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.
Highlights
The tangram, known as a Chinese puzzle (Goodrich 1817), is a collection of seven√polygons, called tans: five isosceles right triangles, two with legs of length 1, one with 2 and length 1 two and w√i2tha2n,daasnqaunagrelewoifthπ4 sides of
Dissection of a polygon P into pieces P1, . . . , Pk is given if P is the union of all pieces Pi, 1 ≤ i ≤ k, and if no two pieces have interior points
We comment briefly the situation for triangular and quadrangular tangrams (Sect. 2), and we provide a topological tool that permits a systematic approach to pentagons (Sect. 3)
Summary
The tangram, known as a Chinese puzzle (Goodrich 1817), is a collection of seven. √polygons, called tans: five isosceles right triangles, two with legs of length 1, one with 2 and length 1 two and w√i2tha2n,daasnqaunagrelewoifthπ sides of. Theorem 1 (Wang and Hsiung 1942) There exist, up to isometry, exactly 13 convex tangrams: one triangle, six quadrangles, two pentagons and four hexagons. This motivates the question for other natural classes of tangrams. Theorem 2 There exist, up to isometry, exactly 53 simple pentagons that are tangrams: two convex ones, 20 non-convex lattice ones and 31 non-convex non-lattice ones. These pentagons are given in detail in Sect. The present paper is an extended version of Pohl (2019)
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