It’s not much of an exaggeration to say that the Dutch sculptor Rinus Roelofs organizes his life around the annual Bridges conference, an international event celebrating synergies between math and art. In July, the conference drew more than 400 artistically minded mathematicians and mathematically minded artists to the Eindhoven University of Technology in the Netherlands.
Dr. Roelofs’s main contribution to the program was a half-ton, eight-foot-tall metal sculpture called “Dancing Cubes.”
“I do not call my art ‘mathematical art,’” Dr. Roelofs said in a video interview about a week later. “It’s art about mathematics. Every artist has a favorite subject, things you want to talk about. For me, that’s math.”
“Dancing Cubes” is a riff on the geometric notion of duality: a relationship by which one form emerges, or can be constructed, from another. “If you have one object, you can make another object, the dual of the first object, and if you put the two together you have a new construction,” Dr. Roelofs said.
Duality played a part in Dr. Roelofs’s dual Ph.D. in architecture and mathematics, which he obtained in 2020 at age 66. His artistic practice had always been research-driven, and in 2012, while creating a sculpture with equilateral triangles, he made a number of mathematical discoveries. For instance, he found a new series of uniform polyhedra based on helical rods. “They look like spiky, twisted cylinders,” said Doris Schattschneider, a mathematician and professor emerita at Moravian University who was on Dr. Roelofs’s thesis committee.
Dr. Schattschneider, an expert on math-art intersections, noted that an elegant example of duality involves the cube and the regular octahedron, a three-dimensional geometric solid with eight faces, each an equilateral triangle. These two polyhedra, she said, “are called geometric duals because both have 12 edges, but the cube has six faces and eight vertexes while the octahedron has eight faces and six vertexes.”
“Rinus doesn’t just do this classic display of duality, growing one construction inside the other,” Dr. Schattschneider said. “He rounds the edges and sometimes does a twist, so the end result is an intriguing sculpture.”
For “Dancing Cubes,” Dr. Roelofs used a technique of his own devising — called an extrusion transformation — to extend his creation from classical geometry to innovative art. The sculpture is composed of only two elements: a right-angled triangle (30 copies) and a rounded triangular variation (12 copies). “One needs the other to exist,” Dr. Roelofs said. “If you take away one, the sculpture will fall apart.”
The artwork earned its name because when it is assembled small cubes emerge along the central axis. “It’s a little surprise,” Dr. Roelofs said.
A larger art exhibition of similar wonders is the centerpiece of every Bridges conference. This year, the exhibition drew 184 submissions. “This is the biggest and best Bridges ever,” Dr. Schattschneider said in an email from Eindhoven.
Here is a sampling of a few of the other pieces from the exhibition:
“Gradient of Grain” was created by Edmund Harriss, a mathematician and artist at the University of Arkansas, using a computer-controlled router. The radial carvings run along the so-called gradient, perpendicular to the tree’s growth rings. The growth rings serve as inputs that direct the carving algorithm. “When using digital manufacturing to render geometry, the mathematics is often imposed on the material,” Dr. Harriss noted in his artist’s statement. “As a material, however, wood talks back, revealing patterning of grain, whatever the geometry does.”
Kanata Warisaya, a doctoral student in architecture and engineering at the University of Tokyo, said in an interview that “the fun of geometry” motivated him to create “Spherical Hinged Tessellation feat. TMK122.” Nevertheless, he envisioned this squeezable sphere as a stress toy: “I find it relaxing.” His inspiration was an art-science class led by Tomohiro Tachi, a professor of computational origami at the University of Tokyo, and Asao Tokolo, an artist.
“Scissor Tessellation” is by Seri Nishimoto, also a doctoral student at the University of Tokyo. The piece is constructed from a repeating pattern of 48 scissorlike units (four rows by 12 columns), each scissor with an arm-to-length ratio of four centimeters to seven centimeters. “The repetition of the same unit creates an interesting, nontrivial transformation into a curved surface,” Ms. Nishimoto said.
Jill Borcherds, a retired secondary school math teacher in Stevenage, England, created a colorful “Fabric Hexaflexagon.” A flexagon is a flat, folded geometric structure; when flexed along the folds, its various faces emerge. “I tried to bring out the connections between art and maths as a way of engaging students,” Ms. Borcherds said. “Those who struggled were motivated when careful measuring, calculating and constructing produced a beautiful design or 3-D object. So often I have witnessed the sense of wonder appearing when the third face of a hexaflexagon appears.” And then the fourth, fifth and sixth.
“Doo-dah” is a digital creature discovered by John Winston Garth, a computer scientist in Athens, Ala. His artist’s statement explains: “The term ‘computerrarium’ describes it well. Conway’s Game of Life is just that — an environment with rules that mimic life’s, boiled down to their simplest form: birth, death and survival. This discovery, ‘Doo-dah,’ is an example of a precise combination of cells generated by software tailored for the job. Hidden creatures exist in this environment, finding a balance within the typical randomness.”
This video shows two copies of the same gizmo, one white and one green, made by Henry Segerman, a mathematician at Oklahoma State University. Called “Expanding (3, 4, 5) Triangle,” the mechanisms are constructed from expanding racks, a system of circular gears (pinions) meshing with linear gears (racks). The white version is fully-contracted, and the green version is fully expanded. “When any part of the triangle is pulled or pushed, the entire structure gets bigger or smaller,” Dr. Segerman said. Generally, the ultimate dream with expanding mechanisms is to have a solid block of material when the racks are contracted, and then, when the racks are expanded, for all the material to be optimally deployed in a strong and stable structure.
David Reimann, a mathematician and computer scientist at Albion College in Michigan, described “Square Root of Two” as a sort of algorithmic topiary. The digits of the square root of two are outlined — in the root system’s negative space — along the diagonal of a square, conveying the relationship between a unit square and the length of its diagonal.
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