Author: Ed Klotz, PhD
Over the course of my career in the mathematical optimization software industry, I’ve noticed an interesting (but somewhat paradoxical) phenomenon: Mathematical optimization is all around us, but nobody sees it.
Not many people actually appreciate the impact of this powerful AI problem-solving technology, which is used by leading enterprises across more than 40 different industries and plays a key part in keeping our world running and ensuring – for example – that:
In these and other instances, mathematical optimization operates behind the scenes and is utilized to automatically generate solutions to huge, highly complex business problems – and this is most likely why most people are not aware of its importance.
But mathematical optimization is also present in many areas of our everyday lives and is used in numerous real-world applications that are more tangible than the large-scale, industry-based examples highlighted above.
Indeed, if you really look for mathematical optimization, you will find it in many unexpected places in our day-to-day world.
And so – in order to build awareness of the presence of mathematical optimization in our lives and enable greater understanding of the power of this sophisticated AI technology – I decided to write a series of blogs that uncover and showcase these (often hidden) examples of mathematical optimization applications in our everyday lives.
In this blog, we will take a look at the use of mathematical optimization in the art world.
It always surprises me how many people dislike mathematics, but are art afficionados. Many people apparently view these subjects as mutually exclusive. However, these disciplines actually share a significant amount of common ground, as we shall see in this piece.
The Art of Optimization
You may be surprised to learn that mathematical optimization tools and techniques can be leveraged to create works of art.
Of course, the use of mathematical methods in the artistic process is nothing new as many artists over the years have infused mathematics into their work – from the ancient Greeks with their stone mosaics to the Dutch graphic artist M.C. Escher, who utilized geometry extensively in his drawing and prints.
Using mathematical optimization in the creative process, however, is a somewhat novel and cutting-edge approach – but there are a handful of artists out there today who are employing this AI technology in innovative ways to produce their artworks.
Foremost among these practitioners is Robert Bosch, who is a professor of mathematics at Oberlin College & Conservatory and a professional artist who harnesses the power of mathematical optimization to create what he calls “Opt Art”, which is described in detail in his book of the same name.
Bosch, who says that “the creation of a piece of artwork can be considered a problem to be solved”, draws on mathematical optimization techniques to generate pictures, portraits, sculptures, and other works of art.
For example, he has a series of continuous line artworks that were constructed by solving instances of a classical mathematical optimization problem: the Traveling Salesman Problem (or TSP).
The TSP – which is similar to vehicle routing problems that are very common in the logistics industry – seeks to find the shortest route between a set of points (or – to be more precise – the optimal itinerary for a salesman who has to visit multiple cities and then return home).
To create his “TSP art,” Bosch follows these steps:
- Start with a target image (such as Michelangelo’s “The Creation of Adam” or Andy Warhol’s “Campbells’ Soup Cans” or even an emoji).
- Convert the target image into a series of dots.
- Use mathematical optimization techniques and tools to automatically determine the best way to draw a continuous line to connect the dots and complete the picture. The dots are considered as cities, and the optimal route that visits each dot once yields a continuous line drawing of the original image. The accuracy of the image improves as the number of dots is increased. More dots result in a more challenging TSP problem to solve, so the speed of the mathematical optimization solver is important. Bosch uses the Concorde TSP solver, a specialized TSP code that can call Gurobi as a subroutine when needed.
Below are some examples of Bosch’s “TSP art”:. I have a a print of the Warhol self portrait in my living room.
Robert Bosch’s TSP artwork based on Andy Warhol’s self-portrait
Robert Bosch’s TSP artwork “What’s Inside? This is!” which is based on Andy Warhol’s “Campbells’ Soup Cans”
Robert Bosch’s TSP artwork “The Hands,” which is based on Michelangelo’s The Creation of Adam
Robert Bosch’s TSP artworks based on emojis
Robert Bosch’s TSP sculpture “Embrace”
Another example of how Bosch uses mathematical optimization to drive his artistic process is his “Domino Mosaics.” These highly elaborate artworks are formed by leveraging computer software and mathematical optimization techniques to determine the optimal design of highly elaborate mosaics using sets of dominoes.
Robert Bosch with his “Gauss17” Domino Mosaic
Inspired by the intersections between mathematics and art, Robert Bosch applies mathematical optimization methods and technologies to create artworks of deep sophistication and enduring beauty.
Bosch’s art enables us to truly visualize and appreciate the incredible problem-solving power of mathematical optimization.
In the next blog in this series, I will explore the use of mathematical optimization in another area where you may not expect to find it: Puzzles.