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

• 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

Circularity of Square: Upper and Lower Bounds

Rr

δ

One-Piece Partition Upper Bound Single-Angle Lower Bound

ir

θr

a

b

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

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

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

Citations

Mirela Damian and Joseph O’RourkePartitioning Regular Polygons into Circular Pieces I: Convex PartionsApril 2003http://arxiv.org/pdf/cs/0304023v1

Claudia Obermaier and Markus WagnerTowards an Evolved Lower Bound for the Most Circular Partition of a SquareMay 2009http://cs.adelaide.edu.au/~markus/pub/2009cec.pdf