Translative Packing of Unit Squares into Equilateral Triangles
نویسندگان
چکیده
Every collection of n (arbitrary-oriented) unit squares can be packed translatively into any equilateral triangle of side length 2.3755 ̈ ? n. Let the coordinate system in the Euclidean plane be given. For 0 ≤ αi ă π{2, denote by Spαiq a square in the plane with sides of unit length and with an angle between the x-axis and a side of Spαiq equal to αi. Furthermore, let T psq be an equilateral triangle with sides of length s and with one side parallel to the x-axis. A collection of unit squares admits a translative packing into a set C if there are mutually disjoint translated copies of the members of the collection contained in C. The question of packing of unit squares into squares or triangles (with the possibility of rotations) is a well-known problem (see [1], [3], [4] and [9]). Some upper bounds concerning translative packing (without the possibility of rotations) of unit squares into a square are given in [6]. Covering problems are discussed in [7]. In this note, we propose the question of translative packing of squares into an equilateral triangle. Denote by tn the smallest number t such that any collection of n unit squares Spα1q, . . . , Spαnq admits a translative packing into T ptq. The problem is to find tn for n “ 1, 2, 3, . . . . Claim 1. t1 “ ? 2` a 2{3 « 2.23. Proof. Let λ1 “ ? 2` a 2{3. Obviously, Spπ{4q cannot be packed translatively into T pλ1 ́ q for any ą 0 (see Fig. 1, left). The squares Spπ{12q as well as Sp5π{12q cannot be packed translatively either. Consequently, t1 ≥ λ1. 2010 Mathematics Subject Classification: 52C15.
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