Pondering an Artist's Perplexing Tribute to the Pythagorean Theorem

The cover illustration of the January 2009 issue of The College Mathematics Journal (CMJ) has perplexed—even disturbed—a number of people. The cover features a photo of a 1972 work by prominent contemporary artist Mel Bochner titled Meditation on the Theorem of Pythagoras.


The artwork references the idea of relating the lengths of the sides of a 3-4-5 right triangle to the areas of the squares on those sides. To create the piece, Bochner used chalk and hazelnuts placed directly on the floor, materials that would have been familiar to Pythagoras. The construction represented his response to a visit to a temple in Metapontum, the city where Pythagoras purportedly died.

For some, the artwork represents a mathematical bungle. A contributor to the 360 blog, for example, wrote recently: "If the intent was to illustrate a right triangle, then the arrangement of hazelnuts is off." 360 is the unofficial blog of the Nazareth College math department in Rochester, N.Y., and features contributors from the college, local rival St. John Fisher College, and elsewhere.

"To my eye," the commenter continued, "the hazelnut grids look exactly like pins on a Geoboard, or lattice points in the plane. And given that perspective on this image, we see a 2-3-4 triangle, an obtuse triangle, and squares of area 4, 9, and 16." So, from that viewpoint, the mathematics isn't correct.

The commenter then presented his own version of what the artist might have meant in illustrating "the fundamental ideas of relating the lengths of the sides of a triangle to the areas of the squares on those sides using readily available materials."

In the end, the critic wrote, "I find Bochner's Meditation rather confusing, and to some extent disappointing."

In response, CMJ Editor Michael Henle of Oberlin College noted that Bochner's creation is still "marvelously evocative" of the Pythagorean Theorem, a common thread that links several articles in the January issue of CMJ. "This is, after all, a work of art, not mathematics," he added.

Moreover, the Pythagorean Theorem "is more than a piece of mathematics," Henle said. It is "also a piece of our culture and history over 2500 years old." The theorem, he noted, "still says what it says and Meditation on the Theorem of Pythagoras says something different."

Mel Bochner had his own rejoinder. He had visited the deserted temple on a cold and wet day in 1972, finding it little more than a few reconstructed columns, some ancient debris, and scattered building stones. Nonetheless, he strongly sensed the presence of Pythagoras and had the urge to commemorate that feeling.

Remembering his 10th-grade geometry (3² + 4² = 5², or 9 + 16 = 25), Bochner gathered 50 small stones from the temple debris and laid them down. But when he created his pattern, he found that he had three stones left over. Finally, it dawned upon him that the surplus came from counting the corners of the triangle twice.

"What I had stumbled upon was that physical entities (stones) are not equatable with conceptual entities (points)," Bochner said, "or the real does not map onto the ideal."

That's "why the title of the work is Meditation on the Theorem of Pythagoras and not simply Theorem of Pythagoras," Bochner noted, "and also why art is not an illustration of ideas but a reflection upon them."

Bochner welcomed the rediscovery of this "discrepancy" so many years after he had created the artwork. Yet he also wondered "about the unwillingness to assume that I already knew what they had just discovered (do mathematicians still think all artists are dumb?) and not take the next step and ask themselves if it might have been intended to be 'confusing.'"

Henle is pleased with the ongoing debate. "This is the kind of thing I hoped would happen," he said.