Editing Close-packing of equal spheres (section)

=== Cannonball problem ===
{{Main|Cannonball problem}}
[[Image:Fortres Monroe 1861 - Cannon-balls.jpg|thumb|Cannonballs piled on a triangular ''(front)'' and rectangular ''(back)'' base, both [[Face-centered cubic|FCC]] lattices.]]
The problem of close-packing of spheres was first mathematically analyzed by [[Thomas Harriot]] around 1587, after a question on piling cannonballs on ships was posed to him by Sir [[Walter Raleigh]] on their expedition to America.<ref>{{cite encyclopedia |title=Cannonball Problem |encyclopedia=The Internet Encyclopedia of Science |first=David |last=Darling |url=http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html }}</ref> 
Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground.  Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base. 
[[File:Pyramid of Snowballs Large.png|thumb|Collections of [[snowball]]s arranged in pyramid shape. The front pyramid is hexagonal close-packed and rear is face-centered cubic. ]]

The [[cannonball problem]] asks which flat square arrangements of cannonballs can be stacked into a square pyramid. [[Édouard Lucas]] formulated the problem as the [[Diophantine equation]] <math>\sum_{n=1}^{N} n^2 = M^2</math> or <math>\frac{1}{6} N(N+1)(2N+1) = M^2</math> and conjectured that the only solutions are <math>N = 1, M = 1,</math> and <math>N = 24, M = 70</math>. Here <math>N</math> is the number of layers in the pyramidal stacking arrangement and <math>M</math> is the number of cannonballs along an edge in the flat square arrangement.
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