one sphere
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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.
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