Bochner’s Meditation on the Theorem of Pythagoras


The January 2009 issue of The College Mathematics Journal has a Pythagorean theme.  While the articles consist of the usual mix of varied mathematical topics, most of the smaller sidebar inserts contain quotes from books or articles about Pythagoras, and the issue concludes with reviews of recent books by Eli Maor and by Christoph Riedweg on the Pythagorean Theorem and the life of Pythagoras, respectively.

The front cover of the journal has a photograph of a piece created by the artist Mel Bochner, his “response to a visit to a temple in Metapontum”, the city where Pythagoras is said to have died.  Media:  chalk and hazelnuts.  (You can also find a different image of this work, dating to 1972, as the 16th image in the slide show of “Selected Works: 1966 – 2008” on Bochner’s website.)

bochner-pythagI love the simplicity:  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.

I do have one nit to pick, though.  If the intent was to illustrate a right triangle, then the arrangement of hazelnuts is off.  To my eye, 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.

I suspect that what was intended was something akin to the following:


Here we can view the diameter of each hazelnut as being our unit of length, so that the circular area taken up by each hazelnut suggests the unit of area (the circumscribing square) .

This image differs from Bochner’s piece in a critical way:  Bochner has arranged his hazelnuts with relatively large gaps between each nut, while in the schematic I’ve abutted them to one another, as one would do if the diameter of each nut was a unit of length, and the nuts were being used as a measurement device.

The large amount of space between the nuts is akin to lattice points in the plane, in which it is the gap itself which constitutes the unit of length, and the vertices (or hazelnuts) are our attempt at an approximation to ideal points in the plane.

If the triangle is meant to be a 3-4-5 triangle, the corresponding lattice image would be as follows:


In the end, I find Bochner’s Meditation rather confusing, and to some extent disappointing.

11 Responses to “Bochner’s Meditation on the Theorem of Pythagoras

  1. Dave Richeson Says:

    Good eye on the cover art! It looks so right, but really very wrong.

  2. Michael Henle Says:

    Even as Mel Bochner’s “Meditation on the Theorem of Pythagoras” shows a 2-3-4 triangle (not a right triangle) while displaying the squares associated with a 3-4-5 right triangle, it is still marvelously evocative of the Pythagorean Theorem. This is, after all, a work of art, not mathematics.
    There is nothing wrong here. This is not a mathematical treatise. To criticize Bochner’s work because it contains a triangle that is not a right triangle is like criticizing Guernica because Picasso did not realistically draw the fires set by the German bombs.
    Let us remember that the Pythagorean Theorem is more than a piece of mathematics but also a piece of our culture and history over 2500 years old. The Pythagorean Theorem still says what it says and “Meditation on the Theorem of Pythagoras” says something different.

  3. mel bochner Says:

    nietzsche once said “one writes not only to be understood, but also to be misunderstood.” the same can be said about the motivation for making works of art, but this strikes me as the moment to try to make this particular work “un-misunderstood”.
    when i visited the “temple of pythagoras”, in metaponte, on a cold and wet day in 1972, it was completely deserted. it isn’t much of a temple, just a few reconstructed columns, plus some ancient debris and building stones lying around. but, for whatever reason, i strongly sensed the presence of pythagoras there, and i had the urge to commemorate that feeling. what better way, i thought, than to lay out a simple demonstration, in stones, of his eponymous theorem? so, remembering my 10th grade geometry (3 squared + 4 squared = 5 squared, or 9 + 16 = 25) i picked up, from a pile of debris in the middle of the temple, 50 small stones. i laid them down and found that i still had 3 remaining. figuring a mistake had been made when i initially gathered them up, i recounted 50 stones, and laid them out again. but, again, there was a surplus of three. at first i was baffled, until it dawned on me that the surplus was due to the fact that the corners of the triangle were being counted twice, ie., they were overlapping. what i had stumbled upon was that physical entities(stones) are not equatable with conceptual entities (points). or,the real does not map onto the ideal. which is why the title of the work is “Meditation on the Theorem of Pythagoras” and not simply “Theorem of Pythagoras”. and also why art is not an illustration of ideas but a reflection upon them.
    i am pleased that after all these years someone was able to discover this “discrepancy” for themselves (although i have written about it elsewhere in art publications). that said, i do wonder 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”…

  4. jd2718 Says:

    You know, I looked at the title and the photo, and was slightly disappointed. I thought you were more discerning. But then I read. Faith instantly restored.

    I think you make a very sharp point.

    Hazel nuts (not just the meat) look like acorns. Are they closely related? Can we eat acorns?

    Noisettes. Findik. Those I knew. I googled up more translations, there seem to be dozens of unrelated words, even within language families.


  5. Ξ Says:

    This has been a very interesting post to follow. I’ll confess to initially looking at the picture through the eyes of a math teacher rather than through the eyes of an artist, and I appreciate reading what went into it, and thinking of stones versus points and also points versus lines.

    Jonathan, we can eat acorns but they have to be prepared a special way, if I’m remembering right. There are still rocks with holes right near where I grew up that were created from the members of the Chumash Tribe pounding acorns into pulp (I think then boiling water was involved, to rid the acorns of tannin).

  6. relaxation Says:

    Although you do need to boil acorns to remove the tannin, it’s worth pointing out that tannin itself is not harmful and actually has antibacterial properties.

  7. mantra meditation Says:

    This is one interesting meditation – meditating on the theorem of Pythagoras. Very interesting.

  8. Dumdum Says:

    To me, out of context, the picture represents a facet of natures balance. My first impression was that of the growth symmetry of a tree. I’d like to think of this the next time I might sketch a tree.

  9. binaural beats Says:

    Interesting stuff. I am not sure I see what you see but I tend to look at it more from an artists point of view than from a math point of view and I do like the idea of the natures balance as mentioned in the above comment.

  10. beginners meditation Says:

    A very interesting article that beginners in the art of meditation can read about. It is inspiring how you associate math and meditation. Thanks for this wonderful piece of article.

  11. Tom Says:

    Well actually the arrangements of the hazelnuts (or pebbles) in Bochner’s work into just those configurations – rather than the ones you suggest – is the whole point of the work! It is precisely at the corners of the triangles/squares where the relationship between object and concept, abstraction and materiality, begins to break down and become destabilised. The configuration that you suggest would be an illustration of an idea, nothing more. This is what makes these works by Bochner so great – they are simultaneously simple, yet intellectually complex and thought-provoking.

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