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We know and accept that a line in a 2-d space has zero width. However, this causes a an issue as explained here.

Suppose that we have a closed area $A$ of White color, and we draw one colored line (of any other color other than White) crossing it. The consumed area by the line from the area of the shape is zero and the remaining White area (without the are consumed by the line) of the shape is still $A$. Now if we draw a very large number of colored lines crossing the White closed area, we know that the total free area $A$ will be affected. This can be simply simulated by pencil and paper. We can say that:

White Area = $A$ - (Number of Lines) * (Area of each colored Line)

But since the "Area of each colored line" is zero, this means that the White area value will never be affected regardless how many lines cross it. However this is contrary to what we can experiment with a paper and a pencil.

How can this be represented mathematically and justified?

A sample with few lines is shown below.

enter image description here

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    $\begingroup$ Well, does your pencil also draw zero-width lines? $\endgroup$
    – Adayah
    Commented May 16, 2018 at 8:50
  • $\begingroup$ @Adayah, Good point, so are you suggesting that infinite number of lines will have zero width? $\endgroup$
    – NoChance
    Commented May 16, 2018 at 8:57
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    $\begingroup$ What do you mean by the width of an infinite number of lines? An arbitrary set of lines doesn't always form the shape of a 'thicker' line, so how would you measure the width of that possibly odd shape? The total area of them, however, will be zero, provided that there are countably many lines. Otherwise we don't know even that, for instance, the whole square can be covered with sufficiently many lines. $\endgroup$
    – Adayah
    Commented May 16, 2018 at 9:04
  • $\begingroup$ @Adayah, I see your point, maybe it is better (for me) to consider the line as a "Logical Concept" not a physical one. I wish I had this phrase made in any of the line definitions I have read. $\endgroup$
    – NoChance
    Commented May 16, 2018 at 9:20

2 Answers 2

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The first thing you need to realise is that when we speak of lines, surfaces or regions in mathematics, it is something much different from what we may mean by those terms elsewhere.

Now if we draw a very large number of colored lines crossing the White closed area, we know that the total free area $A$ will be affected. 

No it won't be affected one bit.

Thus, you arrive at your "contradiction" because you're still thinking of your lines as having some thickness. No, they don't. When we make diagrams, those pictures are only a convenient representation or symbol for mathematical objects. So when you draw a line with a pencil, for example, you have strictly speaking only drawn a figure or symbol of a line, not and never the line itself, which can never be seen or observed. This manner of language may be confusing at first, but once you're aware of this distinction between mathematical objects and their representations, it disappears. Thus we routinely speak of the graph of a function as the function, even though the graph is just a representation of the function. This same abuse (but convenient) of terminology is prevalent throughout the discipline, but it doesn't give much problems.

So, no matter how many lines you lay across the region, you have not covered anything. Indeed, the region is made up of infinitely (not merely countably infinitely, but uncountably infinitely) many line segments, so that no number of sequentially laid lines will take up any part of it. This follows from the fact that the real numbers are much more (in a definite sense) than the rationals. This follows from Cantor's work, where he showed that whereas the rationals can be put into a one-one correspondence with the positive integers, the real numbers cannot be so associated. Indeed, one can put a line segment into correspondence with a square, so that in a way your square contains as many points as its side. This is crazy, of course, but welcome to set theory.

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  • $\begingroup$ Thank you for your answer. You made a good point about the fact that a closed area contains INFINITE NUMBER OF POINTS, and to fill a closed are within a closed area, it would take infinite number of lines. $\endgroup$
    – NoChance
    Commented May 16, 2018 at 9:11
  • $\begingroup$ @NoChance To be clear, not even infinitely many lines laid one by one are sufficient. We need as many lines as there are points in $(0,1)$, which is something we cannot imagine doing. We will always miss out some line, no matter how we try. You should also note that area is a measure, and depends only on the length, not the thickness, of the constituent lines. $\endgroup$
    – Allawonder
    Commented May 16, 2018 at 9:19
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    $\begingroup$ thanks for the explanation. $\endgroup$
    – NoChance
    Commented May 16, 2018 at 9:22
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Given the answer above and the comments, I came to realize that a line with zero length is not a physical entity in 2-d space. It is a logical entity/concept/shape only. It is a representation of a special relationship between points. Accordingly, the zero length assumption makes sense, and infinite lines will never occupy an area in 2-d space. Thanks for the help of the contributors.

No thanks to my 3rd grade math teacher!

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