  In three-dimensional computational topology, the theory of normal surfaces is
a tool of great theoretical and practical significance. Although this theory
typically leads to exponential time algorithms, very little is known about how
these algorithms perform in "typical" scenarios, or how far the best known
theoretical bounds are from the real worst-case scenarios. Here we study the
combinatorial and algebraic complexity of normal surfaces from both the
theoretical and experimental viewpoints. Theoretically, we obtain new
exponential lower bounds on the worst-case complexities in a variety of
settings that are important for practical computation. Experimentally, we study
the worst-case and average-case complexities over a comprehensive body of
roughly three billion input triangulations. Many of our lower bounds are the
first known exponential lower bounds in these settings, and experimental
evidence suggests that many of our theoretical lower bounds on worst-case
growth rates may indeed be asymptotically tight.
