Archive for January 28th, 2009

Bochner’s Meditation on the Theorem of Pythagoras

January 28, 2009

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.