Let's say I'm on this map, and I want to go from A to B:
---------------
A | | | | |
---------------
| | | | |
---------------
| | | | |
---------------
| | | | |
---------------
B
I can go right then down:
---------------
A +---|---|---|-+ |
---------------
| | | | | |
---------------
| | | | | |
---------------
| | | | | |
-------------|-
B
The distance will be 2 (1 horizontal + 1 vertical)
I can also go in zig-zag like this:
---------------
A |-+ | | | |
-|------------
| +-|-+ | | |
-----|---------
| | +-|-+ | |
---------|-----
| | | +-|-+ |
-------------|-
B
Whis is also 2 (1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4)
If I'd go diagonally though,
---------------
A | \ | | | |
---------------
| | \ | | |
---------------
| | | \ | |
---------------
| | | | \ |
---------------
B
the distance is √2 of course.
Now what happens if I make the grid infinitely smaller?
Going in the zig-zag pattern, each intermediary step is infinitely small, but you still get (number of subdivisions) steps going right, and (number of subdivisions) steps going down, and the total sum is still 2, not √2.
But when the grid is infinitely small, I was of the impression (erroneously, obviously), that it was the equivalent of a straight diagonal line.
So, in a way, my question is, how come a zig-zag line where each 'zig' and 'zag' is infinitely small is not equivalent to a straight line?
