# Science and Culture: Armed with a knack for patterns and symmetry, mathematical sculptors create compelling forms

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When he was growing up in the 1940s and 1950s, teachers and parents told Helaman Ferguson he would have to choose between art and science. The two fields inhabited different realms, and doing one left no room for the other. “If you can do science and have a lick of sense, you’d better,” he recalled being told, in a 2010 essay in the *Notices of the American Mathematical Society* (1). “Artists starve.”

Ferguson, who holds a doctorate in mathematics, never chose between art and science: now nearly 77 years old, he’s a mathematical sculptor. Working in stone and bronze, Ferguson creates sculptures, often placed on college campuses, that turn deep mathematical ideas into solid objects that anyone—seasoned professors, curious children, wayward mathophobes—can experience for themselves.

Mathematics has an intrinsic aesthetic—proofs are often described as “beautiful” or “elegant”—that can be difficult for mathematicians to communicate to outsiders, says Ferguson. “It isn’t something you can tell somebody about on the street,” he says. “But if I hand them a sculpture, they’re immediately relating to it.” Sculpture, he says, can tell a story about math in an accessible language.

Bridges between art and science no longer seem outlandish nor impossible, says Ferguson. Mathematical sculptors like him mount shows, give lectures, even make a living. They teach, build, collaborate, explore, and push the limits of 3D printing. They invoke mathematics not only for its elegant abstractions but also in hopes of speaking to the ways math underlies the world, often in hidden ways.

## Celebrating Nature

Math sculptors embed mathematical concepts in their designs. Ferguson’s *Umbilic Torus SC*, a 65-ton, 28-foot tall bronze sculpture at Stony Brook University commands attention because it stands like a looming, precision-sliced donut. But to a mathematician, it’s a nod to representation theory: the study of different ways to look at structures in abstract algebra, and Ferguson’s area of specialty. Mathematicians call this shape a “twisted toroid,” and it is connected to the group of two-by-two invertible matrices. The math doesn’t stop there. Look closely at the surface and you’ll see a raised Peano-Hilbert space-filling curve, a pattern introduced more than 100 years ago that transforms a tortuous, one-dimensional line into a two-dimensional area.

But many math sculptors also aim to connect their pieces to the observable, natural world. A gypsum sculpture by David Bachman, for example, looks just like a seashell in both shape and color. But it’s wholly artificial: Bachman first drew—on a computer screen—a curve representing the profile of the shell. Then, he used a computer program to generate equations to describe the spiral shape seen on the outside, as well as the twisting internal structure. Bachman sent his design to a 3D printer, and a convincingly real shell emerged layer by layer. It’s so convincing that the first thing people do when they see it, he says, is put it to their ear. “I wanted to prove the point that you can create a very natural looking thing with mathematics,” he says.

Seeking to evoke another aspect of the natural world, math sculptor Robert Fathauer has been building ceramic pieces based on fractals: geometric shapes, often observed in natural phenomena that are characterized by their self-similarity. Zoom in or out of fractals and you still see the same basic structure.

Fractals appear in nature in various guises. Trees, for example, have branches, which have smaller branches, which have leaves, which contain ribs and veins that also look like branches. Fractal patterns can also be found in coastlines and coral reefs. One of Fathauer’s recent fractal-inspired pieces has a circular base that rises and divides into three lobes, each of which itself divides into three lobes. Fathauer, a former researcher at NASA’s Jet Propulsion Laboratory who now lives and works in Phoenix, Arizona, created paper models for each division, and molded the clay by hand to get the effect right.

The sculpture, called *Three-Fold Hyperbolic Form*, exhibits another of his inspirations: negative curvature, a concept in geometry and topology that describes a surface curving in two different directions from every point. Saddles, for example, have negative curvature. So does the space between two hands about to engage in a handshake, a geometric idea exploited by Ferguson in a smaller bronze sculpture called *Invisible Handshake.*

Negative curvature isn’t just a mathematical curiosity; it represents a challenging idea in geometry. Hyperbolic geometry is a non-Euclidean geometry; its shapes and laws are generated when Euclid’s fifth postulate—which implies certain rules about parallel lines in a plane—is relaxed. A hyperbolic shape embedded in three Euclidean dimensions will have negative curvature anywhere.

This is what Fathauer achieved with his *Three-Fold Hyperbolic Form*. It’s not just an abstract vision of a coral reef. Formulated with non-Euclidean math, it represents a geometrical perspective that’s usually hidden from observers.

## Sculpture as Outreach

Beyond aspiring to math and nature-inspired forms, math sculpture offers the opportunity to engage people who ordinarily steer clear of mathematics. In the mid-90s, for example, Fathauer was inspired by his mathematical background to start designing toys and puzzles for students, creating tools that show math isn’t all about equations and calculations and numbers; it’s also a way of thinking about patterns, symmetry and structure. Fathauer’s toy company is called Tessellations, named after mathematical patterns made without gaps or overlaps, something like the art of M. C. Escher. In addition to tessellation coloring books, Fathauer’s company produces interactive toys, such as foam conic sections and puzzles that, when solved, reveal trigonometry proofs.

Also aiming to educate, George Hart starts his sculptures with an underlying foundation of Platonic solids, but pushes the limits of geometry by combining forms and adding flourishes, curves, and symmetries. Hart’s inspiration is classical, hearkening back to mathematical solids first described by ancient civilizations. But his methods and materials are modern. “I always begin with mathematical things that interest me, in some sense, and try to find some way to turn them into something physical,” Hart says.

Often building with everyday materials, Hart has built a polyhedron out of playing cards and a mobile called *No Picnic*, consisting of symmetric orbs made from forks, knives, and spoons. In 2011, he designed a snub dodecahedron—a 92-faced solid with 60 vertices—made entirely of paperclips. Then Hart challenged his fans to build their own.

Creating geometric sculptures, Hart notes, requires advance and careful planning. “You don’t just randomly come up with a shape that’s going to fit together; you have to think about how the parts go,” he says. “It’s a puzzle I like to pose to myself. Each of my sculptures is a puzzle for me to assemble the pieces.”

Or for his audience: Hart leads groups in constructing their own math sculptures from the pieces he designs and brings. Math departments invite him in to remind mathematicians about the humanistic side of their field. Art departments invite him to show how mathematical ideas drive architecture and design. “Sometimes in the arts, they think math is just for counting or science or whatever,” he says.

Hart uses computer modeling software to work out the geometries of his ideas, and he often uses precision laser cutters to produce the pieces. Then it’s up to his audience to figure out the patterns and fit the pieces together.

“What’s the pattern, where should the next one go? That’s a big part of how mathematicians think,” says Hart, a cofounder of the National Museum of Mathematics, in New York City. “You see examples of something and try to understand the overall structure, so you can predict how it goes in the future. I try to recreate that method of thinking in a fun event where I have people build a sculpture with me.”

Mathematical sculpture, says Ferguson, can “bring people around” to math. In 2001, the National Science Foundation hosted an exhibition of his work in Washington, DC. The audience included members of Congress. Ferguson recalls that one visiting National Science Foundation staff member remarked that the sculptures bore little resemblance to the equations, theorems, and rules he’d been forced to learn in school. “I think I succeeded in communicating something about mathematics he had no idea about,” he says. “That was important.”

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- Mathematics