Within the Hilbert scheme of curves in projective space is a subscheme of curves that are "tropical" in the sense of Tevelev: they interact well with the coordinate subspaces. I will explain why, from the point of view of tropical and logarithmic geometry, this locus ought to be the principal open cell in another moduli space, of which the Hilbert scheme is only an approximation. This "logarithmic Hilbert scheme" was recently constructed in work with Davesh Maulik (MIT) and is the core of a new theory of logarithmic Donaldson-Thomas invariants. The story touches another major character in the story of polyhedral and algebraic geometry: the secondary polytope of Gel'fand-Kapranov-Zelevinsky. I'll try to give some sense for why.