08 September 2012

Dürer fails his drawing assignment

Surely, you've seen Dürer's Melencolia engraving dozens of times. There is definitely something wrong with the perspective of the ladder: it's leaning uneasily against the far side of the building, yet its feet seem to be to the left of it. But that is not the issue here.

Did you ever carefully watch the bell in the upper right corner? I never did before I read Daniel Silver's Slicing a Cone for Art and Science in the latest issue of American Scientist. Let's have a closer look.

Look carefully at the opening of the bell: its right end is wider than its left end. Wow! Didn't Dürer ever watch a coin or a mug sideways? There is no way to ever see a circle flattened to an oval, as Dürer has drawn it. It's invariably an ellipse, with two axes of symmetry. Dürer knew it was an ellipse all right, but he had the wrong idea of what an ellipse looked like. The fact that he translated the word "ellipse" by "egg line" (eyer linie) reveals his mistake: he was convinced that cutting a cone with a plane resulted in a curve which was wider where the cone was wider. He even provides an elaborate technique how to accurately draw die linie elipsis, and sure enough, it's egg shaped:

(Here the whole book, courtesy Google.) So here we have a renaissance artist of the highest rank, dedicating a whole book on how to apply mathematics to art, and adopting technical artifacts to improve precision:

Yet he didn't see what any child can see: that circles compressed conserve symmetry. It's a mystery. Did he prefer mathematics (wrong mathematics, in his case) over observation? If so, let's give Albrecht the right mathematics.

We want to cut a fixed cone with an arbitrary plane. Equivalently, we'll intersect an arbitrary cone with the fixed plane z=0. A cone standing on its head along the z-axis has an equation of the form (1). Tilting the axis to an arbitrary direction requires two rotations around orthogonal axes, and moving the vertex to an arbitrary point requires an additional translation. These transformations replace x with something like ax+by+cz+d and likewise for y and z. This results in an equation of the form (2). In fact, this says little more than that a cone, just like a sphere or a cylinder, is a quadratic surface. It intersects the plane z=0 along a curve with equation (3). By an appropriate rotation one makes the xy-term disappear. (Consult any book on plane analytic geometry.) We are left with an equation of the form (4), which can be rewritten as (5). One more translation, and we are done: (6) is the standard equation of our conic section. Evidently, the equation remains unchanged when x is replaced with -x, or y with -y. The first of these two symmetries is the one lacking in Dürer's bell where replacing x with -x does make a difference.

By this easy piece of mathematics any high school student beats Dürer, an accomplished renaissance man of the highest skills. No, an ellipse does not get wider where the cone does— common knowledge, firmly established at least since Apollonius (died c. 190 bC). Yet Dürer was unaware. Apparently, it's not all that evident if you have to find out by yourself. This seems to be a good place to recall what von Neumann said:

Young man, in mathematics you don't understand things.
You just get used to them.

P.S. In the Failed Celebrities department you might also enjoy Einstein fails his Calculus homework.