Sphere packing in a sphere
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions.
Number of inner spheres | Maximum radius of inner spheres[1] | Packing density | Optimality | Diagram | |
---|---|---|---|---|---|
Exact form | Approximate | ||||
1 | 1.0000 | 1 | Trivially optimal. | ||
2 | 0.5000 | 0.25 | Trivially optimal. | ||
3 | 0.4641... | 0.29988... | Trivially optimal. | ||
4 | 0.4494... | 0.36326... | Proven optimal. | ||
5 | 0.4142... | 0.35533... | Proven optimal. | ||
6 | 0.4142... | 0.42640... | Proven optimal. | ||
7 | 0.3859... | 0.40231... | Proven optimal. | ||
8 | 0.3780... | 0.43217... | Proven optimal. | ||
9 | 0.3660... | 0.44134... | Proven optimal. | ||
10 | 0.3530... | 0.44005... | Proven optimal. | ||
11 | 0.3445... | 0.45003... | Proven optimal. | ||
12 | 0.3445... | 0.49095... | Proven optimal. |
References
- ^ Best packing of m>1 equal spheres in a sphere setting a new density record
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Packing problems
- Bin
- Set
- In a circle / equilateral triangle / isosceles right triangle / square
- Apollonian gasket
- Circle packing theorem
- Tammes problem (on sphere)
- Apollonian
- Finite
- In a sphere
- In a cube
- In a cylinder
- Close-packing
- Kissing number
- Sphere-packing (Hamming) bound