Translative Packing of Unit Squares into Squares
Janusz Januszewski
Institute of Mathematics and Physics, University of Technology and Life Sciences, Kaliskiego 7, 85-789 Bydgoszcz, Poland
Correspondence should be addressed to Janusz Januszewski,januszew@utp.edu.pl
Received 28 March 2012; Revised 24 May 2012; Accepted 24 May 2012
Academic Editor: Naseer Shahzad
Copyrightq2012 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.
Every collection of narbitrary-orientedunit squares admits a translative packing into any square
of side length√2.5·n.
1. Introduction
Letibe a positive integer, let 0 ≤ αi < π/2,and let a rectangular coordinate system in the
plane be given. One of the coordinate system’s axes is calledx-axis. Denote bySαia square in the plane with sides of unit length and with the angle between thex-axis and a side ofSαi equal toαi. Furthermore, byIsdenote a square with side lengthsand with sides parallel
to the coordinate axes.
We say that a collection ofnunit squaresSα1, . . . , Sαnadmits a packing into a setC if there are rigid motionsσ1, . . . , σn such that the squaresσiSαiare subsets ofCand 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 intoI2, but they cannot be packed intoI2−for any >0. Three and four unit squares can be packed intoI2as wellsee Figure1a. Obviously, two, three, or four squaresS0can be translatively packed intoI2. If eitherα1/0 orα2/0, then two squaresSα1andSα2cannot be translatively packed into
I2. The reason is that for everyα /0,the interior of any squareSαtranslatively packed into
I2covers the center ofI2 see Figure1b.
The problem of packing of unit squares into squareswith possibility of rigid motions is a well-known probleme.g., see1–3. The best packings are known for several values of
n. Furthermore, for many values ofn, there are good packings that seem to be optimal. In this paper, we propose the problem of translative packing of squares. Denote bysn
a b Figure 1
a
Q3
Q2
Q1
λ1
λ2
√
5
b
Figure 2
a translative packing intoIs. The problem is to findsnforn1,2,3, . . .. Obviously,sn>√n.
By4, Theorem 7, we deduce that limn→ ∞sn/√n1. We show that
sn≤√2.5·n. 1.1
2. Packing into Squares
Example 2.1. We haves1
√
2. Each unit square can be translatively packed intoI√2, but it is impossible to translatively packSπ/4intoI√2−for any >0.
Example 2.2. We haves2
√
5see5. Here, we only recall that two squares:Sarctan 1/2 andSarctan 2cannot be translatively packed intoI√5−for any >0see Figure2a.
Example 2.3. We haves42
√
2. Four squaresSπ/4admit a translative packing intoI2√2
see Figure3, where√2/2 ≤ λ ≤ 3√2/2. In Figure3band Figure4a, we illustrate the cases whenλ√2 andλ√2/2, respectively. By these three pictures, we conclude that four squaresSπ/4cannot be translatively packed intoI2√2−, for any > 0. Consequently,
s4 ≥ 2 √
2. On the other hand, four circles of radius √2/2 can be packed into I2√2 see Figure4b. Since any squareSαican be translatively packed into a circle of radius√2/2,
it follows thats4≤2 √
2.
Lemma 2.4see5. Every unit square can be translatively packed into any isosceles right triangle
λ
a b
Figure 3
a b
Figure 4
Theorem 2.5. Ifn≥3, thensn≤√10√5/2√3·√n.
Proof. LetSα1, Sα2,andSα3be unit squares and put
λ1
1 2
√
10√5. 2.1
Three congruent quadrangles Q1, Q2,andQ3, presented in Figure 2b, of side lengths
λ1, √
5,√2.5, andλ2 λ1− √
5, are contained in Iλ1. Since the length of the diagonal of
Sαiis smaller than√2.5, by Lemma2.4we deduce thatSαican be translatively packed
intoQi fori 1,2,3. Consequently, the squaresSα1, Sα2,andSα3can be translatively
packed intoIsand
s3≤λ1 √
10√5 2√3 ·
√
3. 2.2
Now assume that 4≤n≤16.
Denote bymnthe smallest numberssuch thatncircles of unit radius can be packed
intoIs. The problem of minimizing the side of a square into whichncongruent circles can be packed is a well-known question. The values ofmnare known, among others, forn≤16 see Table 2.2.1 in6or7. We know that
m44, m5<4.83, m6 <5.33, m7<5.74, m8 < m96,
m10<6.75, m11 <7.03, m12< m13 <7.47, m14< m15< m168.
a b Figure 5
Since each unit square is contained in a circle of radius√2/2, it follows thatnunit squares can be translatively packed intoI√2mn/2. It is easy to verify that
sn≤ √
2 2 mn<
√
10√5 2√3 ·
√
n, 2.4
forn4,5, . . . ,16.
