What is a Line? A Definition and Explanation

[trambolin]

"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)

I gave you an example, how do you define a line in the set of continuous functions? You are only giving us vague geometric analogies but not actually defining something.

"(…) 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.

So, a line namely the z-axis itself cannot be defined, if I am looking towards the x-y plane because it has no length but depth. See our point? It immediately creates confusion on the simplest examples if you insist on having physical values in the definition. We are creating an unnecessary expectation by using physical analogies which we must avoid when we are trying to communicate with these ideas. We can not and should not shape the idea before we convey to others because every individual has a different imaging device in the brain.

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.

We have an underlying habit of generalizing two dimensions to three i.e starting from length then width and depth etc. We are used to look at the flatland from the top. But we can also look at it from the +ve x-axis and see things becoming bigger or smaller and also moving up and down. There is absolutely no constraint for doing not so. But it is just not the habit.

"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.

Sorry I don't understand what you say here.

 

[equilibrum]

Only if you first defined "direct displacement"!

daniel rey m. 's explanation is similar to what i was trying to convey but really,anything goes. A line is an axiom anyway,no?

 

[daniel rey m.]

"A line is an axiom anyway,no?" -- equilibrium

No, not according to Hilbert's axioms for geometry. D H, our friendly neighborhood PF Mentor here, has furnished us a link to them, and Hilbert says that "point", "line" and "plane" are "undefined terms".

" So, a line namely the z-axis itself cannot be defined, if I am looking towards the x-y plane because it has no length but depth. See our point?" -- trambolin

That is so in a 3-D context, but not when defining a line as an ideal, isolated entity, which is how it's done when teaching math on a first, elementary approach.

" We are used to look at the flatland from the top. But we can also look at it from the +ve x-axis and see things becoming bigger or smaller and also moving up and down." -- trambolin

I'm still trying to understand that. Do you mean to say look at it edge-on, as when you place a rigid piece of cardboard horizontally before your eyes and all you see is a line? In that case all you'll see of Flatland will also be a line, not things going up and down, since in that case they'd be jumping out of Flatland and falling back into it. From no point of view would you see anything there moving up and down. From above you'd see them moving within its two dimensions, on the same plane, somewhat like an airplane passenger looking down.

"Sorry I don't understand what you say here." -- trambolin

I couldn't've expressed myself more clearly in that last paragraph. It's a paragon of conciseness and clarity! Maybe the expression in Latin is causing the confusion? Meditate on its opposite --"a posteriori"-- and then maybe you'll grasp the idea in all its shining glory. Sorry, I refuse to be your virtual dictionary.

 

[trambolin]

"Sorry I don't understand what you say here." -- trambolin

I couldn't've expressed myself more clearly in that last paragraph. It's a paragon of conciseness and clarity! Maybe the expression in Latin is causing the confusion? Meditate on its opposite --"a posteriori"-- and then maybe you'll grasp the idea in all its shining glory. Sorry, I refuse to be your virtual dictionary.

Thanks, that's very kind of you.

 

[HallsofIvy]

A line is an axiom anyway,no?

I'm not sure what that even means. A "line" is a geometric object. An axiom is a statement. How can an object be a statement? I think what you intend is what I said earlier- that "line" is an undefined term in geometry.

 

[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.