Sculpture is often seen as one of the highest forms of art. To sculptor Helaman Ferguson, it is also one of the highest forms of mathematics. How does he mix art and math to create his works?

Sculptor Helaman (HEEL-a-mon) Ferguson did not plan on merging math and art into a single career. But being born to artistic parents, growing up in the family of a stepfather who was a stone mason, and completing a doctoral degree in mathematics all planted powerful seeds.

"I was set up to do both math and art," says Ferguson from his home in the Maryland suburbs outside of Washingon, D.C. "I took some graduate courses in sculpture while I was finishing my Ph.D. at the University of Washington in Seattle. I had thought that art and science had to be in different compartments."

Ferguson learned quickly that this separation was not necessary. He began integrating mathematical principles into his paintings and eventually into his sculptures. "I'm really letting mathematics out of the closet," he says. "It has an aesthetic component. And even though math was created in response to practical problems, it has developed in directions which are not just practical but which are beautiful, interesting, and fun."

For his sculptures, which are displayed around the world, he often uses granite. "It will last a long time, even in our polluted society," he explains. "What also appeals to me is that this kind of stone is worthless, or not worth much, until you do something with it."

Ferguson adds value to the stone, ironically, through the mathematical process of subtraction. He uses diamond-tipped chain saws to cut away unwanted pieces of the granite block. The saws get smaller and smaller as he refines the shape of the sculpture. He switches to a variety of lighter and lighter grinding tools to finish and polish the work.

"You are taking away less and less," says Ferguson, "and this 'limit' idea of subtracting smaller and smaller pieces is the basis of calculus."

This approach also gives new meaning to the idea of "checking your work" when you are subtracting. "Your mistakes are costly," Ferguson observes. To avoid those mistakes, he makes extensive use of computer designs that apply complex mathematical formulas to the works he is creating.

"In a computer, you can build mathematical models, ask questions, and get answers," Ferguson says. "That's very important in designing bridges, for example. When I'm doing a 57-ton piece that will be standing in a fountain, I also have to be able to decide whether or not the piece is going to stand up."

Most of Ferguson's sculptures become elegant drawing boards for his explorations of mathematical principles—from negative Gaussian curvature to non-Euclidian geometry. If these names sound complicated, the shapes and dimensions of the artworks themselves offer a living math lesson.

Take the sculptures, Costa Surface (which is made of bronze and aluminum) and Snow Costa (which actually was made of snow for a special, temporary exhibition). Both were built around negative Gaussian curvature, in which the various curves of the objects form "saddle points." These archlike areas provide stability as well as an attractive form to the sculptures. "No matter how you turn the structure, you will always have an arch supporting it," Ferguson points out.

This stability, he adds, is the same one that allows a snow igloo to survive harsh Arctic snows and winds. It also lets a curved potato chip in a bag survive the shipping process without breaking.

Then there are Ferguson's ventures into non-Euclidian geometry. "Everybody studies Euclidean geometry in class, where the angles of a triangle always add up to 180 degrees," he says. "In non-Euclidean geometry, the sum of the angles can be anywhere between 0 and 180 degrees."

This phenomenon is made possible by the curved space of non-Euclidean geometry. (The Euclidean version uses a flat two-dimensional grid.) And Ferguson's Hyperbolic Quilt, pictured below, illustrates the world of non-Euclidean geometry and the angles and shapes that can inhabit it.

In the case of the Hyperbolic Quilt, all of these shapes are pentagons. The sum of their angles is less than the 540 degrees you would expect to find in Euclidean geometry. Because of their shape and angles, these pentagons could not exist on a flat surface in a side-by-side (or tessellated) formation. To prove the point, the leather quilt built by Ferguson will not lie flat, except one pentagon section at a time.

Among the locations where Ferguson's sculptures are displayed in the United States:

• University of California, Mathematical Sciences Research Institute, Berkeley, California
  
  
• University of St. Thomas, St. Paul, Minnesota
  
  
• National Council of Teachers of Mathematics, Reston, Virginia
  
  
• Burton Science Center, Smith College, Northampton, Massachusetts

Learn More

• In the Destination MATH: MSC V activity Calculating the Volume and Surface Area of a Right Cylinder, you can calculate the volume of bronze needed for making a sculpture of a person. (This activity is currently available on CD. An online version will be available soon.)
  
  
• The Tangible Math activity Tessellating Regular Polygons lets you explore tessellations within Euclidean geometry.

• The Riverdeep story, "The Art Angle," looks at an innovative museum where the art collection serves as an uncoventional math textbook.
  

More Links

• Visit helaman ferguson sculpture, which contains a gallery of the artist's works and explanations of the mathematics behind them.
• Find out how another sculptor uses mathematics in his works at Symbolic Sculpture and Mathematics.

Related Resources

• Get the book, Helaman Ferguson: Mathematics in Stone and Bronze, by Helaman and his wife, Claire Ferguson.