Blog

Math, Art, and the World

——Watched "OPT Art by Dr. Robert Bosch"

Two years ago, when I first tried to apply Genetic Algorithm to find optimal feature sets for EEG data, I learnt about traveling salesman problem (TSP). I was fascinated by the "Mona Lisa's Smile" which was created by applying TSP, although I didn't understand the detailed method. The first time I noticed an image created by solving integer programming (IP) would probably be a year ago. In a corner of a store where I was shopping, I came across a framed picture, with a lot of sub figures splicing together. Since it was right by my foot, I scrutinized those small figures, and found that there were only five or six basic figures, such as portraits, scenery, and animals, repeatedly, creating the whole picture. Some of them were bright and some were dark, I cannot see what it tells, until I walked far away and looked back, I found that the small figures disappeared, and all I could see was a man's portrait, with the varying colors decently sketching his outlines. I was very curious about how the artist know where to put those small figures, and I guessed that it must be some computer programs that automatically matched the figures with the original portrait.

Until now, after learning Optimization, I understand better the way that these pictures are created through mathematical representation. What makes me more excited and also illustrated by Dr. Bosch in the seminar is that pictures created using the idea of IP seems to be very diversified and beautiful. When using well designed basic patterns, picture can be made simple and in high quality, and the computational time dramatically decreases. This seems happen to indicate that no matter how complex the world is, how confusing the phenomenon is, the truth hiding behind are always supported by simple rules. Using basic and simple logic to present and simplify the complexity, and exhibits artistic beauty; that is the charm of math.

The seminar given by Dr. Bosch discussed how to use optimization methods to create picture portraits and sculptures. He described the detailed steps in creating a domino picture, and used a simple assignment model to demonstrate how integer programming (IP) works and how IP helps with domino arrangement. Then he extended the problem to how to present the same picture differently, instead of dominoes, using other patterns such as tile-based art, triangle shaped, and 3D cubic based art. Finally he showed pictures created by TSP, and mentioned related works done by other artists.

As explained in the lecture, to "dominize" a target picture, we first need to chop it into squares of pixels, and then replace each square with a single "megapixel", represented with a number between 0 to 9, where 0 means darkest black and 9 means brightest white (consider the dominoes '9' has the most white dots, and '0' is all black). The most important and interesting part is using IP to determine the best possible arrangement of dominoes. We define binary variables to express whether each double '9' domino is placed in orientation v/h with the top left square covering square (i,j). The constraints include that each domino must be used s times (depending on how many sets), and each square must be covered exactly once. After the IP is solved, we can find the arrangement for all dominoes. The created pictures, look abstract when staring up close, but make sense when keeping a certain distance. It reminds me of a famous Chinese poet, living in the sixth century, who made a poem saying that "We never know the real look of Lu Mountain, as we are living in it." Sometimes we stuck at some point, it might help if we jump out and look back from a larger scale, we may find more surrounding information that putting together make a whole story. It is also very interesting that the more sets of basic patterns we use, the more accurate the created picture resembles the original one; suppose the created pictures are zoomed to the same size. In this case, resolution is higher, and each single pattern becomes less distinguishable. Result is: our eyes are so perfectly fooled that we may not even be able to tell that there are several little monkeys on Monroe's nose. We lose the details while we are obtaining the details!