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# feb22a - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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s s s 6.055 / Art of approximation 24 3 3 r = . 4 π The bisection path is one-sixth of a circle, so its length is 2 π r π 3 3 π 3 l = = = . 6 3 4 π 12 The best previous candidate (the first picture) has length 1 / 2 = 0 . 707 . . . . Does the mess of π and square roots produce a shorter path? Roll the drums . . . : l = 0 . 67338 . . . , which is less than 1 / 2 . So the circular arc is the best bisection path so far . However, is it the best among all possible paths? The arc-length calculation for the circle is messy, and most other paths do not even have a closed form for their arc lengths. Instead of making elaborate calculations on every path, of which there are many, try symmetry , which is the mathematical principle for the three methods in this part of the book. To use symmetry, replicate the triangle six times to make a hexagon, thereby replicating the candidate path as well.
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feb22a - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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