  We give an explicit bijective correspondence between between nonzero pairs of
complex numbers, which we regard as spinors or spin vectors, and horospheres in
3-dimensional hyperbolic space decorated with certain spinorial directions.
This correspondence builds upon work of Penrose--Rindler and Penner. We show
that the natural bilinear form on spin vectors describes a certain
complex-valued distance between spin-decorated horospheres, generalising
Penner's lambda lengths to 3 dimensions.
  From this, we derive several applications. We show that the complex lambda
lengths in a hyperbolic ideal tetrahedron satisfy a Ptolemy equation. We also
obtain correspondences between certain spaces of hyperbolic ideal polygons and
certain Grassmannian spaces, under which lambda lengths correspond to
Pl\"{u}cker coordinates, illuminating the connection between Grassmannians,
hyperbolic polygons, and type A cluster algebras.
