The Colloquial ▲

The Triangle Offense, the Triangular Trade, the Bermuda Triangle. None of these evoke the  equilateral upwards-pointing three-sided polygon that Lupyan finds is the most common prototypical representation of the triangle. Yet in each the concept of the triangle informs meaning, similar to drum circle or checkout line. To define a triangle as just a shape is to ignore the societal context it has gained, even if it was originally because of its rigidly agreed-upon definition. Why is this?

One theory would be that these terms rely upon a faulty understanding of the definition of the triangle. Perhaps historians describing ships' non-linear paths across the ocean thought that a triangle's sides could be curved. Of course, I do not find this convincing. Consider a shape that looked just like these ships' paths: a triangle whose edges are replaced by faint curves that start and stop at the same places. Even if that is not a mathematical triangle, I would not hesitate to call that a triangle, depending on the context.

So it seems that a concept can still be recognized as such if one of its features is relaxed. However, this does not apply to all features. After all, adding another side to a triangle would make it no longer be a triangle, even in the most socially-abstracted, non-mathematical sense. Something about the curvature of the sides makes it more 'triangular' than the number of sides. I propose two potentially co-existing mechanisms for this.

The first is encroachment on the ground of another concept. Adding a side to a triangle makes it a quadrilateral, and the set of quadrilaterals and triangles is widely understood to be disjoint; I would guess that set relations are among the most fundamental concept relations we have. Of all of the features a basic shape (as society has defined a basic shape) can have, sidedness is among the most important because it provides us with a heuristic for distinguishing basic shapes from one another. Whenever we are asked what type of shape something is, we use this heuristic, ignoring whether or not the line might be curved, or dotted or even just implied. If we were faced with shapes with curved sides enough that we had to add word-concepts for them into the language, curvature would suddenly not be an acceptable trait of a 'triangle'.

The second is pliability of the original concept. The line is the mundane concept most associated with a side of a shape, and in a mundane context, a line can be anything that is thin and starts at one place and ends at another. That is not to say that the curved line is just as 'linear' as a straight line, but that it is similar enough to the original idea that the concept would be well-understood.