kepler's conjecture

[...]... we find the formulation of his famed conjecture: A sphere surrounded in a certain way by twelve other spheres represents the tightest possible packing Kepler didn’t prove this statement, he just announced it Undeterred by such imperfections, Kepler then went on to describe a remarkable fact Such a packing can be built up from two different arrange- 22 KEPLER’S CONJECTURE ments, one with a quadratic,... manner of packing achieves the best possible density: 100 percent of the line 1 One can also define a curved line as a one-dimensional object Then the spheres would be pieces of the curved line 4 KEPLER’S CONJECTURE is filled with matches and there is no space left over in between This is so obvious that even mathematicians do not require a proof Let us move to two dimensions Here the problem is to place... be able to fit only three spheres into a square of edge-length four, while four spheres would fit into it when using the square packing Something similar is also true in three dimensions, and Kepler’s conjecture, the subject matter of this book, refers to space that has no borders, that is, that extends to infinity We saw that the hexagonal packing is denser than the square packing in (a) (b) (c) (a)... a New Year’s gift to his friend Wacker von Wackerfels It was called The Six-Cornered Snowflake, and in it he described a method of packing balls as tightly as possible This marks the birth of Kepler’s Conjecture We will have more to say about snowflakes and their relationship to the packing of cannonballs in the next chapter Let us use melons as an illustration If melons were cube-shaped, everything... why, the reader may ask, did nature evolve round melons (assuming, for illustration purposes, that melons are perfectly round 2 Japanese farmers have figured out how to grow cubic watermelons 6 KEPLER’S CONJECTURE objects)? And why are so many other fruits and vegetables approximately round? Well, nature did not worry about limited space on ships or in the holds of aircraft, but it did worry about moisture... indicate that nothing is as simple as it 4 Physicists, on the other hand, do worry See what Per Bak has to say about the stability of sandpiles in How Nature Works (New York: Copernicus, 1996) 8 KEPLER’S CONJECTURE looks, especially in mathematics In 1883 the crystallographer William Barlow (1845–1934) pointed out that there is not just one good way to stack melons, but two Barlow was a self-educated scientist... o’clock with the singing of psalms) and, later, at Maulbronn In 1589, half a year after his father’s final departure, the newly graduated Baccalaureus entered college with the idea of eventually 12 KEPLER’S CONJECTURE Johannes Kepler becoming a man of the cloth But, as was the custom at the time, Kepler had to take two years of classes in the faculty of arts of Tübingen University before embarking on the... from the childless uncle The young boy was sent to study Latin so that he could eventually become a lawyer and enter Denmark’s civil service But at age thirteen, Tycho witnessed an event that 14 KEPLER’S CONJECTURE would shape his career: he observed a partial eclipse of the sun that had been predicted for that day The openmouthed boy decided then and there that astronomy would be his profession But first... master and his new assistant did not turn out to be an easy one Brahe gave Kepler the task of figuring out the movements of the planets, which, as he had already discerned, did not follow 16 KEPLER’S CONJECTURE circular orbits The assistant was supposed to do this based on Brahe’s own observations of the stellar positions, but without getting full access to the data Only when the master felt like it,... emperor already owed the imperial mathematician Desperately looking for funds, Kepler rode 500 kilometers on horseback to the town of Regensburg There he hoped to collect the emperor’s out- 18 KEPLER’S CONJECTURE standing debt from the Reichstag, the assembly of German dignitaries, who should have honored the emperor’s promises The trip took nearly four weeks Foul autumn weather got the better of Kepler . alt="" KEPLER’S CONJECTURE How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World George G. Szpiro John Wiley & Sons, Inc. KEPLER’S CONJECTURE How. Kepler’s sphere-packing conjecture is just such a problem—it looks simple at first sight, but reveals its subtle hor- rors to those who try to solve it.” I first met Kepler’s conjecture in 1968,. been the first person to state this conjecture and went on to say that together with Fermat’s famous theorem this was one of the oldest unproven mathemati- cal conjectures. I then forgot all about