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We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise all the closed oriented 4-manifolds that have complexity $\leq 1$. These are precisely the 4-manifolds that are generated by a certain set of 20 blocks, that is some basic 4-manifolds with boundary consisting of copies of $S^2 \times S^1$, plus connected sums with some copies of $\mathbb {CP}^2$ with either orientation.

Four-manifolds with Shadow-complexity Zero