Walls, Windows, and Riemann Sums

Many years ago, while I was in high school, I got my first broad view of mathematics from a beautifully illustrated book in the LIFE Science Library. One photograph from this book that has remained lodged in my mind is of the Kresge Auditorium at the Massachusetts Institute of Technology.

Kresge Auditorium (designed by Eero Saarinen), Massachusetts Institute of Technology, Cambridge, Mass.

The photo accompanies a discussion of integration (finding totals by stuffing areas bounded by curves with rectangles). The caption suggests that the nearly rectangular windows of the auditorium's glass front fills the area under the curving roof, "architecturally illustrating the technique by which calculus finds the area under a curve."

Technically, the method of using rectangles to approximate the total area under a curve is known as a Riemann sum, named for mathematician Bernhard Riemann (1826-1866).

Monona Convention Center (designed by Frank Lloyd Wright), Madison, Wisconsin.

I still can't look at a wall without, at some point, thinking of it as an integral (or Riemann sum), filling in the area between the ground (axis) and the curve defined by the roof.

Memorial Center (Mini-Dome), East Tennessee State University, Johnson City.

Reference:

Bergamini, D., and the editors of LIFE. 1963. Mathematics. Time Books.

Photos by I. Peterson

Möbius Continuum

A complicated, twisty form stands guard at the entrance to the National Air and Space Museum on the National Mall in Washington, D.C. Created by architect and sculptor Charles O. Perry and installed in 1976, the sculpture is called Continuum.

Perry describes his artwork as a Möbius surface of seven saddles, designed to represent the continuum of the universe. In effect, despite its confounding twistiness, the sculpture has just one continuous surface and one edge—just like a Möbius strip.

"In this case, the edge of the sculpture portrays the path of a star as it flows through the center of the sculpture's 'black hole' into negative space-time and on again into positive space," Perry explains.

Like many of his sculptures, Continuum reflects Perry's desire to explore the paradoxes and enigmas posed by scientific discovery and to express the solemn beauty of scientific ideas and the attendant quest for knowledge.

"The difference between the artist and the scientist is that the scientist uses intuition to look toward finding the facts, while the artist uses intuition to intrigue others," Perry contends. "Thus, I remain here on the edge of science, encouraged to make these sculptures that attempt to speak of the orders of our universe."

Calligraphic Möbius by Charles O. Perry, Crystal City, Virginia.

Möbius surfaces appear astonishingly often in Perry's sculptures, sometimes in barely recognizable form. Perry finds topological puzzles written in the forms that emerge, seemingly unbidden, from his creative mind. What happens, for example, when two Möbius surfaces come together, then intertwine?

Helix Möbius Mace by Charles O. Perry, Crystal City, Virginia.

The amazingly different ways in which artists can present a Möbius surface are reminders that a topological form retains its essential character—in this case, its one-sidedness—no matter how much the figure is deformed, just as long as it isn't punctured or torn. Such infinite flexibility offers a vast playground for creative reconfiguration of an intriguing shape.

"The breadth of expression possible with mathematics as a discipline is almost endless," Perry insists.

Early Mace by Charles O. Perry, Peachtree Center, Atlanta, Georgia.

The basis for this twelve-foot stainless-steel sculpture is the form created when the arcs on the face of a sphere are inverted, along the lines of the stitching on a baseball.

Perry's monumental sculptures can be found in many locations around the world. If you happen to encounter one, take a close look and try to decipher its geometric message.

Icosaspirale by Charles O. Perry, in front of One Maritime Place (Alcoa Building),

San Francisco.

"Just as it sets my brain off to hear Bach, so the exquisite natural laws of form strike a chord in me," Perry says. "My pieces are ordered unto themselves, as if space had a set of rules similar to the counterpoint of music."

"At times, the works will have a single generating order which will grow to completion and stop, while others will have overlays of mutating effects," he continues. "Most, however, are given intertwining themes which are meant to be played upon by sunlight. Many have some form of transparency to them. All have a sequential movement."

