While reading Feynman’s autobiography a few days back, I came across a funny story. As an MIT undergraduate, he explained to a fellow student in a drawing course a “special” property of the French curves they used: “At the lowest point on each turn, no matter how you turn it, the tangent is horizontal. His fellow students thought apparently thought this was remarkable and spent some time playing around with the curve, despite having already taken calculus.

I ran into the same situation today. While working out an exercise on an ordinary differential equation in the plane, I wondered if a smooth curve ${c: I \rightarrow \mathbb{R}^2}$ whose tangent vector ${\dot{c}(t)}$ is always pointing in the same direction as ${c(t)}$ lies on a straight line through the origin. A reparametrization by arc length allowed me to figure it out quickly.

A few minutes later, though, I realized something. The condition above was just that the tangent vectors of ${c}$ are parallel—so ${c}$ is then a geodesic with respect to the usual connection on ${\mathbb{R}^2}$, up to reparametrization! And geodesics in this case are just straight lines.  So it should have been immediately obvious to someone who had spent a month blogging about nothing (and immersed in) but differential geometry.

This was probably a bad example, but I think the point stands that perhaps books ought to emphasize these kinds of little insights more to avoid creating this kind of fragility. Amusingly, Feynman says he found a similar problem (albeit obviously with more complicated material) when a graduate student at Princeton talking to people who worked there.

Next time I will go back to actually blogging about mathematics.