## July 6, 2007

### Binary Frieze

Like a surreal piano keyboard, it stretches in a narrow band, high along the walls of the cavernous lobby of the Ernst & Young office building in downtown Toronto. Created in 1993 by artist Arlene Stamp, this artwork is based on a visual representation of binary numbers.

A base 2, or binary, number system has only two digits: 0 and 1. In place of the 1s, 10s, 100s, and 1,000s columns (representing powers of 10), the columns of a binary number represent powers of 2. The right-hand binary digit is in the 1s column, the next digit to the left is the 2s column, the next digit farther to the left is in the 4s column, and so on. So the base-10 number 7 would be 111 in base 2; 20 would be 10100; and 255 would be 11111111.

Using black squares to represent 0s and white squares to represent 1s, Stamp filled in a strip of graph paper that was 8 squares wide and 256 squares long with the sequence of binary numbers from 00000000 to 11111111. To her amazement, the resulting pattern resembled a fractal, with each section branching into smaller and smaller fingers.

"Suddenly, I could see the inextricable link between pictures of fractals generated by computers and simple binary numbers, which underlie the structure of the computer itself," Stamp says.

For the Ernst & Young building, Stamp constructed her "Binary Frieze" out of vinyl floor tiles, which were cut by hand, then assembled and glued onto aluminum supports. To perturb the relentless linearity of the fundamental binary number pattern, she also introduced a wave effect and an element of unpredictability by tilting and overlapping adjacent rectangles in certain sections of the mosaic.

The wave effect results from moving a rectangle through two functions, the first determining the width of overlap between adjacent positions of a rectangle and the second determining the degree of tilt. "The interesting thing about these interrelated functions is that they never fall into synchronization, so you cannot predict a particular position ahead," Stamp says. "Each new position of the rectangle is dependent on the previous step and the system flips into reverse when it reaches a particular limit, causing an overall wave pattern."

At the time, Stamp says, "I was very interested in the possibilities for non-repeating patterns in public spaces as a way of enlivening large areas, like floors and walls, that are so often covered up with simple repeating motifs."

"The expansive walls of the Ernst & Young lobby offered me an unusual opportunity to explore nonperiodic pattern over a large area, visible from a distance—perfect for the material presentation of the unfolding fractal pattern of binary numbers," she adds.

A second nonrepeating mosaic design, created by Stamp for the Downsview subway station in Toronto the same year, is based on the digits of pi.

References:

Peterson, I. 2002. Tiling with pi. Math Horizons 10(November):11-15.

______. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

______. 2000. Sliding pi. MAA Online (June 5).