December 15, 2012

Umbilic Torus, Writ Large

Stony Brook University has a new landmark—a gracefully contoured, intricately patterned ring that rises 24 feet above its granite base. Installed and dedicated in October, this bronze sculpture is a visual testament to the beauty of mathematics.

Created by noted sculptor and mathematician Helaman Ferguson, the sculpture was constructed from 144 bronze plates, each one unique and formed in a sandstone mold carved to order by a computer-controlled robot at Ferguson's Baltimore warehouse studio. Baltimore Sun article.

Helaman Ferguson stands with a completed portion of his Umbilic Torus SC sculpture in his Baltimore warehouse studio.

Titled Umbilic Torus SC, this sculpture is a giant version of one of Ferguson’s signature pieces, Umbilic Torus NC, created in 1988.

One rendition of Umbilic Torus NC, 27 inches tall, stands in the lobby of MAA headquarters in Washington, D.C.

The basic underlying form is a torus, but with a roughly triangular rather than a circular cross section.

This assemblage of bronze plates shows the curved triangular cross section of the sculpture.

The triangular cross section has three inwardly curving sides, which correspond to a curve called a hypocycloid. In this case, the curve is the path followed by a point on the circumference of a small circle that, in turn, is rolling inside a circle three times as wide. The result is a curve with three cusps, known as a deltoid.

As shown in this model, the Stonybrook sculpture's granite base shows this curve.

Imagine sweeping this curved triangle through space while rotating it by 120 degrees before the ends meet to form a loop. The result is one continuous surface, and the three cusps, as seen in the cross section, lie on the same curve. In other words, a finger tracing the cusp-defined rim travels three times around the ring before ending up back at its starting point. The term “umbilic” in this context refers to the particular way in which the torus is twisted to give this property.

The sculpture’s surface is covered by an approximation of a surface-filling curve know as the Peano-Hilbert curve. After a few steps, the pattern looks like an intricate but highly regular maze.

After four stages (iterations), the Peano-Hilbert curve begins to look like a maze.

Rendered in bronze, it gives the sculpture a distinctive surface relief pattern—a continuous trail that echoes Mayan pictographic writing or ancient Chinese bronze vessels. Ferguson adapted this pattern to curved contours of his sculpture.

Commissioned by Jim Simons and the Simons Foundation, Umbilic Torus SC took nearly two years to complete. The project involved not only Ferguson but also a team of engineers, welders, programmers, and others, who had to cope with one challenge after another. Even the problem of moving the massive sculpture from Baltimore to Stony Brook caused much head scratching and required considerable ingenuity to solve.

The official dedication (video) of the sculpture took place on October 25, 2012.


Ferguson, C. 1994. Helaman Ferguson: Mathematics in Stone and Bronze. Meridian Creative Group.

Ferguson, H. 1990. Two theorems, two sculptures, two posters. American Mathematical Monthly 97(August-September):589.610.

Photos by I. Peterson

December 3, 2012

A Normal Haystack

This gracefully contoured haystack at the Ikalto Monastery in the country of Georgia has a symmetric shape that resembles the characteristic "bell" curve of a normal distribution—with a readily visible y-axis poking through the top.

Photo by I. Peterson

November 22, 2012

Crafting a Penrose Tiling

An array of wooden tiles assembled into an intriguing pattern flecked with stars forms a striking contrast to the regular arrangement of bricks making up the wall on which it hangs on the third floor of Avery Hall, home of the mathematics department at the University of Nebraska-Lincoln.

Constructed by Nebraska mathematician Earl S. Kramer from diamond-shaped cherry and maple tiles and installed in March 2005, the wall piece represents a patch of one of the infinite number of ways in which to arrange fat and skinny diamonds into an aperiodic pattern characteristic of a Penrose tiling.

The two types of tiles for assembling such a Penrose tiling are rhombs (each rhomb has four sides of equal length) with acute angles of 36 and 72 degrees. Matching rules specify the ways in which these rhombs must be assembled edge to edge to create an aperiodic tiling (one in which the tiling cannot be lifted and placed back onto itself with all points displaced but still looking the same).

