Imagine that there is a mystery closed curve about the origin. Allow two parallel lines to approach the origin from diametrically opposed directions, and have them stop where they first become tangent to the mystery curve. Suppose you do this from all pairs of directions from (0,π) to (π,0), and find that the lines stop the same distance apart everywhere. What is the mystery curve? (Don’t worry, this isn’t a post about mathematics!) You thought of a circle, right? Is that the only answer?

Many years ago, when the professor in a class on differential geometry asked that question, and provided the answer (“No”), I became fascinated by mathematical counterexamples. In this case, the Realeaux triangle is the simplest counterexample; British coins and Wankel engine pistons use these shapes:

*Realeaux triangle (Wikimedia Commons image)*

You can also make such shapes with more sides, and get rid of the sharp corners (brainteaser: how?) to get smooth equal-width curves (the generic name for such shapes). The tiny shock of seeing your assumptions undermined, and your gut-level conviction overturned, provides a high that I call an UnAha! experience.

There is an odd exhiliration to be found in showing your “gut” instinct that it is wrong. The gut is driven towards conviction. It provides to decisions a sense of moral certitude; “rightness” lives in the gut. Counter-examples keep the capacity for doubt alive. Of course, you can then go back and re-examine where your certitude is coming from, and get deeper insights as a result. I talked about this in my previous pieces, How to Define Concepts, and Concepts and Prototypes.I once knew a woman whose rhetorical style was entirely gut-driven: all declaration of (and provocation of) *pathos* and appeals to *ethos*. Her signature phrase was “I just know.” She would stare pityingly at those of who labored to add *logos* to the argument. Luckily for her, she seemed to find an ennobling grace in defeat.

I dont understand. How do you define a tangent at the C-0 corners? I think you mean distance between the parallel lines when they cant go any further (in other words, length of the object in any direction is constant). Also I dont quite understand how (0,Pi) defines a pair of directions?

Oops strike what I said in parens. Distance between the lines is correct.

i.e. for 0, pi, lines approach from east and west respectively.

you can’t define tangents at corners obviously, but that’s the brainteaser… part. There’s a way to draw ‘triangular’ equal width curves with smooth corners .

Topology and Analysis seem full of counterintuitive ideas. There are two excellent Dover books – counterexamples in Analysis and counterexamples in Topology – to back that statement up.

Personally, the most unAha! moment I ever had is discovering that rational numbers are countable. Another one is the construction of continuous, monotone increasing, nowhere differentiable functions!