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?

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?

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

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.

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 $\{(0, t) \mid 0 \le t \le 1\}$?

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.

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.

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?

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.

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.