Mathematics and sculpture
Mathematics and sculpture intersect in fascinating ways, with mathematical principles often guiding the creation and stability of sculptures. The term "sculpture" originates from the Latin word for "to carve," and sculptures can be crafted from various materials such as wood, metal, glass, and even living plants like bonsai. They can take multiple forms, including free-standing pieces and reliefs, and can be static or kinetic, challenging viewers' perceptions. Mathematical concepts are frequently represented in three dimensions through sculpture, providing visual insights into complex ideas that may be difficult to grasp. Notable examples include the work of mathematicians who create sculptures that embody geometric principles and other mathematical forms, such as Möbius loops and zonohedra. Collaborative projects, like the Hyperbolic Crochet Coral Reef, bridge art and mathematics to address broader societal issues, while initiatives by mathematicians-turned-sculptors demonstrate the aesthetic appeal and educational potential of mathematical artworks. Overall, the fusion of mathematics and sculpture creates a unique dialogue between disciplines, inviting exploration and appreciation from diverse audiences.
Mathematics and sculpture
SUMMARY: Mathematics may be necessary to assure the stability of a sculpture and sculptures can represent mathematical concepts in three dimensions.
The word “sculpture” comes from Latin sculpere, meaning “to carve.” Sculptures can be made from variety of materials, including wood, metal, glass, clay, textiles, or plastic that is carved, cast, welded, cut, or otherwise formed into shapes. Topiary and bonsai are living sculptures. Modern sculptors even experiment with light and sound. Additionally, sculptures may be free-standing objects or appear as reliefs on surfaces like walls.
The Taj Mahal, one of the most recognizable structures on Earth, includes many geometric reliefs. Sculptures can be static or kinetic, like Rube Goldberg contraptions, and projection sculptures change appearance when viewed from different sides. The outdoor Penrose tribar sculpture in East Perth, Australia, appears to be the illusory figure developed by Roger Penrose when viewed from the correct angle. While mathematical forms have long been used to create sculpture, mathematicians have come to embrace this incredibly flexible art form to investigate many mathematical concepts that might otherwise be difficult to visualize. Many mathematical sculptures are quite aesthetically pleasing in addition to being highly functional in clarifying and representing mathematical ideas. Displays of mathematical sculptures are now a regular part of many art exhibitions and mathematics conferences.
Mathematical Sculptures
Researchers who explore higher degrees of dimensionality often find it challenging to represent these concepts to people whose everyday perception is three-dimensional. Mathematician Adrian Ocneanu’s work includes modeling regular solids mathematically and physically. His “Octatube” sculpture, on display in Pennsylvania State University’s mathematics building, maps a four-dimensional space into three dimensions using triangular pieces bent into spherical shapes. “Octatube” is conformal; the angles between faces and the way the faces meet are uniform. It was sponsored by Jill Grashof Anderson, whose husband was killed on September 11, 2001. Both graduated with mathematics degrees in 1965. Mathematician Nigel Higson said, “For professionals the sculpture is very rich in meaning, but it also has an aesthetic appeal that anyone can appreciate. In addition, it helps to start conversations about abstract mathematical concepts—something that is generally hard to do with anyone other than another expert.”
Other concepts explored by mathematical sculptures include minimum variation surfaces, such as spheres, toruses, and cones, which humans tend to judge to be aesthetically pleasing because of their constant curvature; zonohedra, a class of convex polyhedra with faces that are point-symmetrical polygons, such as parallelograms; and Möbius loops, Klein bottles, and Boy’s surfaces, named for mathematicians August Möbius, Felix Klein, and Werner Boy, respectively. Sculptures on exhibit at the Fermi National Laboratory, like “Monkey-Saddle Hexagon,” focus in part on saddle-shaped minimal surfaces.
Mathematicians Who Sculpt
Art and mathematics have been intertwined for centuries, and many historical sculptors such as Leonardo da Vinci were also mathematicians. Cubist sculptors explored many new perspectives on dimension and geometry. Spouses Helaman and Claire Ferguson have created and written extensively about mathematical sculpture. Helaman developed the PLSQ algorithm for finding integer relationships, considered by many to be among the most important algorithms of the twentieth century. He creates his award-winning sculptures to represent mathematical discoveries, and the pair’s worldwide presentations have been praised for their accessibility and for initiating dialogue among multiple disciplines.
