# How do curves consist of points?

According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in other words there is nothing, then how is it possible to draw any curve? The thing I could imagine is that maybe a point is not as Euclid thought. I mean a point can be thought of as a small line segment whose length is approaching zero but never becomes exactly zero. In this way, we can say that curves consist of points (the line segment with length approaching zero). But then it breaks the fact as given by Euclid that a point has no dimensions. Please help me to get out of this dilemma.

• You are wrong by thinking that the union of $n$-dimensional objects must be $n$-dimensional as well. Think of $\mathbb{R}^2$: It is the union of all lines $\mathbb{R}(x,0)$ which are parallel to the $y$-axis. These lines are all one-dimensional objects, but their union $\mathbb{R}^2$ is a 2-dimensional vector space. May 30, 2016 at 5:34
• Ah okay, then I agree with the others: We live in 3-dimensional space, and however you "draw" something, it will in fact be 3-dimensional. The point is that "drawing" is hard to be defined/modeled in a mathematically correct way. you are merely moving a pen around space, and through some mechanism the ink is coming out and sticking on a sheet of paper after a certain period of time. So as soon as you want to imagine a (not really existing) pen that draws single points, it raises the question of how you even define that. May 30, 2016 at 6:35
• But isn't this the same as saying: How can the integral of a function over an interval $[a,b]$ have a positive value if the integral of the same function over ${a}$ (as a "interval with length 0") is zero? How can the probability that an (arbitrary) integer is even be 1/2 if the probability for an arbitrary integer to be 0 equals 0? And I think your dilemma is also a similar problem as that the union of $n$-dimensional sets can build a $n+k$ dimensional vector space. May 30, 2016 at 7:43
• I think the point is: Just because the points are lines of length zero, it doesn't mean that they do not exist. The "length" is just a function assigning every line some value. Usually you probably define it such that it always takes non-negative values, but of course there may be objects with length 0, which Euclid defines as points. May 30, 2016 at 7:45
• @user306288 No, it means the "paradox" was based on faulty assumptions and/or understanding to begin with. May 31, 2016 at 9:03

Intuitive interpretation:

A curve has an infinity of points. So many that the amount counteracts the abscence of dimensions, like an $0\times\infty$ undeterminacy.

More precisely, a curve still has no dimension transversally, but a finite (or infinite) dimension longitudinally.

You can think of it as the limit of a necklace of pearls (points of finite area) in contact, getting smaller and smaller but more and more numerous. In the end, an infinitely thin but continuous string remains.

• IMO, your final paragraph is more important than the initial paragraph -- if you forget how points are related, then their union is always a zero dimensional discrete space, no matter how many.
– user14972
May 30, 2016 at 18:44
• @Hurkyl : I agree with you on this. How can 0*infinity counteract the absence of dimensions. It is undefined first of all. The second sentence does indeed make sense. May 31, 2016 at 11:56
• @Yves Daoust , Could you please explain, how to get a line segment out of a set of points. I am hopelessly stuck at understanding infinities. May 31, 2016 at 12:00

Euclid did say that that "A point is that which has no part". It is nothing more that a whisper of an indication that "You are here." You can't really see a point since there is nothing there.

"...if point has no dimensions, i.e. in other words there is nothing,
then how is it possible to draw any curve?"


If you're thinking in terms of "connecting the dots", it isn't physically possible. Pick a ridiculously small positive number, $\delta$. It doesn't matter how small. It is a fact that there are as many points in the interval $(0, \delta)$ as there are points in the universe. There is no way you can physically enumerate all of the points in the smallest of curves. I don't believe that Euclid had our understanding of infinity, but I think he was aware of its paradoxical abundance.

But you don't have to draw curves. They are just a set of points. A graph is just a representation of that set and only serves to fuel our intuition.

Euclid's definitions of point, line, and segments, have zero functionality. They may sound nice but they don't really say anything useful.

If you want to know what a point really is, propositions like the following are much more useful.

"Two distinct points determine a unique line."

"If two distinct lines intersect, then they intersect at a single point."

Having reread your question for the upteenth time, it occurs to me that you may be thinking of infinitesimals. Infinitesimals are basically numbers that are smaller in magnitude than any real number but are not equal to zero. You may want to check out THIS and THIS.

• So what should I conclude from all of this ? May 30, 2016 at 10:38
• I know about zeno paradox . So at the end I should conclude that what I asked is a paradox only?!! May 30, 2016 at 11:43
• @user306288 I think the point is that intuition is fine, but when it starts to get confusing (like in this case) stick to the rigorous (modern) definition of points and lines. Euclid's definition may be intuitive but is clearly not rigorous.. "A point is that which has no part", how are you supposed to work with that? What does part even mean? Is the null set a point then? And so on..
– Ant
May 30, 2016 at 13:22
• @user306288 - A line is not a physical thing. When you talk about the point with coordinate $(2,3)$, the point is the location, not what is at that location. When you talk about the line defined by $x+y=1$ you are talking about an infinite set of pairs of numbers that point at the graph and say "here, here, here, ...". May 30, 2016 at 15:00

You are confusing theory with parctise.
A point does not exist in the real universe.
Anything you draw will have some dimensions.

• I am shocked if point has no existence then what is its worth? May 30, 2016 at 5:32
• As you have already mentioned a line segment with length rending to 0. But this is within the realm of logic/maths and may not apply to the real world. May 30, 2016 at 5:34
• Well even straight lines or perfect circles do not exist in the real world. But we can use them to model real world curves/lines etc. May 30, 2016 at 5:35
• "I am shocked if point has no existence then what is its worth?" Negative numbers have no existence, so what are they worth? May 30, 2016 at 8:21
• Yeah, though sometimes it's good to have a look at the bigger picture, the place which inspired maths. May 30, 2016 at 19:32

A curve is completely determined by two facts:

• Knowledge of all of the points lying on the curve
• Knowledge that the curve is drawn on the Euclidean plane

When it's said that a curve is made out of points, one really means to include in the latter fact too, or something similar (e.g. a topology or a metric on the collection of points).

