE Albrecht received his initial training in painting and woodcutting in Germany.