Eclipse by Charles O. Perry, Hyatt Regency Hotel, Embarcadero Center, San Francisco.

References:

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

Photos by I. Peterson

An Irresistible Edge

Based on a trapezoid split into two triangles, the National Gallery of Art's East Building features walls that meet at odd angles, eschewing the right angles of more conventional structures.

One particularly sharp corner, visible to the right of the East Building's H-shaped, west-facing façade, has attracted a lot of attention. There, two walls meet at 19 degrees to form the apex of a narrow triangle. This 19-degree "fin" rises 107 feet from ground to roofline. If you look closely at the corner, you'll see a smudge darkening the lavender-pink marble.

Over the years, so many people have felt the urge to touch the unusually sharp corner that countless hands have deposited their oils on the marble to create a dark stain. It stretches over a span of about two feet, tapering off at its upper and lower ends.

In effect, the stain is a population distribution, representing all the people who have visited and touched the corner, given that most people naturally reach out to touch the wall just below shoulder height.

Normal distribution (bell curve).

The fin's stain is just one of many instances in which human use can leave its mark on objects in the environment. Such usage or wear patterns can often tell you something about the population or behavior responsible for the marks. See, for example, "Statistical Wear."

Statistician Robert W. Jernigan of American University has long studied such patterns and collected images that illustrate statistical ideas. His blog, "Statpics," includes a wide variety of such examples.

One of my favorite examples from my own travels is a stone staircase in Wells Cathedral, England, where centuries of foot traffic have worn characteristic concavities into the steps.

Photos by I. Peterson

Splitting a Trapezoid

In Washington, D.C., the National Gallery of Art's East Building, which opened to the public in 1978, features a façade that teases the eye. Designed by architect I.M. Pei, the massive structure is a regimented assemblage of vast walls, skewed polygons, and sharp edges. Walls unexpectedly meet at acute or obtuse angles instead of commonplace right angles.

To someone used to the relentlessly omnipresent right angles of more typical structures, viewing the East Building can be disconcerting. The relationships among angle, position, and perspective, born of long experience, no longer apply. One has to learn afresh how to view the building.

In commenting on the inspiration for his East Building design, Pei recalled, "I sketched a trapezoid on the back of an envelope. I drew a diagonal line across the trapezoid and produced two triangles. That was the beginning."

The trapezoid itself arose from the geometric shape of the building site, a plot bounded by four streets, with one running diagonally. Pei's design turned the trapezoid into an isosceles triangle and a smaller right triangle, a result possible because the wide side of the trapezoid was precisely twice the length of the opposite, parallel side.

Dividing the isosceles triangle into two right triangles adds to the unaccustomed abundance of acute and obtuse corners in the resulting structure. Indeed, serving as the structure's basic motif, triangles abound throughout the building.

References:

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

Photos by I. Peterson

LeWitt's Pyramid

Sol LeWitt's Four-Sided Pyramid, installed in 1999 at the National Gallery of Art Sculpture Garden in Washington, D.C., consists of concrete blocks precisely stacked to form a stark, eye-catching pyramid. In bright sunlight, the blindingly white blocks and shadows play curious visual tricks on the eye as you view the structure from different angles.

Although the blocks are rectangular, each one equivalent to two cubes attached side by side, LeWitt's structure can look like a huge pile of cubes from some viewpoints. The contrasting white blocks and dark shadows can also create a flip-flopping (isometric) optical illusion, where it isn't clear whether a given vertex is an inside or outside corner.

LeWitt used cubes and multiples of cubes, arranged in myriad ways, as basic components in many of his constructions, both solid and open. Repeating patterns and geometric regularity were also key elements of his art.

LeWitt's approach was to come up with a concept for each structure, often presented as a set of instructions that assistants could then use to construct the object. In the case of Four-Sided Pyramid, a team of engineers and stone masons, in collaboration with LeWitt, built the structure according to his plan.

The stepped shape of the terraced pyramid alludes to the setback design characteristic of many New York skyscrapers, including the Empire State Building. It also references the ziggurats (temples) of ancient Mesopotamia.

