  We usually think of 2-dimensional manifolds as surfaces embedded in Euclidean
3-space. Since humans cannot visualise Euclidean spaces of higher dimensions,
it appears to be impossible to give pictorial representations of
higher-dimensional manifolds. However, one can in fact encode the topology of a
surface in a 1-dimensional picture. By analogy, one can draw 2-dimensional
pictures of 3-manifolds (Heegaard diagrams), and 3-dimensional pictures of
4-manifolds (Kirby diagrams). With the help of open books one can likewise
represent at least some 5-manifolds by 3-dimensional diagrams, and contact
geometry can be used to reduce these to drawings in the 2-plane.
  In this paper, I shall explain how to draw such pictures and how to use them
for answering topological and geometric questions. The work on 5-manifolds is
joint with Fan Ding and Otto van Koert.
