  The Thurston metric on Teichmuller space, first introduced by W. P. Thurston
is an asymmetric metric on Teichmuller space defined by $d_{Th}(X,Y) = \frac12
log\sup_{\alpha} \frac{l_{\alpha}(Y)}{l_{\alpha}(X)}$. This metric is geodesic,
but geodesics are far from unique. In this thesis, we show that in the
once-punctured torus, and in the four-times punctured sphere, geodesics stay a
uniformly-bounded distance from each other. In other words, we show that the
\textbf{width} of the geodesic envelope, E(X,Y) between any pair of points $X,Y
\in T(S)$ (where $S = S_{1,1}$ or $S = S_{0,4}$) is bounded uniformly. To do
this, we first identify extremal geodesics in Env(X,Y), and show that these
correspond to stretch vectors, proving a conjecture of Huang, Ohshika and
Papadopoulos. We then compute Fenchel-Nielsen twisting along these paths, and
use these computations, along with estimates on earthquake path lengths, to
prove the main theorem.
