Translative Packing of Unit Squares into Squares

Translative Packing of Unit Squares into Squares

Translative Packing of Unit Squares into Equilateral Triangles

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 tr...

Packing Unit Squares in Squares: A Survey and New Results

Let s(n) be the side of the smallest square into which we can pack n unit squares. We improve the best known upper bounds for s(n) when n = 26, 37, 39, 50, 54, 69, 70, 85, 86, and 88. We present relatively simple proofs for the values of s(n) when n = 2, 3, 5, 8, 15, 24, and 35, and more complicated proofs for n=7 and 14. We also prove many other lower bounds for various s(n). We also give the ...

Packing Unit Squares in a Rectangle

For a positive integer N , let s(N) be the side length of the minimum square into which N unit squares can be packed. This paper shows that, for given real numbers a, b ≥ 2, no more than ab− (a + 1− dae)− (b + 1− dbe) unit squares can be packed in any a′ × b′ rectangle R with a′ < a and b′ < b. From this, we can deduce that, for any integer N ≥ 4, s(N) ≥ min{d√Ne, √ N − 2b√Nc + 1 + 1}. In parti...

On packing squares into a rectangle

We prove that every set of squares with total area 1 can be packed into a rectangle of area at most 2867/2048 = 1.399. . . . This improves on the previous best bound of 1.53. Also, our proof yields a linear time algorithm for finding such a packing.