What happens to a point when you rotate a line?

Question

If I have a line on an $XY$ grid:

o---o---o---o---o
a   b   c   d   e

And I rotate it like so:

        o
       /  e
      o
     /  d
    o
   /  c
  o
 /  b
o
 a

What is considered to have happened to point $'e'$? Did it move from $x_5-y_1$ to $x_3-y_5$? Did it disappear and a new point created?

And if I change a line like:

            o
           /  e
          o
         /  d
o---o---o
a   b   c 

Did the same thing happen (in terms of point movement)? Or something else?

To be clear, I'm not looking for how to do this in math or geometry, but rather what is considered to be the effect? Change of point or creation of point?

Answer 1

Interesting question, especially if you think in terms of "real" world. Let's use electrons for your points. In quantum mechanics you cannot distinguish between electrons. So you can either move an electron, or destroy it and create a new one. In both cases the physics is the same. There is no difference between the two processes. In many cases one would write the transformation from the initial state to the final state as a linear combination of the "rotation" and "annihilation-creation" processes.

You can also imagine this: let's start with a paper disk, and put 5 paper pieces on it, such that they form a line. Can you tell if you rotated the disk say 45 degrees, or you remove the pieces of paper, and put five identical ones in the new position? If not, then it does not matter what process happened, and you can only describe the transformation from one state to the other by the "rotation".

Answer 2

There are many ways to answer this question $-$ You question is somehow opinion based and might be closed $-$ but at least I try to give an answer to create some intuition for you. This answer is based on the geometry in $\Bbb R^2$.

Think about your points on the paper as eternal and fixed in place. A point has coordinates $(x,y)$ and these numbers state exactly where this point can be found, and will forever be.

A line is made from some of these points. This is no process of assembling the line from the points, but your line is nothing but the points on it (like a forest is nothing but the trees in it). Mathematically there is no meaning in "rotating a line", but just in the (already) rotated line. You cannot move the line because you cannot move the points from which it is made of. Instead your rotated line is a completely new line which (as we say) looks like the former line, but "rotated". The rotated line may be made from other points than the former one. If the old line and the new line intersect, then this intersection is a point which is contained in both lines. Note that you never "created" this new line. It was always there. You just ignored it until we started talking about it.

Note that no points have been created nor destroyed in the process. Points are always there. But you might ignore them until you need them. When you draw a line on paper (which is a non-perfect realization of an abstract idealized entity), you just highlight some of the points which you want to mark as being "on the line". You have not created the points.

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The general philosophy

We can apply this philosophy to everything in mathematics. A mathematical object (a point, a line, a number, a 24-dimensional symplectic manifold, ...) has an identity, it is static and eternal. Either it exists, or it does not. We never create anything, we just discover it, or start paying attention to it. We never modified an object, we just redirected our attention to another object which differs in a certain aspect from the former one.

A rotated line can never be the same line than the former one, because if they were the same, they would agree in every property. But they do not $-$ they have different orientations.