I don’t see how a STRAIT line can be defined by its length!
‘The shortest distance between two points’ that well known statement, for me, leaves out the word unobstructed.
Also what if I place two dots on a piece of paper and use a ‘strait-edge’ to draw a line to connect the dots, then I lay the paper around a cylinder so the dots
become closer together is it still a strait line? A curved strait line maybe? But no longer the shortest distance between the two dots!
Deformation, yes but the line itself is unaltered. Hope you get my drift.
My post was:
How do we know if a line is straight? What is it judged by?
Can we prove a line is strait? If we use an instrument we need the prove that that is strait also. Haven’t gone into any research, math or otherwise
it was just idle thougth.
[The question of what it means to be straight can only be answered relative to the space you are considering! In your example you wrap the space around a cylinder, changing the space, but then consider the surface of the cylinder as a subspace of the 3-dimensional space it lies in. The original line was straight in the classical sense, which can be phrased as “any two points on the line are as far apart in the line as they are in the plane. That is no longer true on the cylinder since once you wrap far enough round you can find points on the wrapped line which are closer on the cylinder than they are in the line itself. That is you can get from one point to the other faster using a short cut on the surface of the cylinder. It is even more obvious that you can take shortcuts once you are allowed to cut across the cylinder in the 3 dimensional space. However, what is true on the cylinder is that short enough segments of the wrapped line are still distance minimising, so the wrapped line is “locally” a geodesic” from the point of view of the cylindrical surface. That is definitely not true one you consider the corkscrew line in three dimensional space. Harry]