Finally, assume thatn >16. We use two lattice arrangements of circles. There exists an integerm≥4 such that either
m2< n≤m2m or m2m < n≤m12. 2.5
Obviously,m2 circles of radius√2/2 can be packed intoI√2m see Figure 5a.
Moreover,m2mcircles of radius√2/2 can be packed intoI√2m√2/2 see Figure5b; it is easy to check that
√
3m2·
√ 2 2 < √ 2m √ 2
2 , 2.6
providedm≥4.
Ifm21≤n≤m2m, then
sn≤ √ 2m √ 2 2 ≤ √
2·√n−1
√
2 2 <
√
10√5 2√3 ·
√
n. 2.7
Ifm2m1≤n≤m12, thens
n≤√2m1. Sincem2m1≤n, it follows that
m≤1/2√4n−3−1/2. Thus,
sn≤ √ 2 1 2 √
4n−31 2
<
√
10√5 2√3 ·
√
n. 2.8
sn≤
1 π 2k
√
21k n2k2−4k2. 2.9
Proof. Assume that Sα1, . . . , Sαn is a collection of n unit squares. Letk be the greatest
integer not over√4n, letηπ/2k, and put
ζ 1η√21k n2k2−4k2. 2.10
For eachi∈ {1, . . . , n},there existsj ∈ {1,2, . . . , k}such that
j−1η≤αi < jη. 2.11
Putϕαi−j−1η. We have 0≤ϕ < η. Moreover, letλ3andλ4denote the lengths of
the segments presented in Figure6a. Since
λ3λ4 cosϕsinϕ≤1ϕ <1η, 2.12
it follows thatSαi is contained in a squarePi with side length 1η and with the angle
between thex-axis and a side ofPiequal toj−1η. We say thatPiis aj-square.
To prove Lemma2.7, it suffices to show thatP1, . . . , Pncan be translatively packed into
Iζ. Denote byAjthe total area of thej-squares, forj 1, . . . , k. Obviously,
k
j1
Ajn 1η2. 2.13
Put
hj Aj
ζ−2√2 1η2
√
2 1η, 2.14
forj1, . . . , k. Observe that
k
j1
hj A1· · ·Ak
ζ−2√2 1η 2k
√
Figure 6
Thus,
k
j1
hj n 1η
2
ζ−2√2 1η 2k
√
2 1η. 2.16
The equation
n 1η2
x−2√2 1η2k
√
2 1ηx 2.17
is equivalent to
x2−x·2√2 1ηk1 8k−n 1η20. 2.18
It is easy to verify thatxζis a solution of this equation. Consequently,
k
j1
hj n 1η
2
ζ−2√2 1η2k
√
2 1ηζ. 2.19
We divideIζintok rectanglesR1, . . . , Rk, whereRj is a rectangle of side lengthsζ
andhj. Since the diagonal of eachPiequals√21ηand
Aj
ζ−2√2 1ηhj−2√2 1η, 2.20
it follows that all j-squares admit a translative packing into Rj for j 1, . . . , k see
Figure6b. Hence,P1, . . . , Pn,and consequentlySα1, . . . , Sαncan be translatively packed
intoIζ. This implies thatsn≤ζ.
Theorem 2.8. Letnbe a positive integer. Then
sn≤√n
√
2π 2
4
√
nO1, 2.21
asn → ∞.
Proof. Letkbe the greatest integer not over√4n. Since
n2k2−4k2<n2k2≤
References
1 H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer, New York, NY, USA,
1991.
2 E. Friedman, “Packing unit squares in squares: a survey and new results,” The Electronics Journal of
Combinatorics # DS7, 2009.
3 F. G ¨obel, “Geometrical packing and covering problems,” in Packing and Covering in Combinatorics, A.
Schrijver, Ed., vol. 106 of Mathematical Centre tracts, pp. 179–199, 1979.
4 H. Groemer, “Covering and packing properties of bounded sequences of convex sets,” Mathematika,
vol. 29, no. 1, pp. 18–31, 1982.
5 J. Januszewski, “Translative packing two squares in the unit square,” Demonstratio Mathematica, vol.
36, no. 3, pp. 757–762, 2003.
6 G. Fejes T ´oth, “Packing and covering,” in Handbook of Discrete and Computational Geometry, J. E.
Good-man and J. O’Rourke, Eds., pp. 19–41, CRC, Boca Raton, Fla, USA, 1997.
Submit your manuscripts at
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Mathematics
Journal ofHindawi Publishing Corporation http://www.hindawi.com
Differential Equations
International Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex Analysis
Journal of Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Optimization
Journal ofHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014 International Journal of
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporation http://www.hindawi.com Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Discrete Mathematics
Journal ofHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014