The complete characterization of tangram pentagons

The old Chinese puzzle tangram gives rise to serious mathematical problems when one asks for all tangram figures that satisfy particular geometric properties. All $13$ convex tangram figures are known since 1942. They include the only triangular and all six quadrangular tangram figures. The families of all $n$-gonal tangram figures with $n \ge 6$ are either infinite or empty. Here we characterize all $53$ pentagonal tangram figures, including $51$ non-convex pentagons and $31$ pentagons whose vertices are not contained in the same orthogonal lattice.