The particular tiling pattern depicted in the wall piece is one of two Penrose rhomb arrangements that have the dihedral automorphism group d5, featuring rotations of order five and reflections across a line, readily apparent in the design.

For another artistic representation of a Penrose tiling, see "Tessellation Tango."


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

Photos by I. Peterson

November 17, 2012

Plaster Models of Mathematical Surfaces

The shapes of surfaces captured the imagination and attention of many mathematicians during the nineteenth century. To unfold the visual secrets compressed and hidden within the shorthand of algebraic expressions, geometers drew pictures, fashioned models, and even wrote manuals on how to visualize or construct geometric forms.

These models and drawings were not only a source of pleasure but also a valuable tool for probing a slew of exotic geometric structures. Indeed, during the latter part of the nineteenth century, no university mathematics department could count itself at the forefront of research and pedagogy without owning a set of plaster models depicting an array of geometric forms. Firms, particularly in Germany, specialized in crafting these teaching aids.

Rudolf Clebsch's diagonal cubic surface.

Over the years, the models largely disappeared from view, supplanted by more abstract representations and new directions in mathematical research and pedagogical approach. More often than not, they ended up gathering dust in closets or simply in the trash.

I was reminded of this history on a recent visit to the mathematics department at the University of Nebraska in Lincoln. Retrieved from storage and exhibited in illuminated display cases, these models now serve as a link between the rich early history of geometric visualization and modern, computer-based approaches to representing the intricacies of geometric forms.

You can find collections and displays of plaster mathematical models at a number of universities, including the University of Illinois at Urbana-Champaign, University of Arizona, HarvardUniversity, Hebrew University of Jerusalem, and others. The original, hand-crafted models are also being recreated using digital fabrication technology (see and inspiring the work of artists (see

These graceful plaster models bring together the logically abstract and the visually concrete in mathematics. They are vivid testimonials to the work of nineteenth-century geometers and to the beauty of mathematical forms. They represent elegant milestones in the struggle by mathematicians to elucidate the fundamental principles of geometry.


Peterson, I. 1990. Islands of Truth: A Mathematical Mystery Cruise. W.H. Freeman.

Photos by I. Peterson

August 19, 2012

Ocean Park

"The 'Ocean Park' paintings of contemporary artist Richard Diebenkorn glow with a soft, hazy light. Translucent, luminous colors wash over barely visible skeletons of horizontal, vertical and diagonal lines. Each canvas stands as a window onto an abstract landscape—a serene sea or a stretch of open land."

I wrote those words in 1986 to start off an article for Science News magazine about an attempt to develop a set of rules—a shape grammar or formal description—that would allow someone to analyze the structure of a set of paintings and to generate similar images. Such a computer-mediated effort would, at the same time, provide insights into an artist’s working process.

My article focused on the pioneering work of computer scientist Russell A. Kirsch and art historian Joan L. Kirsch to create such a design grammar (or computer model) for the "Ocean Park" paintings of Richard Diebenkorn (1922-1993). His work lent itself to such an effort because Diebenkorn painstakingly layered his paintings, with thin washes of paint overlapping each other and an underlying framework of lines dividing the canvas into rectangular and triangular areas. His technique made it possible to trace the order in which he put down the elements of his composition.

I was reminded of that long-ago reporting and writing project when an expansive exhibition devoted to Diebenkorn's "Ocean Park" artworks recently arrived at the Corcoran Gallery of Art in Washington, D.C. It was thrilling for me to see this extensive collection, ranging from large canvases to small drawings and painted cigar-box lids, in the sky-lit, wide-open galleries of the Corcoran.

I had seen individual paintings before, but viewing works that spanned Diebenkorn's Ocean Park period (he lived in the Ocean Park neighborhood of Santa Monica, California, from 1967 to 1988), vividly revealed how his vision and technique evolved over the years.

Unable to capture the colors, texture, or brushwork of a completed Diebenkorn painting, the Kirsches concentrated on the geometric framework on which the artist draped his paint. They discovered that Diebenkorn's line patterns were tightly constrained—every line depended on every other line.

