Hypersphere Packing
Draw unit 
-spheres in an -D space centered at all 
coordinates. Then place an additional Hypersphere atthe origin tangent to the other Hyperspheres. Then the central Hypersphere is contained insidethe Hypercube with Vertices at the centers of the other spheres for between 2 and 8.However, for 
, the central Hypersphere just touches the Hypercube of centers, and for 
, theHypersphere is partially outside the hypercube. This can be seen by finding the distance from the origin to the centerof one of the Hyperspheres
The radius of the central sphere is therefore
. The distance from the origin to the center of the boundinghypercube is always 2 (two radii), so the center Hypersphere is tangent when 
, or , and outside for .The analog of face-centered cubic packing is the densest lattice in 4- and 5-D. In 8-D, the densest lattice packing ismade up of two copies of face-centered cubic. In 6- and 7-D, the densest lattice packings are cross-sections of the 8-Dcase. In 24-D, the densest packing appears to be the Leech Lattice. For high dimensions (
-D), thedensest known packings are nonlattice. The densest lattice packings in -D have been rigorously proved to havePacking Density 1, 
, 
, 
, 
, 
,
, and 
(Finch).
The largest number of unit Circles which can touch another is six. For Spheres, themaximum number is 12. Newton 
considered this question long before a proof was published in 1874. The maximumnumber of hyperspheres that can touch another in -D is the so-called Kissing Number.
References
Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hermit/hermit.html Gardner, M. Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 89-90, 1966.
- Stieltjes' Theorem
- Stieltjes-Wigert Polynomial
- Stirling Cycle Number
- Stirling Number of the First Kind
- Stirling Number of the Second Kind
- Stirling's Approximation
- Stirling Set Number
- Stirling's Finite Difference Formula
- Stirling's Formula
- Stirling's Series
- Stirrup Curve
- Stochastic
- Stochastic Calculus of Variations
- Stochastic Group
- Stochastic Matrix
- Stochastic Process
- Stochastic Resonance
- Stokes Phenomenon
- Stokes' Theorem
- Stolarsky Array
- Stolarsky-Harborth Constant
- Stolarsky's Inequality
- Stomachion
- Stone Space
- Stone-von Neumann Theorem