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Problem 59: Most Circular Partition of a Square
 Statement
 What is the optimal partition of a square into convex pieces
such that the circularity of the pieces is optimized?
The circularity of a polygon is the ratio
of the radius of its smallest circumscribing circle to the radius
of its largest inscribed circle. Thus circular pieces have
circularity near 1, and noncircular pieces have circularity greater
than 1. An optimal partition minimizes the maximum ratio over all
pieces in the partition.
 Origin
 [DO03a]
 Status/Conjectures
 Open.
 Motivation
 This is a type of ``fat'' partition.
 Partial and Related Results
 It is known from [DO03a] that the equilateral triangle
requires an infinite number of pieces to achieve the optimal
circularity of 1.5, and that for
all regular kgons, for k≥5,
the onepiece partition is optimal.
The square is a difficult intermediate case.
It is known that the optimal ratio lies in the
narrow interval
[1.28868, 1.29950].
The upper bound is established by the 92piece partition shown
in Figure 2.
Figure 2:
92piece partition achieving 1.29950

It is conjectured in [DO03a] that, as with the
equilateral triangle case, no finite partition achieves the
optimal ratio, but rather optimality can be approached as
closely as desired as the number of pieces goes to infinity.
 Categories
 packing; meshing
 Entry Revision History
 J. O'Rourke, 16 Aug. 2004.
 DO03a

Mirela Damian and Joseph O'Rourke.
Partitioning regular polygons into circular pieces I: Convex
partitions.
In Proc. 15th Canad. Conf. Comput. Geom., pages 4346, 2003.
arXiv:cs.CG/030402.
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The Open Problems Project  September 19, 2017