Tiling a Circular Disc with Congruent Pieces
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Tiling a Circular Disc with Congruent Pieces ´ ad Kurusa , L´angi Zsolt and Viktor V´ıgh Arp´ Abstract. In this note, we prove that any monohedral tiling of the closed circular unit disc with k ≤ 3 topological discs as tiles has a k-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer, and Guy in 1994. Mathematics Subject Classification. 52C20, 52C22. Keywords. Tiling, dissection, monohedral, topological disc, Jordan region.
1. Introduction A tiling of a convex body K in Euclidean d-space Rd is a finite family of compact sets in Rd with mutually disjoint interiors, called tiles, whose union is K. A tiling is monohedral if all tiles are congruent. In this paper, we deal with the monohedral tilings of the closed circular unit disc B 2 with center O, in which the tiles are Jordan regions, i.e., are homeomorphic to a closed circular disc. The easiest way to generate such tilings, which we call rotationally generated tilings, is to rotate around O a simple curve connecting O to a point on the boundary S 1 of B 2 , where by a curve we mean a continuous map of the interval [0, 1] to the Euclidean plane. The following question, based on the observation that any tile of such ´ Kurusa’s research was supported by NFSR of Hungary (NKFIH) under grant numbers A. K 116451 and KH 18 129630, and by Ministry for Innovation and Technology of Hungary (MITH) under grant TUDFO/47138-1/2019-ITM. Z. L´ angi’s research is supported by the NFSR of Hungary (NKFIH) under grant number K-119670, the J´ anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, and grants BME IE-V´IZ TKP2020 ´ and the UNKP-19-4 New National Excellence Program by the Ministry for Innovation and Technology. V. V´ıgh’s research was supported by NFSR of Hungary (NKFIH) under grant number K 116451, and by Ministry for Innovation and Technology of Hungary (MITH) under grant TUDFO/47138-1/2019-ITM. 0123456789().: V,-vol
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a monohedral tiling of B 2 contains O, seems to arise regularly in recreational mathematical circles [13]. Question 1. Are there monohedral tilings of B 2 in which not all of the tiles contain O? The answer to Question 1 is affirmative; the usual examples to show this are the first two configurations in Fig. 1. The following harder variant is attributed to Stein by Croft, Falconer and Guy in [2, last paragraph on p. 87]. Question 2 (Stein). Are there monohedral tilings of B 2 in which O is in the interior of a tile? A systematic investigation of monohedral tilings of B 2 was started in [7] by Haddley and Worsley. In their paper, they called a monohedral tiling radially generated, if every tile is radially generated, meaning that its boundary is a simple curve consisting of three parts: a circular arc of length α and two other curves one of which is the ro
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