Posted by: Alexandre Borovik | February 19, 2009

Meditation on the Theorem of Pythagoras

From MAA Online:

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.


Mel Bochner: Meditation on the Theorem of Pythagoras

Mel Bochner: Meditation on the Theorem of Pythagoras



  1. A very 1980s American pop-culture catchphrase was to recover from a particularly embarrassing blunder by asserting, “I meant to do that!” If those around were polite, they’d make a show of believing what everybody knew to be false.

  2. The artist himself said that he was very puzzled when, planning to use 9 +16+25=50 stones, he discovered that 3 had been left over, but eventually realised where they came from. The set up for the very first and spontaneous installation, apparently, was the most romantic: ruins of a temple associated with Pythagoras, on a rainy day. Imagine this scene: a stormy night, lightnings and thunder, a torrent soaked artist stands on his knees by his impromptu installation in the shadow of a temple and looks, in puzzlement, at three extraneous stones in his hand.

    But we have to admit, Bochner registered some level of mathematical thinking. Unfortunately, the theorem itself lies at a deeper level. It remains unclear whether the artist ever reached it.

    Mel Bochner’s official website explains that this is a reproducible installation: “hazelnuts on floor / size determined by installation”. Fare enough, the Theorem of Pythagoras is also reproducible, this is the whole point of theorems.

    I love the whole affair, it is full of unintentional and therefore even sweeter irony.

  3. Perhaps the artist knew right away that there was a flaw in his illustration, or perhaps he’s only justifying it now to fend off the critics; I’m willing to give him the benefit of the doubt. But if his intention was to invite the same thoughts in the viewer that were prompted in him by the three remaining stones, why didn’t he include the extra stones in the image? If there were three stones off to one side, I think more viewers would readily conclude that the artist was aware that his image was not depicting a 3-4-5 triangle and solve the problem.

  4. Oh dear! Perhaps “4+9=16” is Bochner’s Theorem…. or maybe it’s intended as a meditative paradox or koan.

  5. […] February 21, 2009 in Uncategorized I learned about this exhibition at Heckscher Museum of Art, Huntington, NY, April 19, 2008 – June 22, 2008, only as a result of my search for more Mel Brochner musings on Pythagoras (my quest was inspired by his unintentionally outgageous Meditation on the Theorem of Pythagoras.  […]

  6. […] more Mel Brochner musings on Pythagoras (my quest was inspired by his unintentionally outrageous Meditation on the Theorem of Pythagoras. His later work is an attempt of mathematical […]

  7. I find it interesting that the error here arises from the use of different scales in the placement of the pebbles on the sides of the triangle (one scale for the pebbles on the hypotenuse, and another scale for the pebbles on the other two sides of the triangle). Realizing that there are different scales being used in the figure is, I would argue, not an inherently mathematical ability. Physicists and economists are usually better at such measurement tasks than are pure mathematicians. A person could master a great deal of 20th century mathematics, perhaps even all of it, without ever being able to see that different scales are being used here. Certainly nothing in my pure mathematical training has prepared me to notice such differences.

  8. peter: yours is a very interesting and deep comment. what you describe is a feature of affine geometry: different directions have different and incomparable scales. The Pythagoras Theorem, is, of course, the cornerstone of metric geometry.

    In two dimensions, affine geometry is symplectic: it allows to introduce the concept of area, which is a symplectic (or skew-symmetric) bilinear form. Metric geometry is the geometry of a symmetric bilinear form.

    It is a thesis of Israel Gelfand that teaching geometry should follow its natural logical division and clearly separate symplectic geometry from metric geometry. He had even started to write a textbook of elementary geometry where area is introduced before (universal) distance; at the first stage of development, every line has its own unit of length. At the next stage, a circle as a geometric figure and compasses as a drawing tool ensure the unified standards of measurement.

    Perhaps, Mel Bochner’s obsession with measurement (a theme and title of his many works) is not that naive.

  9. I think this image should be compared with the following :

    * how many equally spaced knots should a closed rope have so that you can use the Pythagorean theorem to construct a right angle with it ? Should you do something special on angles ?

    * some popular books about the history of mathematics claim that this kind of rope was used by egyptians for land surveying and masonry. But what kind of precision can we expect from this method ?

    * The computation are worth doing. Which Pythagorean trouble would you choose ?

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