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The spread of a finite set of points is the ratio between thelongest and shortest pairwise distances.  We prove that the Delaunaytriangulation of any set of $n$ points in~$\Real^3$ with spread$\Delta$ has complexity $O(\Delta^3)$.  This bound is tight in theworst case for all $\Delta = O(\sqrt{n})$.  In particular, theDelaunay triangulation of any dense point set has linear complexity.We also generalize this upper bound to regular triangulations of$k$-ply systems of balls, unions of several dense point sets, anduniform samples of smooth surfaces.  On the other hand, for any $n$and $\Delta = O(n)$, we construct a regular triangulation ofcomplexity $\Omega(n\Delta)$ whose $n$ vertices have spread $\Delta$.

Dense Point Sets Have Sparse Delaunay Triangulations or ?... But Not Too Nasty?