  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.
