Rigorous packing of unit squares into a circle

Area of Science:

• Computational Geometry
• Optimization Problems
• Applied Mathematics

Background:

• Packing problems involve fitting shapes into a container.
• Rotating unit squares in a circle is complex due to orientation freedom.
• Previous optimal solutions were only proven for packing one or two squares.

Purpose of the Study:

• To determine the smallest circle capable of containing three non-overlapping, rotatable unit squares.
• To develop a rigorous method for solving complex packing problems with rotational freedom.
• To establish optimal arrangements for packing three unit squares.

Main Methods:

• Modeling the packing problem as a constraint satisfaction problem (CSP).
• Employing computer-assisted methods with interval arithmetic techniques.
• Decomposing the CSP into subproblems by constraining square centers to overcome computational complexity.

Main Results:

• Rigorous enclosures for all optimal arrangements of three unit squares in a circle were found.
• The method successfully solved the packing problem for three squares, extending previous results.
• The approach required solving 6, 43, and 12 subproblems for 1, 2, and 3 squares, respectively.

Conclusions:

• The developed interval arithmetic-based method provides a generalizable solution for packing any number of rotating unit squares into the smallest circle.
• This study advances the understanding of optimal packing configurations for rotatable objects.
• The computer-assisted approach offers a robust way to solve complex geometric optimization problems.