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

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: α 1 =180/√(5), α 2 =180/√(7), α 3 =180/√(11), α 4 =360-α 1 -α 2 -α 3.

4. N=2 Proper CongruenceMirror CongruenceYear K. EriksonO(n 3 logn)Not bounded1996 Rote et al.O(n 2 logn)N/A2007 Rote et al.O(n 3 ) at leastO(n 3 )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(n 3 ) algorithm

5. N=2 At least O(n 2 ) for string matching Congruence of polylines from two partitioned polygons is detected by string matching in constant time, with O(n 2 ) preprocessing and space. Theorem: Given a string R of length n, an n by n table H of integers in the range 1... n 2 can be computed in time O(n 2 ) 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/tilin gs.pdf

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

10. References 1.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.

11. Thanks!