  A fan $F$ is \emph{endpoint-homogeneous} if for any two endpoints $e,e'$ of
$F$, there is a homeomorphism $h: F \rightarrow F$ such that $h(e) = e'$.
  We prove there are uncountably many distinct homeomorphism types of
endpoint-homogeneous smooth fans. To do this, we associate to each such fan $F$
a topological invariant, in the form of a characteristic subset $EPG(F)
\subseteq [0,1]$ describing how the endpoints of $F$ limit onto any given blade
of $F$. We describe precisely all the uncountably many different $X \subseteq
[0,1]$ that can arise as $EPG(F)$ for some endpoint-homogeneous smooth fan $F$.
We also prove the existence of $\frac{1}{n}$-homogeneous smooth fans for all $n
\geq 5$.
