  A fundamental question for simplicial complexes is to find the lowest
dimensional Euclidean space in which they can be embedded. We investigate this
question for order complexes of posets. We show that order complexes of thick
geometric lattices as well as several classes of finite buildings, all of which
are order complexes, are hard to embed. That means that such d-dimensional
complexes require (2d + 1)-dimensional Euclidean space for an embedding. (This
dimension is in general always sufficient for any d-complex.)
  We develop a method to show non-embeddability for general order complexes of
posets which builds on properties of the van Kampen obstruction.
