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Q&A

Comments on How do mathematicians measure area for points?

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How do mathematicians measure area for points?

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There are two points (adjacent or not adjacent) on top of an infinite plane; how to measure how much area they catch together, as a "set".

Because that if we put a minimal border around the two points that border would be made from ten points by itself and would catch its own ten-point-area, it won't be clear what is really a part of what:

  • One set for all points together
  • Two different sets: One for the two points and one for the border points
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1 comment thread

Needs a little more clarity (10 comments)
Needs a little more clarity
r~~‭ wrote over 2 years ago

I believe there's a lovely question in here somewhere but it hasn't emerged yet. Some questions for you: What do you mean by ‘dot’? Do you mean the same thing as ‘point’, or are you using a different word intentionally to signify something else? Is a ‘minimal border around ... two dots’ something other than a line segment connecting two points? If not, how does a line segment ‘catch’ an area? With the usual meaning of terms like ‘point’, ‘line segment’, and ‘area’, the area of a point is 0, as is the area of a line segment (indeed, of an entire line). Is your question really something like: how is it possible that something like a circle can have area when it's made of area-less points?

deleted user wrote over 2 years ago · edited over 2 years ago

Thanks r~~

Yes, indeed I meant "points" (in Hebrew it's נקודה as in dot, so I was biased from that).

Is a dot's area really 0? I mean, must it be defined as zero? For example, let's compare the "physical void" to the "mathematical plane"; in the physical void, a most elementary particle won't be "zero" but "1" (catches the area of 1 in the void) so why not say that a dot catches the area of 1 in the plane?

Peter Taylor‭ wrote over 2 years ago

The area of a circle is $\pi r^2$, and a point is the limit of a circle as $r \to 0$.

r~~‭ wrote over 2 years ago

I recommend editing your question to be that, if that's what it is. The thing about a minimal border around two points is still imprecise; if that's an important part of the question, explain a bit more, and otherwise remove it.

Peter Taylor‭ wrote over 2 years ago · edited over 2 years ago

This seems to be getting less clear, not more. Take a concrete example: let the two points be $(0, 0)$ and $(0, 1)$. What is the "minimal border" if not the line segment $\{(0, t) \mid 0 \le t \le 1\}$?

Skipping 1 deleted comment.

deleted user wrote over 2 years ago

Peter Taylor‭r I don't know what is t there and I don't know what is | in math so I currently can't say if you wrote a rhetorical question or meant to ask me something aiming to better understand me.

Peter Taylor‭ wrote over 2 years ago

The notation can be read "the set of points $(0, t)$ where the variable $t$ is between $0$ and $1$ inclusive". It's just the line segment (technically not a line, because lines are infinite) between the two points.

DavidCary‭ wrote over 2 years ago

I agree that this sounds like it could be a lovely question with a few more details. What do you mean by a "border ... made from ten points" and the "ten-point-area"? Perhaps you could make a rough sketch (pencil on paper or ASCII art is fine) or gave a link to where this unusual terminology came from? Is a "three-point-area" a triangle? Does this arrangement

o o
o o o o o
o o o

look like a kind of 10-point area?

deleted user wrote over 2 years ago · edited over 2 years ago

DavidCary‭

Thanks,

What do you mean by a "border ... made from ten points" and the "ten-point-area"?

I think I meant to squares instead points.

Is a "three-point-area" a triangle?

I think it can be defined as such

I understand that in essence I confused points with squares:

If we have 2 squares we could put 4 squares on top, 1 square from the left, 1 square from the right and 4 squares on bottom, so then we get a structure of two squares wrapped in 10 squares.

□□□□
□■■□
□□□□

Instead points there could be circles, of course.

I understand that I have defined "point" different than common in mathematics.

Peter Taylor‭ wrote over 2 years ago

That graphic helps a lot, thanks. But if the two initial squares aren't horizontally or vertically adjacent, it's not clear that the border is well defined. You can draw the 8-square box around each one, and then connect appropriate corners of the boxes with straight lines, but deciding which squares to include in the lines has an aesthetic component. E.g. if you do it by drawing a zero-width line between the centres of the squares and asking which squares are at more in than out, there are cases where some squares are exactly 50% in and 50% out.