Congruent Partitions of Polygons

Problem 73 from The Open Problem Project: http://maven.smith.edu/~orourke/TOPP

Lei He

The Problem

• To partition a given polygon P into N congruent pieces(or ‘tiles’) so that the fraction of the area of P not covered by the union of the piece is as small as possible

• Congruence, perfect congruent partition

Observation

• There exist quads with no perfect congruent partition for any N. For example:α1=180/√(5), α2=180/√(7), α3=180/√(11), α4=360-α1-α2-α3.

N=2

Proper Congruence Mirror Congruence Year

K. Erikson O(n3logn) Not bounded 1996

Rote et al. O(n2logn) N/A 2007

Rote et al. O(n3) at least O(n3) 2008

• Given a polygon P, compute a partition of P into two (properly or mirror) congruent polygons P1 and P2, or indicate such a partition does not exist.

• Solved by Rote et al. with an O(n3) algorithm

N=2• At least O(n2) for string matching

Congruence of polylines from two partitioned polygons is detected by string matching in constant time, with O(n2) preprocessing and space.

Theorem: Given a string R of length n, an n by n table H of integers in the range 1 . . . n2 can be computed in time O(n2) for string matching

• At least O(n) for partitionTheorem: Let a pseudo chord denote a line segment whose endpoints are on the boundary of a polygon P. Given a simple polygon P = v0, . . . , vn−1 and a query pseudo chord , with O(n) preprocessing and space, the area of the polygon P(determined by such that either v0 ∈ δ(P) or vn−1 ∈ δ(P)) can be computed in constant time.

Claims(N=2)

Claim: If P is a convex polygon and two partition shapes are also convex pieces, what is highest fraction of the area of P is left over?1. If P is a triangle, the upper bound is roughly 5.6%2. Triangle is not the polygon shape with lowest upper bound

Claim: If P is a convex polygon, and can be partitioned perfectly into N non-convex pieces, it can be also partitioned into N convex pieces.

Claims(N=2)

Related Work

• Open problem 67: Fair Partitioning of Convex Polygons • The tiling problem

Different from partition problem, try to merge polygons instead of partitioning them.http://www.cs.duke.edu/courses/fall08/cps234/projects/tilings.pdf

Summary

Status:little work for N>2

not well developed for N=2Importance:

architecture

References1. K. Erikson. Splitting a polygon into two congruent pieces. The American

Mathematical Monthly, 103(5):393–400, 1996.2. D. El-Khechen, T. Fevens, J. Iacono, and G. Rote. Partitioning a polygon into two

congruence pieces. Proceedings of the Kyoto International Conference on Computational Geometry and Graph Theory, page ?, 2007.

3. D. El-Khechen, T. Fevens, J. Iacono, and G. Rote. Partitioning a polygon into two mirror congruent pieces. In Proc. 20th Canad. Conf. Comput. Geom., pages 131–134, August 2008.

4. R. Nandakumar. Cutting mutually congruent pieces from convex regions. http://arxiv.org/abs/1012.3106, 2010.

5. R. Nandakumar. ’Congruent partitions’ of polygons—a short in-troduction. http://arxiv.org/abs/1002.0122, 2010.

Thanks!