Optimal Partitioning of a Square Problem : Numerical Approach

Abstract

Optimal partitioning of a square problem is the search for the least-diameter way to partition a unit square into n pieces. In 1958, Wanceslas solved the problem for n = 3 in [6]. Later in 1959, Page settled the case n = 5 in [5]. In 1965, Graham studied a similar problem on partitioning an equilateral triangle into n pieces for n up to 15. It is well-known that each optimal partition is composed of polygonal pieces. In 1973, Guy and Selfridge extended the result together with a rigid definition of d n as follows: For S denoted a closed unit square, we define the family Π of all partition π of S into n disjoint subset S i such that ∪S i = S. We denote d n = inf π∈Π max 1≤i≤n sup P,Q∈Si d(P, Q) (0.1) where d(P, Q) is the Euclidian distance between P and Q. The results in [3] give the exact d n for n up to 10 and also gave the numerical bounds of d n for some n. Later in 1986, Jepsen worked independently to verify the case n up to 6 in [4]. However, the proof of each n is independent from the others. In 2013, Chaidee and Wichiramala considered the partition as an embedded planar graph of degree 3 with straight edges. We also proposed an algorithm to generate all of combinatorial patterns, which are 3-regular graph, and optimize each pattern. The result agree with the previous works for n from 4 to 7. The results are shown in [1]. Though the algorithm can be applied for larger n, the efficiency may decrease exponentially. In this work, we improve the algorithm in [1] by considering dual graph perspective. With a similar process, we generate the dual of each combinatorial pattern by triangulating vertex set of the dual graph. After that, we convert each dual back to a combinatorial pattern and then optimize numerically. To make this proceed much faster, we provide the following lemmas to eliminate patterns which do not lead to optimal patterns. Lemma 1. For n ≥ 9, an optimal partition has at least two points on each size of a square Lemma 2. For n ≥ 9, an optimal partition must have an internal pieces inside a square Lemma 3. For n ≥ 9, an optimal partition may not have three consecutive pieces laid from a corner to the opposite corner of a square. Since, for each partition, each piece is polygon, we may find its diameter by max i,j d(P i , P j) where P ′ i s are vertices of the considered piece. In the optimization process, we consider positions of vertices as variables. Therefore, the optimization process used in this work is convex programming. The results from the new algorithm agree with the result in [3] for n = 9 and 10 by using Wolfram Mathematica9.0.1. Moreover, we discover a numerical solution for n = 11 by the proposing algorithm. The numerical result confirm the bound and conjecture proposed in [3].

Full text

Optimal Partitioning of a Square Problem :
Numerical Approach
Supanut Chaidee1,∗and Wacharin Wichiramala2
Department of Mathematics and Computer Science
Faculty of Science, Chulalongkorn University
254 Phayathai Road, Pathumwan, Bangkok, Thailand
1schaidee@hotmail.com
2wacharin.w@chula.ac.th
Abstract
Optimal partitioning of a square problem is the search for the least-diameter way to partition a unit
square into npieces. In 1958, Wanceslas solved the problem for n= 3 in [6]. Later in 1959, Page settled the
case n= 5 in [5]. In 1965, Graham studied a similar problem on partitioning an equilateral triangle into n
pieces for nup to 15. It is well-known that each optimal partition is composed of polygonal pieces.
In 1973, Guy and Selfridge extended the result together with a rigid definition of dnas follows: For S
denoted a closed unit square, we define the family Π of all partition πof Sinto ndisjoint subset Sisuch
that ∪Si=S. We denote
dn= inf
π∈Πmax
1≤i≤nsup
P,Q∈Si
d(P, Q) (0.1)
where d(P, Q) is the Euclidian distance between Pand Q.
The results in [3] give the exact dnfor nup to 10 and also gave the numerical bounds of dnfor some n.
Later in 1986, Jepsen worked independently to verify the case nup to 6 in [4]. However, the proof of each n
is independent from the others. In 2013, Chaidee and Wichiramala considered the partition as an embedded
planar graph of degree 3 with straight edges. We also proposed an algorithm to generate all of combinatorial
patterns, which are 3-regular graph, and optimize each pattern. The result agree with the previous works
for nfrom 4 to 7. The results are shown in [1].
Though the algorithm can be applied for larger n, the efficiency may decrease exponentially. In this
work, we improve the algorithm in [1] by considering dual graph perspective. With a similar process, we
generate the dual of each combinatorial pattern by triangulating vertex set of the dual graph. After that,
we convert each dual back to a combinatorial pattern and then optimize numerically. To make this proceed
much faster, we provide the following lemmas to eliminate patterns which do not lead to optimal patterns.
Lemma 1. For n≥9, an optimal partition has at least two points on each size of a square
Lemma 2. For n≥9, an optimal partition must have an internal pieces inside a square
Lemma 3. For n≥9, an optimal partition may not have three consecutive pieces laid from a corner to the
opposite corner of a square.
Since, for each partition, each piece is polygon, we may find its diameter by maxi,j d(Pi, Pj) where P′
isare
vertices of the considered piece. In the optimization process, we consider positions of vertices as variables.
Therefore, the optimization process used in this work is convex programming.
The results from the new algorithm agree with the result in [3] for n= 9 and 10 by using Wolfram
Mathematicar9.0.1. Moreover, we discover a numerical solution for n= 11 by the proposing algorithm.
The numerical result confirm the bound and conjecture proposed in [3].
Acknowledgement : The authors wish to thank the Development and Promotion of Science and Technology
Talents Project (DPST) by the Institute for the Promotion of Teaching Science and Technology (IPST),
Ministry of Education, Thailand for financial support.
References
[1] S. Chaidee and W. Wichiramala, Numerical Approach to the Optimal Partitioning of a Square Problem,
Proceeding of the Thailand 18th Annual Meeting in Mathematics (2013), 199-206.
∗Corresponding author

Numerical Approach to the Optimal Partitioning of a Square Problem 2
[2] G. Chartrand and P. Zhang, Introduction to Graph Theory, McGraw Hill (2005).
[3] R. K. Guy and J. L. Selfridge, Optimal Covering of the Square, Colloquia Mathematica Societatis Janos
Bolyai (1973), 745-799.
[4] C. H. Jepsen, Coloring Points in the Unit Square, The American Mathematical Monthly 17: 3 ( 1986),
231-237.
[5] Y. Page and J. L. Selfridge, Elementary Problems and Solutions: E1374, The American Mathematical
Monthly 67: 2 (1960), 185.
[6] G. K. Wanceslas and J. Lipman, Elementary Problems and Solutions: E1311, The American Mathe-
matical Monthly 65: 10 (1958), 775.