Re: math question Message #12 Posted by carey on 25 Apr 2012, 9:26 p.m., in response to message #7 by Marcus von Cube, Germany
Hi Marcus,
Great question! Vectors may be helpful in resolving the paradox.
1. Convert all line segments into vectors by inserting arrows.
2. For each pair of horizontal and vertical vectors (i.e., for each zig and zag
along the path) draw diagonal resultant vectors, forming little triangles.
3. For each little triangle, the difference between the distance along the zig-zag (the
sum of 2 sides) and the displacement (the length of the diagonal) is the triangle error.
4. The total error is the sum of all triangle errors along the path. The total error
is the same whether there are few zig-zags (a few large errors) or many
little zig-zags (many little errors). This can be seen by adding all the horizontal arrows
and, separately, adding all the vertical arrows. The vector sums are constant no
matter the size of the zig-zags (if they weren't constant, some paths would
undershoot or overshoot the destination).
Paths of many small zig-zags look smoother and are better approximations to the diagonal in a least squares sense (distance of zig-zag path to diagonal path). However the total error in approximating the length of the diagonal is the same for paths of many large zig-zags or many small zig-zags (few big errors vs many little errors).
A hallmark of a fractal (self-similar, irregular) curve (e.g., a coastline) is that its curve length is not a definite value but depends on the scale used in measuring it. In this problem: (1) the zig-zag curves are not self-similar (i.e., when you zoom in on the coarse zig-zag curve you do not find little zig-zag curves) and, (2) the length of the zig-zag curves is definite and not dependent on the scale used in measuring them. Hence, this does not appear to be a fractal problem.
Edited: 26 Apr 2012, 9:08 a.m.
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