# Just sayin’ (lines and curves)

In mathematics, a line is a kind of curve. Curves don’t need to be straight, but lines always are.

In art, a curve is also called a line. Lines don’t need to be straight, but curves never are.

I’m just sayin’.

h/t Jen Fraze, Mankato, MN

### 10 responses to “Just sayin’ (lines and curves)”

1. I love this. I have conditioned my students to expect a mild electric shock every time they utter the phrase “straight line.” About ten of them, in unison, will reply, “All lines are straight.” It’s like a geometric/catholic mass.

2. Doesn’t Euclid’s Elements (at least the most common English translation) make a distinction between a “straight line” and a more general “line” that may be curved?

3. I’d be in trouble in Chris’ class because I do say “straight line.” OUCH!! So in math, a line is a special curve (like a square is a special rectangle). Whereas in art, a curve is a special line. Did I get that right?

4. Christopher

I think that’s right, Fawn. And well put. Art experts are probably pretty rare visitors here at OMT, so there’s likely no one present to set us straight.

And Roy? You may well be right about that. But would you say in College Algebra that a parabola is a line?

• Currently, no, but I’m starting to wonder whether it might make some small kind of sense, given the popular meaning/usage of “line.” At the very least, I’ll be more careful in the future to make sure to directly emphasize that in mathematics, the word “line” usually refers specifically to a straight line. (I say “usually” because it occurs to me now that besides Euclid’s Elements, another context in which the word “line” can still refer to something curved is in calculating a “line integral.”)

I’m not trying to be contrarian, and I certainly don’t advocate going back to the older usage of the word “line” in mathematics, but I must object to the practice of discouraging specificity in language (e.g. “straight line”), even if it’s redundant to a trained mathematician.

5. Don

Chiming in late to the argument…every year I fight the “lines are straight in math” argument. My students are so used to things like “curved line tools”, the game “line rider”, and other such non-mathematical statements that they think that I am crazy. However, if we are lax in our mathematical interpretation of the word, then what will we call linear equations and linear systems? Should we change them to be first-degree equations or straight, maybe constant rate of change, equations?
With all this said, I wonder why I didn’t ever argue against a line being straight. Is this argument stemming from students having too many experiences in a wide variety of fields, lazy terminology in other fields, or maybe a lack of understanding the need for a specialized vocabulary? I don’t know, but I expect that next year I will discuss rate of change (variable and constant) before discussing our special friend the (straight [cringe]) line.

• Other fields don’t have lazy terminology; they simply have different terminology — apparently roughly the same terminology once used by mathematicians, before it was changed to reduce the need for adjectives (wait, who’s lazy? :))

We don’t need to be lax, but we also shouldn’t cringe at students using phrases that are correct and reinforce a specific meaning, even though they have recently become somewhat redundant within the context of most* mathematics.

* To continue the tradition of counterexamples from my previous two posts, there are “lines” in hyperbolic and spherical geometry that I personally wouldn’t describe as “straight”…

• For what it’s worth, the taxonomy of one-dimensional things is a standard for my high school kids. I.e., they must be able to make the distinction between line/segment/ray, which is why I’m generally such a hardass about it.

To play devil’s advocate about the Poincaré Plane and hyperbolic/spherical geometry, I would say that those lines are still “straight.” Straight is relative to what metric you choose. Poincaré lines only look curved from the Euclidean perspective. If you were a hyperbolic creature, that “curve” would be the shortest distance between any two points thereon. To quote Danielson, “Just sayin’.”

• That is a good point. In fact, the phrase “straight line” is often used by mathematicians in reference to the lines of hyperbolic and spherical geometries (just as in Euclidean geometry, as I mentioned before).