Optimal filling of shapes

Area of Science:

• Computational Geometry
• Spatial Optimization
• Geometric Algorithms

Background:

• Spatial subdivision problems like covering and packing are well-studied.
• Optimal object placement within defined boundaries is a fundamental challenge.
• Previous methods lacked a unified approach for maximizing interior volume coverage with overlapping objects.

Purpose of the Study:

• To introduce and define 'filling' as a novel spatial subdivision problem.
• To establish theoretical underpinnings for optimal object placement for maximum volume coverage.
• To develop heuristic solutions for practical applications, particularly with disks and polygons.

Main Methods:

• Formulating filling as an optimization problem for object placement.
• Proving theoretical results for n-dimensional spaces using n-balls.
• Developing a heuristic algorithm for disk placement within polygons.
• Analyzing the asymptotic behavior of disk distributions as the number of disks approaches infinity.

Main Results:

• Demonstrated that optimal filling solutions in n-dimensions with n-balls correspond to sets of maximal n-balls.
• Developed a heuristic method for finding maximal disk solutions within polygonal shapes.
• Identified properties of ideal disk distributions in the limit of infinite disks.
• Drew an analogy between ideal disk distributions and energy landscapes.

Conclusions:

• Filling provides a new framework for spatial optimization problems.
• The study offers both theoretical insights and practical heuristics for volume coverage.
• The analogy to energy landscapes suggests potential links to other complex systems.