What is a Line? A Definition and Explanation

[HallsofIvy]

"Only if you first defined 'direct displacement'!" - HallsofIvy

"Movement on a flat surface along the shortest path" (i.e., no beating around the bush, no detours, no bypasses, but linger along the way and watch the landscape if you like)

As I pointed out before, this was posted under "General Mathematics", not Physics. "Movement" is a physics notion- it has no definition in mathematics. This is far from being a valid definition.

"(…) why should anything be compatible with the ordinary 3D dimensions? Why don't you define a line as saying a line = breadth without depth? Why do you start from length and depth comes always the third place?" - trambolin

One couldn't possibly define a line in such a way because breadth implies at least two dimensions. "Breadth" means "the measure of any surface from side to side". Also, "depth" means "the state or degree of being deep", and neither lines nor planes are deep, which is why they are to be seen in the imaginary two-dimensional world someone thought up and called Flatland many years ago. Consequently, as one progresses from the definition of the point, to that of the line and, finally, to that of the plane, one must start out and then proceed as indicated.

What do you mean, "always the third place"? It has to be mentioned in the third case for the very first time because it must be denied when defining a plane.

"That's why we spent two pages just to come to half conclusions." - trambolin

The two pages were ploughed through as a first step so as not to make any comments "a priori" and the warnings concerning context were duly assimilated. The definitions taken from a general-purpose dictionary were offered on the understanding that they are a first approximation to the matter.

 

[chiro]

One way of analyzing what a line is that includes the "straight" and "not straight" varieties is to consider the dimension of an object.

Something that has no dimension is a point. Something that has one dimension would be a line. Something with two dimensions is a plane. Something with three is a hyperplane and so on.

Note that a transformation will exist with simple regions to turn a "straight" line/plane/whatever into a "not straight" line/plane/whatever.

So yeah depending on the dimension of whatever you're describing, that will tell you the minimum number of variables required to describe the said thing, and based on that tell you the kind of object you're dealing with.