In one sense, the problem of finding the densest packing of congruent circles in a square is easy to understand. But on closer inspection, this problem reveals itself to be an interesting challenge of discrete and computational geometry with all its surprising structural forms and regularities. This book summarizes results achieved in solving the circle packing problem over the past few years, providing the reader with a comprehensive view of both theoretical and computational achievements. Typically illustrations of problem solutions are shown, elegantly displaying the results obtained.

Beyond the theoretically challenging character of the problem, the solution methods developed in the book also have many practical applications.

One especially important feature of the book is the inclusion on an enclosed CD of all the open source programming codes used. Since the codes can be worked with directly, they will enable the reader to improve on them and solve problem instances that still remain challenging, or to use them as a starting point for solving related application problems.

Written for: Mathematicians, engineers, and students who are involved either in the design of constrained optimization algorithms, or in packing, covering problems, geometric optimization, or location theory

• Introduction and Problem History
• Problem Definitions and Formulations
• Bounds for the Optimum Values
• Approximate Circle Packings Using Optimization Methods
• Other Methods for Finding Approximate Circle Packings
• Interval Methods for Validating Optimal Solutions
• The First Fully Interval-based Optimization Method
• The Improved Version of the Interval Optimization Method
• Interval Methods for Verifying Structural Optimality
• Repeated Patterns in Circle Packings
• Minimal Polynomials of Point Arrangements
• About the Codes Used

• A. Currently Best Known Results for Packing Congruent Circles in a Square