# Ivars Peterson's MathTrek

April 7, 2003

## Fractured Granite and Fractal Prints

A rectangular slab of polished granite gives an impression of solidity and permanence. With its straight lines and glossy surface, it's an elegant, humanmade artifact meant to stand as a timeless monument or serve as an impermeable skin for a sleek skyscraper.

A fractal stone print.
Courtesy of Nat Friedman.

Breaking a granite slab produces jagged fragments with rough edges, reflecting the raw stone's geological history and structure. Zooming in for a closer view of a broken edge doesn't make the irregularities disappear. Instead, a fractured edge tends to show the same degree of roughness at different magnifications. Indeed, nature features many irregular shapes that are self-similar—that repeat themselves on different scales within the same object.

In the 1970s, mathematician Benoit B. Mandelbrot of Yale University called attention to the self-similar structure of many irregular forms. He coined the word "fractal," based on the Latin adjective meaning "broken," as a convenient label for these shapes.

In working with stone, a sculptor inevitably confronts breakage. When mathematician and sculptor Nat Friedman first looked at the shards of a broken granite slab, however, he saw something more than an unfortunate accident.

A two-sided fractal stone print by Nat Friedman.
Courtesy of Nat Friedman.

Friedman had become aware of how 20th-century sculptors such as Henry Moore (1898–1986) had opened up the solid form to create space so they could explore the interplay of form and light created by the opening. To explore the same sort of interplay in a broken granite slab, Friedman envisioned how he could reassemble the fragments, leaving gaps between the pieces to create tension and highlight the contrast between straight-line cut and fractal fracture.

Friedman sometimes reassembled the physical pieces into a sculpture. He also developed a technique for making prints from such granite assemblages.

To create a fractal stone print, Friedman begins with a one-inch-thick slab of granite, polished on one side and rough on the other. On the rough side, he draws a pattern of straight lines. Tapping with a chisel, he splits the granite along these lines. The result is a division of the initial planar shape into several pieces with ragged fractal edges.

"I have control over the pattern of straight lines that I draw on the back of the stone," Friedman writes in a recent essay. "However, I have no control over the way the stone breaks to form natural fractals, which introduces a certain randomness."

In Nat Friedman's "Pas de Deux," the perimeter straight-line Euclidean geometry plays the supporting role for the interesting inner fractal geometry.
Courtesy of Nat Friedman.

Friedman separates and arranges the broken pieces (perhaps removing some of them) into an interesting pattern laced with open space. "Part of the process is selecting appropriate spacings," he says.

Friedman then rolls black ink on the polished surfaces and places a sheet of thin, porous Japanese paper on the inked stones. When he applies pressure, ink permeates the paper to create a solid black image on the lower surface.

Ink also seeps through the paper to create an interesting gray-black image on the upper surface. "By applying pressure with a burnishing tool along the broken edges, these edges show up as black on the upper surface," Friedman notes. "Thus, one obtains a two-sided print." Folding one side of the paper partially over onto the other side creates a composite image that combines features of both types of prints.

Nat Friedman's "River and Streams," a two-sided print.
Courtesy of Nat Friedman.

Friedman's fractal stone prints can leave a vivid impression of jagged lightning bolts slashing across an otherwise dark sky. They can resemble drawings of crumpled coastlines edging unknown landmasses on an explorer's map or aerial photos of rivers and streams cutting through canyons. They intrigue the eye with stark contrasts in shape and color.

"Visual thinking leads to seeing that mathematical forms also generate art forms," Friedman maintains. An artist can look at a mathematical shape and envision unlimited possibilities, even from a shape as seemingly simple as a tetrahedron, a trefoil knot, a Möbius strip, or a fractal surface. An artist can transform a mathematical idea into an evocative artwork.

References:

Friedman, N. 2003. Fractals bounding negative space: Fractal stone prints. Mathematics Awareness Month. Available at http://mathforum.org/mam/03/essay5.html.

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. New York: Wiley.

Information about the International Society for the Arts, Mathematics, and Architecture, founded by Nat Friedman, is available at http://www.isama.org/.

Mathematics and art is the theme of this year's Mathematics Awareness Month (April 2003). Additional information can be found at http://mathforum.org/mam/03/.