  To each once-punctured-torus bundle, $T_\phi$, over the circle with
pseudo-Anosov monodromy $\phi$, there are associated two tessellations of the
complex plane: one, $\Delta(\phi)$, is (the projection from $\infty$ of) the
triangulation of a horosphere at $\infty$ induced by the canonical
decomposition into ideal tetrahedra, and the other, $CW(\phi)$, is a fractal
tessellation given by the Cannon-Thurston map of the fiber group switching back
and forth between gray and white each time it passes through $\infty$. In this
paper, we study the relation between $\Delta(\phi)$ and $CW(\phi)$.
