Congruent Partitions of Polygons

1. Congruent Partitions of Polygons Lei He Problem 73 from The Open Problem Project: http://maven.smith.edu/~orourke/TOPP

2. 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

3. 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.

4. N=2 • 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

5. 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 partition Theorem: 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.

6. 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.

7. Claims(N=2)

8. 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

9. Summary Status: little work for N>2 not well developed for N=2 Importance: architecture

10. References • K. Erikson. Splitting a polygon into two congruent pieces. The American Mathematical Monthly, 103(5):393–400, 1996. • 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. • 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. • R. Nandakumar. Cutting mutually congruent pieces from convex regions. http://arxiv.org/abs/1012.3106, 2010. • R. Nandakumar. ’Congruent partitions’ of polygons—a short in-troduction. http://arxiv.org/abs/1002.0122, 2010.

11. Thanks!