8 Notably, many well-known examples of fractals show “self-similarity” meaning that they look similar at different scales, such as the Mandelbrot set, Julia set, Sierpinski triangle (Sierpinski gasket), and Sierpinski carpet. Fractal Dimension Let’s think of this intuitively: Our experience with shapes tells us that any bounded one-dimensional object like line segments have a non-zero yet finite length, but zero area. Meanwhile, bounded twodimensional objects like surfaces (e.g. a piece of paper), have infinite “length,” as they can be thought of as shapes that consist of infinitely many lines, yet a non-zero finite area. Then where does the Koch snowflake fit? Considering the snowflake as a hollow shape, its infinite perimeter but zero area suggests a dimension between one and two. This intuitive idea naturally leads us to the concept of non-integer dimension — the fractal dimension, which describes how “complex” a geometric object is. For example, to find out the fractal dimension of the Koch snowflake, let’s recap how dimension works: Think about a two-dimensional square of length 1, we know that by lengthening its sides by two means that we obtain 4 = 22 copies of the original square, and by three obtains 9 = 32 copies. This is exactly because the square is two-dimensional. In general, scaling, for example, a shape of dimension d by a scale factor of c will give cd times its original copy (try this out yourself with a line or cube). Let’s take the upper part of the Koch snowflake now. By how the snowflake is constructed, scaling the snowflake by three gives four of the original copies. Therefore, if we need to assign a dimension d, then by the reasoning above we should require 4 = 3d, in other words d = log 4 / log 3. How about coastlines? Strictly speaking, coastlines are not fractals because fractals are abstract theoretical shapes, but coastlines are fractal-like (that is, having fractal features) to the point where people would approximate their fractal dimension. For instance, the coastline of South Africa, a smooth one in the atlas, has a fractal dimension of 1.02, while those for the border between Spain and Portugal and the west coast of Great Britain are 1.14 and 1.25, respectively [8], meaning that they are more complicated. Conclusion After all, can we measure the length of a coastline? The essence of the coastline paradox is the fact that coastlines are non-rectifiable curves since nowhere in real world is perfectly smooth, and when you zoom in on a map, there are always more jagged parts (usually rocks) which lengthen the coastline. In addition, the length also depends on which standard was taken. As you can imagine, lengths measured at different tide levels can differ significantly because land can become submerged and invisible during high tide. If there is a river, whether and how to include the estuary and its tributaries poses a challenging question. Therefore, a simple answer is: No coastline can be measured objectively. Math Challenge Let’s find the area of a Koch snowflake! Let s be the side length of the initial triangle before any “iteration.” Find the area of the Koch snowflake in the first few iterations in terms of s. What happens when we take the iteration to infinity? (Hint: Infinite geometric series) Answer:
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