Math Problem: How many blocks make up each face of Sol LeWitt's pyramid? Assuming that the pyramid is solid and consists entirely of blocks that are twice as long as they are wide or tall, how many blocks make up the pyramid?

National Gallery of Art in the Classroom: Pyramid Math (pdf).

References:

Lancaster, R., and J. Sandefur. 2005. Sol LeWitt sculpture, four-sided pyramid. Mathematics Teacher 98(February):443.

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

Photos by I. Peterson

Martin Gardner's Generosity

I was saddened to hear this morning that Martin Gardner had died, at the age of 95.

I first encountered Martin's writings when I was in the ninth grade, and I bought a copy of the September 1962 issue of Scientific American. This particular issue featured a series of articles on Antarctica, a topic of study in my geography class. But it was Martin's discussion of divisibility tests that really caught my attention.

Scientific American stayed on my monthly shopping list, and I invariably turned to Martin's "Mathematical Games" column first. Over the years, his remarkably lucid articles introduced me to one wondrous mathematical topic after another: hexaflexagons, fractals, Möbius strips, pentagonal tilings, curves of constant width, the fourth dimension, and so much more. They left an indelible impression, and I often go back to his articles as starting points for my own mathematical excursions.  

In 1986, when I submitted the proposal for my first book to editor Jerry Lyons of W.H. Freeman, Jerry turned to Martin for his assessment of the outline. Martin was kind enough to give it a strong endorsement, noting that, as a longtime subscriber to Science News, he was already familiar with my writing. Two years later, when The Mathematical Tourist was published, Martin provided a quote for the cover.

I wrote to Martin on a number of occasions, looking for advice on where to find information on, say, unusual dice or novel applications of Möbius strips. He always had good advice and often provided me with copies of materials from his own collection.

I met Martin only once, in 1997. My family was vacationing in Asheville, N.C., while I attended MathFest in Atlanta. At the meeting, Ron Graham reminded me that Martin lives in Hendersonville, N.C., very near Asheville, and suggested that I call. When I got to Asheville, I telephoned, and Martin generously invited us to visit.

What a treat! We talked with Martin and his wife, Charlotte, about writing, books, and more. We viewed a variety of mathematical artworks, including a large portrait of Martin constructed of dominoes. He gave me and my two sons, Eric and Kenneth, a tour of his house, particularly the vast collection of papers and objects that he had amassed over the years. He posed puzzles, showed some magic tricks, and pulled out novel gadgets for the boys to try.

I was delighted to see, in person, some of the items that I remembered from long-ago columns. One was a mysterious circuit of switches, wires, and light bulbs that, as I recall, had puzzled me immensely when I had first read about it.

I owe a great deal to Martin Gardner and his generosity and kindness, and I will miss him.

Needle Tower

Slim and graceful, Kenneth Snelson's Needle Tower stretches 60 feet into the sky. The structure looks too delicate to stand so tall, but it's strong enough to withstand severe storms.

Erected in 1968 beside the Hirshhorn Museum and Sculpture Garden on the National Mall in Washington, D.C., this tapered framework of aluminum tubes and stainless-steel cables is an example of a tensegrity structure. The tubes aren't connected to each other. Instead, cables thread through the tubes to hold the assemblage together in perfect balance.

Snelson discovered the underlying principle for such structures in 1948, advocating the term "floating compression" to describe the balance between tension and compression and, in his sculptures, between flexible cables and rigid tubes. R. Buckminster Fuller (1895-1983) coined the word "tensegrity" (combining "tension" and "integrity") for the same idea, and his term stuck. Snelson refers to weaving as the "mother of tensegrity."

Snelson defines "tensegrity" as follows: "Tensegrity describes a closed structural system composed of a set of three or more elongate compression struts within a network of tension tendons, the combined parts mutually supportive in such a way that the struts do not touch one another, but press outwardly against nodal points in the tension network to form a firm, triangulated, prestressed, tension and compression unit."

Snelson's Needle Tower delivers a wonderful geometrical surprise when you venture underneath and look up to see a striking pattern of six-pointed stars.

