Consider an arrow initially pointing from the center of your hamster ball to the point of contact. Now roll to the antipodal point, and note that the arrow has rotated $2\pi$!! and must in fact rotate by $4\pi$ to go all the way around! Then we start to realize that Cartan connections provide a natural description of a spinor! ]]>

Thanks, Blake. It may well be that my most important contribution to Cartan geometry so far is introducing hamsters to the subject. :-)

]]>