George Hart, another mathematician-sculptor, has worked in fields like dimensional analysis. He regularly hosts “sculptural barn raisings,” where people are invited to help assemble large mathematical sculptures and discuss their properties. This includes a traveling sculpture for use at schools and conferences. Hart also uses rapid prototyping technology for mathematics and sculpture work. In 2010, he left Stony Brook University to be chief of content at the interactive Museum of Mathematics, MoMath, in New York City, which opened in 2012. He remained on the board for four years and then pursued a freelance career in art. His website has featured images of his sculptures along with information about where to see his work. He has been dedicated to showing the world that math is cool and to merging the worlds of art and mathematics.
Jay Lagemann, who studied at Princeton University and received his doctorate in logic from Massachusetts Institute of Technology (MIT), began working in sculpting after living a nomadic life, traveling the world and working odd jobs. His passion for mathematics dominated his academic studies, but his love of art grew over the course of his adult life. Eventually, his eighteen-foot tall Swordfish Harpooner was commissioned for the Chilmark's tricentennial in 1994 in Massachusetts. From there, he continued to study sculpture and became inspired by everyday life around him.
Computer-Generated Sculpture
Self-taught artist and mathematician Brent Collins and computer scientist Carlo Séquin created their Fermi mathematical sculpture exhibit as part of a prolific ongoing collaboration. Séquin started researching geometric modeling in the early 1980s and Collins created saddle-form sculptures during the same period, though he only later learned their mathematical names. The Séquin-Collins Sculpture Generator combines the aesthetics of sculpture, mathematical theory, and computer visualization to allow sculptors to rapidly prototype and refine ideas electronically before beginning to work in their chosen medium. A designer can move around and through the model as well as slice and transform it. Some consider the computer images themselves to be “virtual sculpture.” In contrast, some sculptors see computer modeling as too restrictive on the symbiotic processes of design and implementation. Some directions of mathematical sculpture include knots, three-dimensional tessellations, surfaces defined by parametric equations, fractal structures, and models of complex natural entities such as organic molecules.
Other Representations and Projects
The Hyperbolic Crochet Coral Reef project combined mathematics and marine biology to call attention to global warming and other environmental issues using three-dimensional crocheted sculptures of reef lifeforms. Artists create reef components using iterative patterns, which can be permuted to produce a broad variety of lifelike designs. The project is an extension of the hyperbolic crochet work pioneered by mathematician Daina Taimina, who demonstrated that hyperbolic surfaces can be modeled physically.
Some mathematically themed sculptures represent the connections between mathematics and other aspects of society rather than trying to model explicit mathematical concepts. Oakland University’s Department of Mathematics and Statistics has a sculpted ceramic mural called Equation, which was created to explain the development of mathematics and its relationship to the universe and humanity. Though not a mathematician, artist Richard Ulrish stated that he has fond memories of the mathematics courses he took at Oakland.
Bibliography
Abouaf, Jeffrey. "Variations on Perfection: The Sequin-Collins Sculpture Generator." IEEE Computer Graphics and Applications, vol. 18, no. 6, 1998, pp. 15-20. doi:10.1109/38.734974. Accessed 10 Oct. 2024.
Crochet Coral Reef, crochetcoralreef.org/. Accessed 10 Oct. 2024.
Ferguson, Helaman, and Claire Ferguson. “Celebrating Mathematics in Stone and Bronze.” Notices of the American Mathematical Society, vol. 57, no. 7, 2010, pp. 840-850. www.ams.org/notices/201007/rtx100700840p.pdf. Accessed 10 Oct. 2024.
Hart, George W. “Geometric Sculpture.” George W. Hart, www.georgehart.com/sculpture/sculpture.html. Accessed 10 Oct. 2024.
Peterson, Ivars. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley, 2001.
Williams, Riis. “A Math Mind Turns to Sculpture Art for His Life's Mission.” The Vineyard Gazette, 6 July 2023, vineyardgazette.com/news/2023/07/06/math-mind-turns-sculpture-art-his-lifes-mission. Accessed 10 Oct. 2024.