Building on Whitney's classical method of triangulating smooth manifolds, we show that every compact $d$-dimensional smooth manifold admits a triangulation with dual graph of twin-width at most $d^{O(d)}$. In particular, it follows that every compact 3-manifold has a triangulation with dual graph of bounded twin-width. This is in sharp contrast to the case of treewidth, where for any natural number $n$ there exists a closed 3-manifold such that every triangulation thereof has dual graph with treewidth at least $n$. To establish this result, we bound the twin-width of the incidence graph of the $d$-skeleton of the second barycentric subdivision of the $2d$-dimensional hypercubic honeycomb. We also show that every compact, piecewise-linear (hence smooth) $d$-dimensional manifold has triangulations where the dual graph has an arbitrarily large twin-width.