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Closest-Packing or Gravitational Gathering of Spheres

by Bill Lauritzen

 

Applications:  geometry, art, science, mathematics, design, packing, economics, space, architecture. 

Black and white photos by the author, color computer images by my student, Candida Ayala.


  one sphere


two spheres


two spheres gravitationally packed       

    

      
three spheres closest-packed


4 spheres closest-packed (tetrahedron or 4-nook) 


5 spheres


6 spheres (octahedron or 6-nook) 

            
7 spheres

 
7 (side view)


8 (closest-packed and 90-degree packed). Which is more stable?

      
9 spheres


10 spheres (two packing methods)

    
              11 spheres

 


12 (icosahedron or 12-nook)

              
13 (12 around 1) nucleated

   
19 and 19 (in octahedron 6-nook) (14 in a pyramid)

 
35 in tetrahedral (4-nook pattern)


35 (side view) 


42 around 12 around 1 = 55 nucleated

number of spheres in the outer layer = 10F^2 + 2, where F is the frequency or number of layers
 


octahedral shape


7 (6 around 1 in a plane)

 


19= 12 around 6 around 1 in a plane


60 degree versus 90 degree packing. Which packs closer? Which is more stable?


60 degree versus 90 degree packing. Which packs closer? Which is more stable?

 

 


60 degree versus 90 degree packing in space. Which packs closer? Which is more stable?

 


60 degree versus 90 degree packing. Which packs closer? Which is more stable?


1 surrounded by 12 surround by 42 pattern


left: 1 frequency octahedron, right: 2 frequency octahedron, middle: 19 (shown before)


1 and 2 frequency octahedra (12-nooks)


1 through 12 spheres packed


1, 2, 3, 4, 6, and 12 spheres make regular (all-even) shapes.


spheres corresponding to tetrahedron, octahedron, and icosahedron. (4-nook, 6-nook, and 12-nook) 

 

Key words: geometry, art, science, mathematics, design, packing, economics, space, architecture, polyhedron, polyhedra.