February 27, 2012

Endless Ribbon

Endless Ribbon by Max Bill (1953, original 1935), granite, Baltimore Museum of Art.

Swiss artist Max Bill (1908-1994) was a pioneer in sculpting Möbius strips. Starting in the 1930s, he created a variety of "endless ribbons" out of paper, metal, granite, and other materials.

When Bill first made a Möbius strip, in 1935 in Zurich, he thought he had invented a completely new shape.

The artist had been invited to craft a piece of sculpture to hang above a fireplace in an avant-garde house in which everything was to be electric. The idea was to add some sort of dynamic element to increase the attractiveness of an electric fireplace that would need to glow without a mesmerizing dance of flames.

One possibility was a sculpture that would rotate from the upward flow of hot air. Bill's design experiments included twisting paper strips into different configurations.

"After many experiments, I came up with a solution that seemed reasonable," he wrote in a 1972 essay.

"Not long afterwards, people began to congratulate me on my fresh and original interpretation of the Egyptian symbol for infinity and the Möbius ribbon," Bill recalled. "I had never heard of either of them. My mathematical knowledge had never gone beyond architectural calculations, and I had no great interest in mathematics."

Over the years, Bill nonetheless became a strong advocate of using mathematics as a framework for art.

"I am convinced it is possible to evolve a new form of art in which the artist's work could be founded to quite a substantial degree on a mathematical line of approach to its content," Bill argued in a 1949 essay.

As a sculptor, Bill firmly believed that geometry is the principal mechanism by which we try to understand our physical surroundings and learn to appraise relations and interactions between objects in space.

Mathematical art is best defined as "the building up of significant patterns from ever-changing relations, rhythms, and proportions of abstract forms, each one of which, having its own causality, is tantamount to a law unto itself," he insisted.

Bill went on to enumerate some of the perplexities that arise in mathematics and their implications for the artist. One such mathematical conundrum was what he described as the "inexplicability of space."

As an example, he cited the Möbius surface as one "that can stagger us by beginning on one side and ending in a completely changed aspect on the other, and somehow manages to remain that self-same side."

He cited other mathematical perplexities: the peculiar notion of remoteness or nearness of infinity, depending on your perspective, and such counterintuitive, paradoxical concepts as parallel lines that intersect and limitations without boundaries.

In tangling with these ideas, he maintained, mathematical thinking widens the scope of human vision, and art has fresh territories to explore.


Emmer, M. 2000. Mathematics and art: Bill and Escher. In Bridges: Mathematical Connections in Art, Music, and Science Conference Proceedings, R. Sarhangi, ed.

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

Photos by I. Peterson

February 21, 2012

A Flight of Tetrahedra

Standing near the shore of a large pond, the sculpture looks poised for flight. Consisting of an overlapping array of eight tetrahedra, half painted black and half white, it slants steeply into the sky.

The alternating colors and triangular surfaces remind me of the Canada geese that strut nearby, unfolding their wings before they take flight and gather into their ragged V-formations.

Titled Repetitive Graduation and fabricated from painted steel, the large piece was created by Scott Mihalik, a student at the University of North Florida in Jacksonville. Installed in the fall of 2011 as part of a "Sculpture on Campus" program, the artwork stands on the shore opposite the Coggin College of Business.

With this sculpture, Mihalik joins a number of artists who have taken advantage of the visual surprises that, with its sharp angles and four triangular faces, a tetrahedral form offers (see "A Tetrahedral Forest" and "Three Sentinels").

Photos by I. Peterson