  Hamilton in the course of his studies on quaternions came up with an elegant
geometric picture for the group SU(2). In this picture the group elements are
represented by ``turns'', which are equivalence classes of directed great
circle arcs on the unit sphere $S^2$, in such a manner that the rule for
composition of group elements takes the form of the familiar parallelogram law
for the Euclidean translation group. It is only recently that this construction
has been generalized to the simplest noncompact group $SU(1,1) = Sp(2, R) =
SL(2,R)$, the double cover of SO(2,1). The present work develops a theory of
turns for $SL(2,C)$, the double and universal cover of SO(3,1) and $SO(3,C)$,
rendering a geometric representation in the spirit of Hamilton available for
all low dimensional semisimple Lie groups of interest in physics. The geometric
construction is illustrated through application to polar decomposition, and to
the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.
