The complete characterization of tangram pentagons
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The complete characterization of tangram pentagons Sarah Sophie Pohl1 · Christian Richter2 Received: 17 June 2020 / Accepted: 22 August 2020 © The Author(s) 2020
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 ≥ 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. Keywords Tangram · Dissection · Tiling · Pentagon · Lattice Mathematics Subject Classification 52C20 (Primary); 00A08 · 05B45 · 51M04
1 Introduction 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 two with √ 2, a square with sides of length 1 and a parallelogram with sides of length 1 and 2 and an angle of π4 . These seven pieces are arranged, using Euclidean isometries, to form dissections of prescribed or unknown polygons, as in Fig. 1. A dissection (or tiling) of a polygon P into pieces (or tiles) P1 , . . . , Pk is given if P is the union of all pieces Pi , 1 ≤ i ≤ k, and if no two pieces have interior points
To our teachers and friends Prof. Dr. Johannes Böhm on the occasion of his 95th birthday, Prof. Dr. Eike Hertel on the occasion of his 80th birthday and Dr. Carsten Müller for 30 years of service as the school master of the Carl Zeiss Gymnasium Jena.
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Christian Richter [email protected] Sarah Sophie Pohl [email protected]
1
Ringau 40, 37327 Leinefelde, Germany
2
Institute for Mathematics, Friedrich Schiller University, 07737 Jena, Germany
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Beitr Algebra Geom
Fig. 1 The seven tans along with their lattices, dissecting a convex and a non-convex pentagon
Fig. 2 All 13 convex tangrams
in common. A polygon T is called a tangram if T can be dissected into (isometric images of) the seven tans. Tangram puzzles usually ask to find dissections of prescribed polygons (Elffers 1978; Goodrich 1817; Read 1985; Slocum et al. 2004). We do not address aspects of the tangram related to craft, art and design. Fruitful mathematical problems appear when one aims to detect systematically all tangrams satisfying particular geometric properties. Such questions have been posed and studied in several books (e.g. Elffers 1978; Müller 2013, Chapter 7; 2020; Read 1985; Slocum et al. 2004)), papers in mathematical journals (e.g. Gardner 1974a; Graber et al. 2016; Heinert 1998a, b; Read 2004; Wang and Hsiung 1942), private publications (Müller 1997–2014) and contributions to mathematical competitions (Brunner 2014, 2015; Heinert 1996; Pohl 2018, 2019, 2020). The most prominent result of that kind is the following one by Wang and Hsiung (cf. Fig. 2). Theorem 1 (Wang a
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