The Kirsches ended up with about 42 rules that encompassed the logical sequence in which the artist put down his lines and his fondness for nesting forms within forms. They used those rules to generate frameworks for credible simulations of Diebenkorn paintings (Digital Diebenkorn: A video of a system implementing a shape grammar of the artist Diebenkorn's paintings. Refer to the paper: J. Kirsch, Russell Kirsch, and Sandy Ressler: "Computers Viewing Artists at Work," Proceedings of Syntactic and Structural Pattern Recognition, March, 1987).

I was lucky enough to interview Diebenkorn about the Kirsch project. He was gently skeptical.

In the Science News article, I quoted Diebenkorn: "In my work, I'm continually trying to do it differently. For a picture to come to life for me, it necessitates a series of surprises or maybe one big bang of a surprise. That's the crux of my work. It's surprise that keeps it alive for me."

He added, "I'm not sure that the computer allows for that."

The Diebenkorn exhibition was put together by Sarah C. Bancroft of the Orange County Museum of Art (OCMA). The show opened at the Modern Art Museum of Fort Worth in 2011, then appeared at OCMA February 26 to May 27, 2012. It is at the Corcoran until September 23, 2012, and will then travel to other art museums.


Bancroft, S.C., S. Landauer, and P. Levitt. 2011. Richard Diebenkorn: The Ocean Park Series. Prestel USA.

Kennicott, P. 2012. "Richard Diebenkorn: The Ocean Park Series" at the Corcoran Gallery reviewed. Washington Post (June 28).

Peterson, I. 1986. Computing art. Science News 129(March 1):139-140.

Photos by I. Peterson

Geometreks in Madison I: Van Vleck Hall

Van Vleck Hall houses the mathematics department at the University of Wisconsin-Madison. Standing atop Bascom Hill, it consists of an austere tower of offices and meeting rooms and a classroom block burrowed into the hillside.

Designed by the architecture firm John J. Flad & Associates, the complex was dedicated in 1963. It is named in honor of prominent Wisconsin mathematics professor Edward Burr Van Vleck (1863-1943), who was also active in the American Mathematical Society and served at its president (1913-1914). His main mathematical interests were function theory and differential equations.

Sheathed in precast concrete panels, the stark, relentlessly symmetric tower looms over the campus.  Upon closer inspection, however, its austerity is relieved somewhat by geometric patterns that cover three of the four facades of the building’s ground floor.

Made up of arrays of lines, circles, and triangles, these patterns hint at geometric theorems, incompletely sketched in pebbled concrete.

I assume that these decorative facades were part of the original design, but I can find no information on who was responsible for them, what inspired the designs, or how they were conceived and executed.

A hint of geometric theorem?

Intriguing details include this triangle of triangles.

Three of the four sides of the ground floor block are covered with geometric designs.

A bent-paperclip bicycle rack adds another geometric element to the setting.

Photos by I. Peterson

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

January 9, 2012

Twitter Math Journal

Courtesy of Frank Farris, Santa Clara University.
Published in Mathematics Magazine, Vol. 84, No. 4 (October 2009), p. 254.

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January 8, 2012

Pleated Cone

This year's display of mathematical art at the Joint Mathematics Meetings in Boston featured an eye-popping array of three-dimensional structures. One of my favorites was a crinkled surface—an intricately pleated cone—crafted from a single sheet of blue paper.

Titled Pleated Multi-sliced Cone, this beautiful example of geometric origami was the product of a collaboration. Origami artist Robert J. Lang came up with the concept and devised the crease pattern using Mathematica. Artist Ray Schamp printed the intricate pattern on elephant hide paper. Mathematician and origami artist Thomas Hull painstakingly folded the patterned sheet of paper into the final structure.

Pleated Multi-sliced Cone
16" x 16" x 5", elephant hide paper

"Part of the charm of paper folding is its capacity for simple, elegant beauty as well as stunning complexity, all within the same set of constraints," Hull noted in the artist's statement accompanying the artwork.

Pleated Multi-sliced Cone was awarded second place in the exhibition by a panel of judges representing both the Mathematical Association of America and the American Mathematical Society.

Photos by I. Peterson