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Problem 59: Most Circular Partition of a Square

Problem 59: Most Circular Partition of a Square. Chris Brown. The Problem. • What is the optimal partition of a square into convex pieces such that the circularity of the pieces is optimized? • Is the number of partitions required finite?. Circularity.

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Problem 59: Most Circular Partition of a Square

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  1. Problem 59: Most Circular Partition of a Square Chris Brown

  2. The Problem • What is the optimal partition of a square into convex pieces such that the circularity of the pieces is optimized? • Is the number of partitions required finite?

  3. Circularity • The ratio of the radius of the smallest circumscribing circle to the radius of the largest inscribed circle. • Optimized partition minimizes maximum ratio over all pieces in partition. R R R r r r

  4. Circularity of Square: Upper and Lower Bounds One-Piece Partition Upper Bound Single-Angle Lower Bound b R r r i δ r θ a

  5. Damian and O’Rourke: 2003 • Reduced upper bound by solving for partition with γ = 1.29950 • Prove new lower bound γ = 1.28868, dependent on piece adjacent to corner piece • Infinite partitions along boundary can approach lower bounds, but unclear how to fill interior with same aspect ratio

  6. Obermaier and Wagner: 2009 • Attempt to reduce bounds using evolutionary algorithm • Push operator to move vertices, Tile operator to add vertices, Star operator to repair concave pieces • Unable to reduce upper or lower bounds • Convex pieces are necessary on sides to reduce lower bound

  7. Conclusion • Problem remains open • Best complete partition has ratio 1.29950 • Best incomplete partition has ratio 1.28898 • Optimal partition is expected in the range [1.28868, 1.29950] • Conjectured to require infinite partitions

  8. Citations Mirela Damian and Joseph O’Rourke Partitioning Regular Polygons into Circular Pieces I: Convex Partions April 2003 http://arxiv.org/pdf/cs/0304023v1 Claudia Obermaier and Markus Wagner Towards an Evolved Lower Bound for the Most Circular Partition of a Square May 2009 http://cs.adelaide.edu.au/~markus/pub/2009cec.pdf

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