American Mathematical Society

Packing Pennies in the Plane

References

T. Hales, Cannon balls and honeycombs, Notices of the American Mathematical Society(April 2000), 440-449. Tom Hales'home pagehas lots of material on Kepler's conjecture. The full proof of Kepler'sconjecture appears in a series of articles, finishing with oneto appear in the Annals of Mathematics.
J. Kepler, De nive sexangula, 1611.An English translation (by Colin Hardie) was published in 1966 by Oxford Press asThe six-cornered snowflake. This pamphlet is impressive. It is arguable that no one had worked so hard at three dimensional imagingsinceDemocritus and Eudoxus (or their Chineseequivalents) discovered the volume formula for tetrahedra. One curious feature that makes it rather difficult to read is that there are so few images in it and so many attempts todescribe complex 3D phenomena verbally.
I am greatly indebted to the staff at the Fisher Librray and especiallythe librarian, Richard Landon, for their cooperation in producing the images from Kepler's pamphlet that I have used.
E. Klarreich, Foams and honeycombs, American Scientist (March-April 2000).A good popular account of Kepler's and other related conjectures.
R. Peikert,Dichteste Packungen von gleichen Kreisenin einem Quadrat, Elemente der Mathematik 49 (1994), 15-26.A survey of the much more difficult problem of packingin small squares.
C. A. Rogers,The packing of equal spheres, Proceedings of the London Mathematical Society 8 (1958), 609-620.Rogers proves an upper bound for packings in n dimensions whichreduces to Thue's result when n=2. In section 3 he presents the scaling argument(without pictures!) that was used in the last section.
A. Thue, Über die dichteste Zuzammenstellung von kongruenten Kreisenin der Ebene, Christiana Vid. Selsk. Skr. 1 (1910), 1-9.

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