The complete characterization of tangram pentagons

TL;DR

This paper provides a complete mathematical characterization of all 53 pentagonal tangram figures, including convex, non-convex, and lattice-vertex pentagons, expanding the understanding of tangram geometric properties.

Contribution

It offers the first comprehensive classification of all pentagonal tangram figures, including non-convex and non-lattice vertex cases, filling a gap in tangram geometric research.

Findings

Identified all 53 pentagonal tangram figures.

Included 51 non-convex pentagons.

Included 31 pentagons with vertices outside the orthogonal lattice.

Abstract

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.