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To each once-punctured-torus bundle, $T_\phi$, over the circle withpseudo-Anosov monodromy $\phi$, there are associated two tessellations of thecomplex plane: one, $\Delta(\phi)$, is (the projection from $\infty$ of) thetriangulation of a horosphere at $\infty$ induced by the canonicaldecomposition into ideal tetrahedra, and the other, $CW(\phi)$, is a fractaltessellation given by the Cannon-Thurston map of the fiber group switching backand forth between gray and white each time it passes through $\infty$. In thispaper, we study the relation between $\Delta(\phi)$ and $CW(\phi)$.

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane