  It is an amazing and a bit counter-intuitive discovery by Micha Perles from
the sixties that there are ``non-rational polytopes'': combinatorial types of
convex polytopes that cannot be realized with rational vertex coordinates.
  We describe a simple construction of non-rational polytopes that does not
need duality (Perles' ``Gale diagrams''): It starts from a non-rational point
configuration in the plane, and proceeds with so-called Lawrence extensions.
  We also show that there are non-rational polyhedral surfaces in 3-space, a
discovery by Ulrich Brehm from 1997. His construction also starts from any
non-rational point configuration in the plane, and then performs what one
should call Brehm extensions, in order to obtain non-rational partial surfaces.
  These examples and objects are first mile stones on the way to the remarkable
"universality theorems'' for polytopes and for polyhedral surfaces by Mn\"ev
(1986), Richter-Gebert (1994), and Brehm (1997).
