I know this is a physics question but the answer is mathematical.

 

In Stephen Hawking's "A Brief History of Time" he says that there is an equation that explains how a particle will not look the same if one rotates it 360 degrees but it will look the same if it is rotated 720 degrees. Can anyone explain this equation?

Spin 1/2 particles are described by spinors

 

http://www.sjsu.edu/faculty/watkins/spinor.htm

I will echo swansont post. look up representations of the Poincare group, spinors and the Dirac equation.