Ivars Peterson's MathTrek
April 14, 2003
In 1840, the Danish artist Christian Albrecht Jensen (17921870) was commissioned to paint a portrait of the renowned mathematician Carl Friedrich Gauss (17771855). This portrait, showing Gauss at the venerable age of 63, went on display at the Pulkowa Observatory in St. Petersburg, Russia, where it remains to this day.
That painting of Gauss has been copied many times over the years, with variants of the portrait even appearing on bank notes and postage stamps.
One recent rendition of the Gauss portrait was constructed entirely from 48 complete sets of double-nine dominoes. The work of mathematician Robert Bosch of Oberlin College, this remarkable creation and others like it represent the successful application of a novel algorithm for approximating target images as arrays of dominoes.
To create a domino portrait, Bosch starts with an imagea picture of Gauss, Abraham Lincoln, Marilyn Monroe, John Lennon, or some other figureand a certain number of complete sets of dominoes. His goal is to position the dominoes in such a way that, when seen from a distance, the assemblage looks like the target image.
What makes this task especially challenging is the self-imposed requirement of using complete sets. Hence, when Bosch works with 48 sets, he must use precisely 48 blank dominoes, 48 dominoes that have a blank square and a square with one dot, and so on. To make this work requires delicate compromises on the placement of constituent dominoes.
To find optimal positions for the dominoes, Bosch applies a mathematical technique known as integer programming. Long used in operations research, integer programming has aided schedule construction, route selection, and a variety of other large-scale planning tasks.
Bosch's computer program starts by dividing the target image into an array of 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 intermediate values.
A domino is made up of two squares. A complete set consists of 55 dominoes10 that are doubles and 45 that are not doubles. Each double has two possible orientations (horizontal or vertical), and each nondouble has four possible orientations. Clearly, double-blank dominoes would most likely go where the grayscale value is 0, and double-nine dominoes would most likely go where the grayscale value is 9.
Using integer programming, Bosch's software determines the optimal placement and orientation of each of the available dominoes. In effect, it selects the arrangement that most closely matches the numbers that describe the original image. Choosing a larger number of sets of dominoes enables the rendering of a more accurate approximation of the original image.
Bosch has created more than 100 domino portraits. Some examples can be seen at http://www.dominoartwork.com/.
Copyright 2003 by Ivars Peterson
Bosch, R. 2002. Constructing domino portraits. Gathering for Gardner V. April 5-7. Atlanta.
Bob Bosch features his domino portraits at http://www.dominoartwork.com/.
Various portraits of Carl Friedrich Gauss can be seen at http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Gauss.html.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles. Find it at the MAA Bookstore.