Editing Half-integer (section)

==Uses==
===Sphere packing===
The densest [[lattice packing]] of [[unit sphere]]s in four dimensions (called the [[D4 lattice|''D''<sub>4</sub> lattice]]) places a sphere at every point whose coordinates are either all integers or all half-integers. This packing is closely related to the [[Hurwitz integer]]s: [[quaternion]]s whose real coefficients are either all integers or all half-integers.<ref>{{cite journal |first=John C. |last=Baez |authorlink=John C. Baez |year=2005 |title=Review ''On Quaternions and Octonions: Their geometry, arithmetic, and symmetry'' by John H. Conway and Derek A. Smith |type=book review |journal=Bulletin of the American Mathematical Society |volume=42 |pages=229–243 |url=http://math.ucr.edu/home/baez/octonions/conway_smith/ |doi=10.1090/S0273-0979-05-01043-8 |doi-access=free}}</ref>

===Physics===
In physics, the [[Pauli exclusion principle]] results from definition of [[fermion]]s as particles which have [[Spin (physics)|spin]]s that are half-integers.<ref>{{cite book |first=Péter |last=Mészáros |year=2010 |title=The High Energy Universe: Ultra-high energy events in astrophysics and cosmology |page=13 |publisher=Cambridge University Press |isbn=9781139490726 |url=https://books.google.com/books?id=NXvE_zQX5kAC&pg=PA13}}</ref>

The [[energy level]]s of the [[quantum harmonic oscillator]] occur at half-integers and thus its lowest energy is not zero.<ref>{{cite book |first=Mark |last=Fox |year=2006 |title=Quantum Optics: An introduction |page=131 |series=Oxford Master Series in Physics |volume=6 |publisher=Oxford University Press |isbn=9780191524257 |url=https://books.google.com/books?id=Q-4dIthPuL4C&pg=PA131}}</ref>

===Sphere volume===
Although the [[factorial]] function is defined only for integer arguments, it can be extended to fractional arguments using the [[gamma function]]. The gamma function for half-integers is an important part of the formula for the [[volume of an n-ball|volume of an {{mvar|n}}-dimensional ball]] of radius <math>R</math>,<ref>{{cite web |title=Equation&nbsp;5.19.4 |website=NIST Digital Library of Mathematical Functions |url=http://dlmf.nist.gov/ |publisher=U.S. [[National Institute of Standards and Technology]] |id=Release&nbsp;1.0.6 |date=2013-05-06}}</ref>
<math display=block>V_n(R) = \frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}R^n~.</math>
The values of the gamma function on half-integers are integer multiples of the square root of [[pi]]:
<math display=block>\Gamma\left(\tfrac{1}{2} + n\right) ~=~ \frac{\,(2n-1)!!\,}{2^n}\, \sqrt{\pi\,} ~=~ \frac{(2n)!}{\,4^n \, n!\,} \sqrt{\pi\,} ~</math>
where <math>n!!</math> denotes the [[double factorial]].