  These lectures were a part of the geometry course held during the Fall 2011
Mathematics Advanced Study Semesters (MASS) Program at Penn State
(\url{http://www.math.psu.edu/mass/}).
  The lectures are meant to be accessible to advanced undergraduate and early
graduate students in mathematics. We have placed a great emphasis on clarity
and exposition, and we have included many exercises. Hints and solutions for
most of the exercises are provided in the end.
  The lectures discuss piecewise distance preserving maps from a 2-dimensional
polyhedral space into the plane. Roughly speaking, a polyhedral space is a
space that is glued together out of triangles, for example the surface of a
polyhedron. If one imagines such a polyhedral space as a paper model, then a
piecewise distance preserving map into the plane is essentially a way to fold
the model so that it lays flat on a table.
  We only consider the 2-dimensional case to keep things easy to visualize.
However, most of the results admit generalizations to higher dimensions. These
results are discussed in the Final Remarks, where proper credit and references
are given.
