page_title: Close-packing of equal spheres
section_title: Cannonball problem
hierarchical_section_title: Close-packing of equal spheres / FCC and HCP Lattices / Cannonball problem
caption_attribution_description: English: Shot and shell piled under the walls of Fortress Monroe
context_page_description: In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is
  The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only in case of 1, 2, 3, 8 and 24 dimensions.
  Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.
context_section_description: 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. 
  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. 
  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  or  and conjectured that the only solutions are  and . Here  is the number of layers in the pyramidal stacking arrangement and  is the number of cannonballs along an edge in the flat square arrangement.
caption: Cannonballs piled on a triangular (front) and rectangular (back) base, both fcc lattices.
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