  In this paper we construct an uncountable union of line segments $T$ which
has full intersection with the sets $(\{ 0\} \times [0, 1]) \cup (\{ 1\} \times
[0, 1])\subset\mathbb{R}^2$ but has null two-dimensional measure. Further
results are proved on the decay rate of $\mu (T)$ if the line segments
comprising $T$ are replaced with increasingly fine approximations by
parallelograms.
