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Balls in Space (Posted on 2005-12-09) Difficulty: 4 of 5
What is the maximum proportion of space that can be filled with an infinite number of identical spheres?

No Solution Yet Submitted by Andre    
Rating: 4.4000 (5 votes)

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Solution a derivation | Comment 2 of 12 |

Consider the spheres to be of unit radius.  The balls are arranged like oranges on a fruitstand, with the spheres of each layer resting in the indentations among three spheres of the preceding layer.

Each layer is a tessellation of triangles with a sphere centered on each corner.  The triangles have side length 2, and area sqrt(3). Each triangle has parts of three spheres and each sphere is shared among six triangles, so there's an average of half a sphere per triangle.  Thus, there are (1/2) / sqrt(3) spheres per unit area on the layer, or 1/(2 sqrt(3)) or sqrt(3)/6.

Now, how far apart are the layers? The basic unit will be a regular tetrahedron with the center of a sphere at each vertex. The height of this tetrahedron will be the distance between layers. As mentioned previously, the sides of the base have length 2. The apex of the tetrahedron will lie above the center of the base, and we need to find the angle of slant of an edge of the tetrahedron. Consider a right spherical triangle on one of the spheres, where one vertex is on the line connecting a base sphere with the sphere at the apex, another vertex is on a line connecting the base sphere with the next one, and the right-angled vertex is on an angle bisector of the base angle at that sphere.

The base leg of this spherical triangle is 30 degrees and the hypotenuse is 60 degrees. The remaining leg has the angle measure we need. By the law of cosines, cos 60 = cos x cos 30 + sin x sin 30 cos 90. Thus cos x = cos 60 / cos 30 = sqrt(3)/3. Twice the sine of x is the height of the tetrahedron and therefore the distance between layers. As sin x = sqrt(1 - 3/9) = sqrt(2/3), the distance between layers is 2 sqrt(2)/sqrt(3)

As each layer has sqrt(3)/6 spheres per unit area, space has (sqrt(3)/6) / (2 sqrt(2)/sqrt(3)) spheres per unit volume. That's 3 sqrt(2)/24 = sqrt(2)/8.

The volume of a unit sphere is 4 pi / 3, so the volume taken up by sphere per unit volume of space is 4 sqrt(2) pi / 24 = sqrt(2) pi / 6 or about .7404804896930602.

  Posted by Charlie on 2005-12-09 16:43:37
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