## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## May 27, 2007

### Seeing Spots

Every time you look at an image on your TV set or computer screen, you're really looking at a whole bunch of tiny dots, some dark, some light. You usually don't see those dots unless you look very closely or use a magnifying glass. Observed from a comfortable distance, they blend together to give you a recognizable Homer Simpson or a scene from Star Wars.

Illustrators and artists have taken this idea a step further. They've created bug pictures out of tiny pictures, each of which is just a darker or lighter patch in the larger image. They've even made pictures out of dominoes—those black tiles made up of two squares with white dots in them.

A domino with no dots is the darkest tile, and one in which both squares have nine dots is the lightest tile. The trick is to place dominoes within a rectangle in such a way that the light and dark patches—when seen from far enough away—add up to a recognizable image.

Mathematician Robert Bosch of Oberlin College in Ohio has set himself an even bigger challenge. His goal is to create recognizable portraits using complete sets of dominoes. There are 55 two-square tiles in a set of double-nine dominoes. If Bosch works with, say, 24 sets, he must use exactly 24 blank dominoes, 24 dominoes that have a blank square and a square with one dot, and so on.

Clearly, Bosch's method wouldn't work for a snowy Arctic scene. He wouldn't have enough dominoes with many white dots to fill in the picture properly. But it works surprisingly well for portraits of people, which usually have a nice mix of dark and light areas.

Bosch wrote a computer program to help him decide where to put each domino. His program starts by dividing the original black-and-white picture into squares. Some squares are completely white, others are completely black, and the rest are various shades of gray. White squares are assigned the value 9, black squares have the value 0, and gray squares are given in-between values. Clearly, double-blank dominoes would most likely go where the value is 0, and double-nine dominoes would most likely go where the value is 9.

Bosch's software determines where each available domino should go, selecting the arrangement that most closely matches the numbers that describe the original picture. Bosch then builds his domino portrait.

This is a domino version of the famous portrait known as the Mona Lisa. Bosch constructed the portrait from 40 complete sets of dominoes.

Bosch has made more than a hundred, including portraits of Abraham Lincoln, Albert Einstein, John Lennon, and Martin Luther King Jr. You can see some of them at http://www.dominoartwork.com/.

A classroom project involving first- and second-graders produced this domino portrait of Martin Luther King Jr.

Muse, September 2003, p. 34-35.

Images courtesy of Robert Bosch.