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Opt Art

Robert Bosch
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Joel Haack
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Robert Bosch's book Opt Art provides an interesting look at the art of optimization and the use of optimization as a tool to create artworks, several of which are depicted in the many illustrations in the book. Assuming only a first- or second-year college student's mathematical maturity, Bosch describes some of the results of his "more than sixteen-year long obsession with using mathematical and computer-based optimization techniques to create visual artwork."  Throughout the book, he introduces enough ideas from linear programming to enable the reader to appreciate the creativity of a mathematical optimizer to solve linear programming problems, including an introduction to the simplex method, the traveling salesman problem, knight's tours, and other interesting applications.
The first chapter discusses how constraints on artists enhance creativity. In Chapter 2 Bosch introduces Truchet tiles, variable Truchet tiles, and patterns made from them, using the tiles as raw material to create grayscale mosaic portraits of Father Sebastién Truchet. In this chapter, the mathematics required is differential calculus. Flexible star tiles are used to produce a mosaic portrait of Kepler and of van Gogh's Starry Night.
Linear optimization is discussed in Chapter 3 through the LEGO®problem, introducing first geometric solutions in two- and three-dimensions, then the simplex algorithm, and finally the software package Gurobi Optimizer. The need to find integer solutions leads to a discussion of the branch-and-bound algorithm.  No artwork is displayed here.
Instead, the mathematical results of Chapter 3 are put to use in the next chapter, creating mosaics of Frankenstein's monster where the tiles are themselves cartoons. This entails a discussion of linear assignment problems, introduced by a department head's task of assigning faculty to courses with the intent of minimizing the grousing that is expected to occur.
Chapter 5 discusses the more complicated case of using mosaic tiles that consist of full sets of the 55 9x9-dominoes; here a complication is that the tiles are of course rectangles rather than squares. Illustrations show again Frankenstein's monster (using 3 and 48 sets of dominos), but now also the Statue of Liberty (12 sets), Martin Luther King, Jr. (12 sets), and Barak Obama (using 44 sets for the 44th president).
In Chapter 6 we first learn how to solve the traveling salesman problem as an integer linear programming problem, perhaps requiring subtour-elimination constraints, using both Gurobi Optimizer and also the Concorde TSP solver, a sophisticated linear optimization program. After creating point sets for certain designs, continuous line drawings (Hamiltonian circuits) can be created by the TSP technique. The importance of the artist's esthetic sense in selecting the best result is discussed. Artistic examples of the use of this technique include the hands in Michelangelo's The Creation of Adam in the Sistine Chapel, portraits of three Hamiltons (William Rowan Hamilton, Lin-Manuel Miranda, who wrote and played the title role in the musical Hamilton, and the actress Linda Hamilton), and recreations of several photographs and part of the Mona Lisa.
More mathematics and art appear in Chapter 7, where the Jordan Curve Theorem and cubic Bezier curves are discussed. The lovely artwork displayed is reminiscent of Celtic knot designs. The knight's tours discussed in Chapter 8 provide an interesting wrinkle, in that all closed knight's tours have the same length. Instead of minimizing length, here the mathematician/artist seeks to maximize the symmetry of the tour in some way. Three-dimensional sculptures created from the knight's tours provide visual interest.
Chapter 9 discusses the creation of labyrinths from mosaic tiles, imposing a variety of conditions, modeling these as linear optimization problems, then solving. No artwork is displayed here beyond the labyrinth patterns themselves. Side constraints for mosaic patterns are the subject of Chapter 10, including map coloring and pattern matching. The art shown here includes again mosaics of Frankenstein's monster, part of the Mona Lisa, and the girl with the pearl earring from Vermeer's painting by that name. Finally, in Chapter 11, tiles based on patterns formed in Conway's Game-of-Life provide the material for several designs.
Bosch concludes with reflections on the beauty of the modeling for linear programming problems that is enhanced by its utility, contrasting his view with those who believe that there is beauty only in pure, not applied mathematics.
This book would be a useful resource for those with an interest in mathematics and the arts or a supplement for a course in linear programming.   


Joel Haack is Professor of Mathematics at the University of Northern Iowa. He can be reached at
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