ween Integers: ength of a Coastline? 測量海岸線的長度嗎？ defies our expectation that finer measurements yield more accurate results, so obviously, coastlines are not something trivial. Fractals Figure 2a shows the Koch snowflake, an example that can help us understand what’s going on with the paradox. As illustrated in figure 2b, here’s how it’s constructed: 1. Start from a straight-line segment of length 1. 2. Divide it into three line segments of equal length (so each line segment has length 1/3), then replace the middle segment with an equilateral triangle’s two sides. 3. For each existing line segment, perform step two. 4. By repeating this process infinitely, you’ll construct the upper section of the Koch snowflake! To obtain the full shape, simply make three copies of your constructed piece and attach them together in a triangular formation. • Recall that we started with a line of length 1. • The next iteration gives four line segments of length 1/3, making a total of 4/3. • The third step further divides each line segment into four, giving 16 lines of length 1/9, making a total of 16/9… Unlike a smooth curve, which simplifies when zoomed in, the Koch snowflake remains complex at every scale (figure 3). More details will be revealed when you zoom in more, while the pattern shows similarity at every level, making it a “fractal.” To calculate the length of one of its three sides: Figure 2 (a) Koch snowflake and (b) how it is constructed through a series of iterations. Inductively, the total length after each iteration creates a geometric sequence with a ratio of 4/3. Since the ratio is greater than one, the length will go infinitely large. Therefore, the Koch snowflake has infinite perimeter. Curves like the Koch snowflake that have an infinite length are known as “non-rectifiable curve.” The name comes from the fact that those curves cannot be “rectified,” or “straightened out” like a piece of string and then measured. 7 Figure 3 Illustrations showing that (a) a smooth curve simplifies to almost a straight line when zoomed in, while (b) the Koch snowflake remains complex at every scale. (a) (b) By Devandhira Wijaya Wangsa n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 (a) (b)
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