Measuring Coastlines How long is the coastline of Hong Kong? Here are the different answers from different organizations: What causes these inconsistencies in the data? Lewis Fry Richardson (1881–1953), an English mathematician who studied the possible correlation between the length of shared borders and the chance of a war outbreak, was also puzzled by the same kind of inconsistency [4]. He found that Portugal reported its border with Spain as 1,214 km, while Spain claimed 987 km [5]. This discrepancy led to the discovery of the “coastline paradox.” Why do measurements of the same coastline differ? Let’s first understand how the measurement of complex shapes can be carried out systematically. For smooth curves (e.g. no corners) defined by mathematical functions, their exact length can be calculated using calculus. However, for jagged, irregular shapes like coastlines that exist in the real world, no simple formula works. Their complexity requires a systematic approach, such as using fixed-length segments to approximate their length consistently and accurately. Imagine a curve that is smooth but seemingly random (figure 1a). Intuitively, we could use a string resembling its shape. Although this is an accurate method, this approach is practically not feasible when implemented in a computer. This is also not systematic because the string’s shape depends on how it’s placed, making results inconsistent and hard to replicate, especially if we were to apply the method to intricate shapes like coastlines. Instead, we can lay out straight line segments of equal lengths — let's call them “sticks” — to approximate the curve, as shown by the sticks of different lengths in figures 1b and 1c. We then add up the total length of the sticks to get an approximation of the curve’s length. We can expect the approximation to get more accurate as we choose shorter and shorter sticks. Ideally, when the length of each stick is sufficiently short, we get a very good estimate of the length of the curve. Dimension Betw Can We Measure the Len 整數之間的維度：我們能測量 • Central Intelligence Agency of USA: 733 km [1] • World Resources Institute: 955 km [2] • Hong Kong Environmental Protection Department: 1,178 km [3] However, when people use this method to measure the coastline of Great Britain, the result is counterintuitive. Using smaller sticks increases the measured length dramatically; using one-meter sticks will result in a total length over 15,000 km [6], which is even longer than Earth’s diameter (12,756 km) [7]! This (a) (b) (c) Figure 1 (a) A curve that is smooth but seemingly random. (b, c) Straight line segments of equal lengths are used to approximate the curve’s length.
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