There are more sophisticated geometric techniques (e.g. tangent spaces, halos, germs, stalks) that probe the "infinitesimal" shape of the curve at the point;

For example, studying the tangent space to the curve would indeed allow you to say that, at each point where the curve is smooth, it consists of an infinitesimal line.

But just to reinforce my initial point, the shape of that infinitesimal line can be ascertained simply by knowing the curve is being drawn on the plane along with which 'nearby' points lie on the curve.

A point in an n-dimensional Euclidean space is that point which consists of n coordinates which give a distinct location of that point.

A function in Euclidean space essentially provides a rule for defining a set coordinates in space for which that function is true.

You are correct in that the physical realization of a collection of 0-dimensional points is nonsense to create a physical 1-D curve, however, it makes tons of sense when you learn a bit more math in topics like set and measure theory

You are confusing reality and the idealized world of maths. In an ideal world, a point has no dimensions. You could never actually draw a point. In reality, you can. You take a pencil, make a point on a piece of paper and say "that is the point". But the fact that you can see the point already means that is has some dimensions (the ink occupies a non-empty area in our real, three dimensional world). You have not actually drawn the point, but a representation of the (idealized) point.

Similar, if you draw a curve, you need to draw it with some thickness. This means you don't draw the actual idealized curve, but a real life representation of that curve.

Maths is founded in the real world, but goes one step further. Maths discovers the rules behind things, and abstracts them into an idealized world. This ideal world does not exist in reality, but it is nevertheless useful.

For example, infinity is a purely mathematical concept, yet it is tremendously useful for real-life applications, e.g. physics.

• The question is not about curves and points in reality versus ideal curves and points. It is about how "ideal" points, which have zero dimension, can together form an "ideal" curve, which is not zero dimensional. May 30, 2016 at 8:27
• Which is precisely the same question?! Sorry, but if you have a curve, you have some term that all points that lie on the curve have to satisfy. There is nothing else a curve can consist of then the infinite amount of points that lie on it. You have to get rid of the notion grounded in reality that the curve actually is some object you can physically grab. Its not. You just map some numbers to others, and if you let the curve segment be infinitely small you asymptotically approach a point. May 30, 2016 at 9:01
• Not the same question at all. Even someone who perfectly understands that mathematical, "ideal" points and curves can't be "drawn" in reality, might have the same question about the relation between ideal points and ideal curves. That has nothing to do with the difference between physical reality and mathematics. May 30, 2016 at 9:06

As at least one other answer has mentioned, it takes an infinitude of points to make up a curve. (Or to make up any other interval, like the rather simple "curve" of the closed interval [0,1], for example.)

At least one other answer has also mentioned the notion of "measure," and that's probably the more appropriate way to think about the small-ness of a point: it's said to have "measure zero." If you've taken calculus, then you may recall that you can remove a finite number of points from the interval that you're integrating over, and it won't affect the value of the integral. Why? Because each point only has "measure zero," so its contribution to the total value of the integral is negligible, when compared to the infinitely-many contributions made by all the other points in the interval.

Don't forget, when you have a closed interval (or, once you get to topology-level stuff, a "compact" interval), those intervals will have infinitely-many points in them. Like take [0,1] again; there are infinitely many points in that interval. But cut it in half; say, to [0,1/2]. There's still infinitely many points in that interval. Cut it in half again, to [0,1/4]; still infinitely many points in that interval. You can keep cutting it in half as many times as you can imagine, and you'll still be looking at an interval with infinitely many points in there.

All this "infinitely many" stuff that you encounter in math; it may not seem to line up with the real world all that well, but it's necessary for doing calculus (because it's at the heart of the concept of continuity).

This article by Prof. Lawrence Spector helped me a lot and I think it addresses exactly what your confusion is. I'm not sure what the policy is on just posting links, but I don't think i would do a great job paraphrasing what he wrote, so here goes: http://www.themathpage.com/aCalc/apoint.htm

I would also highly recommend reading on the same site: http://www.themathpage.com/areal/real-numbers.htm, about the historical evolution of the real numbers. I think it'll clear up a lot of the questions you probably already have or will have.

• The commonly accepted policy here is to briefly summarize the relevant points of links, so that your answer is self-contained and useful even if the link goes down. But if you've no time to do that, it might be better to just leave a few comments instead of an answer. May 31, 2016 at 11:19

Some other answers have already mentioned the distinction between the ideal geometric world (which we can describe by mathematical rules) and the real world (which our ideal world is intended to approximate).

However, an interesting point is that Euclid never talked about dimension but merely said:

[Euclid Book 1 Def 1] σημειον εστιν ου μερος ουθεν

[a] mark is that of which [there] [is] no part // A point is that which has no part.

It is amusing that we can force a set-theoretic interpretation onto this as:

A point is a non-empty set with no proper subset. // A point is a singleton.

This coincides with the notion that two different straight lines that intersect do so at a point.

If $$A,B$$ are distinct straight lines with an intersection, then $$A \cap B$$ is a point.

In more set-theoretic terms:

If $$A,B$$ are distinct straight lines such that $$A \cap B \ne \varnothing$$, then $$A \cap B$$ is a singleton.

Nice? Furthermore, a line can then be said to be a union of points!