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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 613201, 7 pages
http://dx.doi.org/10.1155/2012/613201
Research Article

Translative Packing of Unit Squares into Squares

Institute of Mathematics and Physics, University of Technology and Life Sciences, Kaliskiego 7, 85-789 Bydgoszcz, Poland

Received 28 March 2012; Revised 24 May 2012; Accepted 24 May 2012

Academic Editor: Naseer Shahzad

Copyright © 2012 Janusz Januszewski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

1. Introduction

Let be a positive integer, let and let a rectangular coordinate system in the plane be given. One of the coordinate system's axes is called -axis. Denote by a square in the plane with sides of unit length and with the angle between the -axis and a side of equal to . Furthermore, by denote a square with side length and with sides parallel to the coordinate axes.

We say that a collection of unit squares admits a packing into a set if there are rigid motions such that the squares are subsets of 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 , but they cannot be packed into for any . Three and four unit squares can be packed into as well (see Figure 1(a)). Obviously, two, three, or four squares can be translatively packed into . If either or , then two squares and cannot be translatively packed into . The reason is that for every the interior of any square translatively packed into covers the center of (see Figure 1(b)).

fig1
Figure 1

The problem of packing of unit squares into squares (with possibility of rigid motions) is a well-known problem (e.g., see [13]). The best packings are known for several values of . Furthermore, for many values of , there are good packings that seem to be optimal.

In this paper, we propose the problem of translative packing of squares. Denote by the smallest number such that any collection of unit squares admits a translative packing into . The problem is to find for . Obviously, . By [4, Theorem 7], we deduce that . We show that

2. Packing into Squares

Example 2.1. We have . Each unit square can be translatively packed into , but it is impossible to translatively pack into for any .

Example 2.2. We have (see [5]). Here, we only recall that two squares: and cannot be translatively packed into for any (see Figure 2(a)).

fig2
Figure 2

Example 2.3. We have . Four squares admit a translative packing into (see Figure 3, where ). In Figure 3(b) and Figure 4(a), we illustrate the cases when and , respectively. By these three pictures, we conclude that four squares cannot be translatively packed into , for any . Consequently, . On the other hand, four circles of radius can be packed into (see Figure 4(b)). Since any square can be translatively packed into a circle of radius , it follows that .

fig3
Figure 3
fig4
Figure 4

Lemma 2.4 (see [5]). Every unit square can be translatively packed into any isosceles right triangle with legs of length .

Theorem 2.5. If , then .

Proof. Let be unit squares and put Three congruent quadrangles , presented in Figure 2(b), of side lengths and , are contained in . Since the length of the diagonal of is smaller than , by Lemma 2.4 we deduce that can be translatively packed into for . Consequently, the squares can be translatively packed into and
Now assume that .
Denote by the smallest number such that circles of unit radius can be packed into . The problem of minimizing the side of a square into which congruent circles can be packed is a well-known question. The values of are known, among others, for (see Table 2.2.1 in [6] or [7]). We know that
Since each unit square is contained in a circle of radius , it follows that unit squares can be translatively packed into . It is easy to verify that for .
Finally, assume that . We use two lattice arrangements of circles.
There exists an integer such that either
Obviously, circles of radius can be packed into (see Figure 5(a)). Moreover, circles of radius can be packed into (see Figure 5(b)); it is easy to check that provided .
If , then
If , then . Since , it follows that . Thus,

fig5
Figure 5

By Theorem 2.5, Examples 2.1 and 2.2, we conclude the following result.

Corollary 2.6. Every collection of unit squares admits a translative packing into any square of area , that is, . Furthermore, .

For , the following upper bound is better than the bound presented in Theorem 2.5.

Lemma 2.7. Let be a positive integer, and let be the greatest integer not over . Then

Proof. Assume that is a collection of unit squares. Let be the greatest integer not over , let , and put
For each there exists such that
Put . We have . Moreover, let and denote the lengths of the segments presented in Figure 6(a). Since it follows that is contained in a square with side length and with the angle between the -axis and a side of equal to . We say that is a -square.
To prove Lemma 2.7, it suffices to show that can be translatively packed into . Denote by the total area of the -squares, for . Obviously,
Put for . Observe that
Thus,
The equation is equivalent to
It is easy to verify that is a solution of this equation. Consequently,
We divide into rectangles , where is a rectangle of side lengths and . Since the diagonal of each equals and it follows that all -squares admit a translative packing into for (see Figure 6(b)). Hence, and consequently can be translatively packed into . This implies that .

fig6
Figure 6

Theorem 2.8. Let be a positive integer. Then as .

Proof. Let be the greatest integer not over . Since by Lemma 2.7, it follows that, for ,

References

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