Translative Packing of Unit Squares into Squares

Every collection of n (arbitrary-oriented) unit squares admits a translative packing into any square of side length √2.5⋅𝑛.


Introduction
Let i be a positive integer, let 0 ≤ α i < π/2, and let a rectangular coordinate system in the plane be given.One of the coordinate system's axes is called x-axis.Denote by S α i a square in the plane with sides of unit length and with the angle between the x-axis and a side of S α i equal to α i .Furthermore, by I s denote a square with side length s and with sides parallel to the coordinate axes.
We say that a collection of n unit squares S α 1 , . . ., S α n admits a packing into a set C if there are rigid motions σ 1 , . . ., σ n such that the squares σ i S α i are subsets of C and that they have mutually disjoint interiors.A packing is translative if only translations are allowed as the rigid motions.
For example, two unit squares can be packed into I 2 , but they cannot be packed into I 2 − for any > 0. Three and four unit squares can be packed into I 2 as well see Figure 1 a .Obviously, two, three, or four squares S 0 can be translatively packed into I 2 .If either α 1 / 0 or α 2 / 0, then two squares S α 1 and S α 2 cannot be translatively packed into I 2 .The reason is that for every α / 0, the interior of any square S α translatively packed into I 2 covers the center of I 2 see Figure 1 b .
The problem of packing of unit squares into squares with possibility of rigid motions is a well-known problem e.g., see 1-3 .The best packings are known for several values of n.Furthermore, for many values of n, there are good packings that seem to be optimal.
In this paper, we propose the problem of translative packing of squares.Denote by s n the smallest number s such that any collection of n unit squares S α 1 , . . ., S α n admits International Journal of Mathematics and Mathematical Sciences a b Figure a translative packing into I s .The problem is to find s n for n 1, 2, 3, . ... Obviously, s n > √ n.

Packing into Squares
Proof.Let S α 1 , S α 2 , and S α 3 be unit squares and put

2.1
Three congruent quadrangles Q 1 , Q 2 , and Q 3 , presented in Figure 2 b , of side lengths λ 1 , √ 5, √ 2.5, and λ 2 λ 1 − √ 5, are contained in I λ 1 .Since the length of the diagonal of S α i is smaller than √ 2.5, by Lemma 2.4 we deduce that S α i can be translatively packed into Q i for i 1, 2, 3. Consequently, the squares S α 1 , S α 2 , and S α 3 can be translatively packed into I s and Now assume that 4 ≤ n ≤ 16.Denote by m n the smallest number s such that n circles of unit radius can be packed into I s .The problem of minimizing the side of a square into which n congruent circles can be packed is a well-known question.The values of m n are known, among others, for n ≤ 16 see Table 2 for n 4, 5, . . ., 16.Finally, assume that n > 16.We use two lattice arrangements of circles.
There exists an integer m ≥ 4 such that either By Theorem 2.5, Examples 2.1 and 2.2, we conclude the following result.
Corollary 2.6.Every collection of n unit squares admits a translative packing into any square of area 2.5 n, that is, For n ≥ 1980, the following upper bound is better than the bound presented in Theorem 2.5.Lemma 2.7.Let n be a positive integer, and let k be the greatest integer not over 4 √ n.Then 2.9 Proof.Assume that S α 1 , . . ., S α n is a collection of n unit squares.Let k be the greatest integer not over 4 √ n, let η π/2k, and put

2.11
Put ϕ α i − j − 1 η.We have 0 ≤ ϕ < η.Moreover, let λ 3 and λ 4 denote the lengths of the segments presented in Figure 6 a .Since 12 it follows that S α i is contained in a square P i with side length 1 η and with the angle between the x-axis and a side of P i equal to j − 1 η.We say that P i is a j-square.
To prove Lemma 2.7, it suffices to show that P 1 , . . ., P n can be translatively packed into I ζ .Denote by A j the total area of the j-squares, for j 1, . . ., k. Obviously,

2.13
Put Thus, The equation It is easy to verify that x ζ is a solution of this equation.Consequently,

2.19
We divide I ζ into k rectangles R 1 , . . ., R k , where R j is a rectangle of side lengths ζ and h j .Since the diagonal of each P i equals √ 2 1 η and it follows that all j-squares admit a translative packing into R j for j 1, . . ., k see Figure 6