This pattern arises naturally out of the requirement that each layer of a tensegrity structure consist of three compression elements (tubes). The sets of three alternate, giving the impression of a six-pointed star as you look up the tower. Snelson's sculptures often show this kind of symmetry.

The elegance of Snelson's tower suggests its use as an aesthetic alternative to conventional communications towers. But tensegrity structures are fairly elastic and flexible. They sway in the wind, which may not be ideal for the antennas and dishes that would top such structures.

Needle Tower recently required some conservation work. A video shows a crew of museum exhibit staff raising the repaired sculpture back into place on Hirshhorn Plaza, clearly demonstrating the structure's strength and flexibility.


Photos by I. Peterson

Infinity in Eight Minutes

From one angle, the sculpture looks like a giant, swooping loop. From another angle, it outlines a pair of wings boldly sketched across the sky.


Eight feet tall and sixteen feet wide, this stainless-steel sculpture slowly rotates atop a black granite pillar in front of the National Museum of American History in Washington, D.C. As it completes each eight-minute revolution, the glistening loop presents one strikingly different view after another.


Designed in 1965 by artist José de Rivera (1904-1985), who titled the piece Infinity, the looped sculpture is based on a mathematical figure known as the Möbius strip. A Möbius strip has just one side and one continuous edge. You can make a model of a Möbius strip by joining the two ends of a strip of paper after giving one end a 180-degree twist.


De Rivera's sculpture is a three-dimensional analog of the usual twisted, rectangular strip. The loop's cross section is an equilateral triangle, and this triangle rotates through 120 degrees before the ends meet to form a complete loop. Instead of three surfaces, the final result has just one continuous surface that runs three times around the loop.

Rotating the triangle through 240 degrees also creates a structure with one surface. Rotating it through 360 degrees produces a twisted structure that has three sides, just as it would have if the loop had a triangular cross section with no rotation.


De Rivera's sculpture was completed and unveiled in March 1967, and its debut was noted in Time magazine. "As the great form revolved majestically for the first time last week," the magazine reported, "the early spring sun glinted off its evolving planes, creating an impression of perpetual motion, a release of boundless and elemental energy."

References:

1967. Sculpture: Infinity in eight minutes. Time (April 7).

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

Shayt, D. 2008. Measuring Infinity: José de Rivera's Smithsonian sculpture on the National Mall. Curator: The Museum Journal 51(April):179-185. doi:10.1111/j.2151-6952.2008.tb00304.x

Photos by I. Peterson

The Street with No Name

In one famous episode of Homer's Odyssey, the wily Greek hero of the Trojan War Odysseus recounts the tale of how he tricked a one-eyed Cyclops named Polyphemus. Trapped in the Cyclops' cave, Odysseus offers the giant three bowls of fine wine from his ship's stores. The Cyclops downs the wine greedily and asks Odysseus his name so that he can thank Odysseus properly. Odysseus replies that his name is Nobody. That's what his mother, father, and friends call him, he insists.

Now drunk, the Cyclops topples to the floor, sound asleep. In the meantime, Odysseus and his men had prepared a stake, which they drill into the giant's eye. Polyphemus, roaring with pain, wrenches the spike from his eye and shouts for help from his neighbor Cyclops. His neighbors, awoken in the night, gather around the cave's mouth. What's the trouble, they ask. "Nobody is trying to kill me," Polyphemus bellows.

"If you're alone," his friends boom back, "and nobody's trying to overpower you now—look, it must be a plague sent by mighty Zeus and there's no escape from that." They lumber off.

I was reminded of this story when I spotted the street sign shown below: No Name Street.

The sign is in the town of Millersburg, Ohio, and you can find this short street marked on Google maps. I don't know its history, but the sign is a neat reminder of the kinds of semantic paradoxes that can arise, in this case because of a collision between the literal meaning of the words (the absence of a name) and the purpose for which the words are used (as a designation for a street).

Reference:

Fagles, R., trans. 1996. Homer: The Odyssey. Penguin.

Photo by I. Peterson