Close-packing of equal spheres | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-08

This is an audio version of the Wikipedia Article:
Close-packing of equal spheres


00:01:17 1 FCC and HCP Lattices
00:01:56 1.1 Cannonball problem
00:04:21 1.2 Positioning and spacing
00:06:45 2 Lattice generation
00:07:25 2.1 Simple hcp lattice
00:16:18 3 Miller indices
00:16:56 4 Filling the remaining space
00:18:30 5 See also
00:18:47 6 Notes
00:18:56 7 External links



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SUMMARY
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In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). 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

  
    
      
        
          
            π
            
              3
              
                
                  2
                
              
            
          
        
        ≃
        0.74048.
      
    
    {\displaystyle {\frac {\pi }{3{\sqrt {2}}}}\simeq 0.74048.}
  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.