Fractal dimension10/20/2023 You can see a list of the lengths of coastlines While there may be some variability between different computed values of the fractal dimension for a coastline, it is MUCH more accurate than a measurement of the perimeter.Īnd yet, people publish values for the lengths of coastlines all the time! But they usually don't tell you how long the rulers they used were, so the value is virtually A more careful analysis of the coastline of Britain Unlike the case with the geometrical fractals, this way of determining the dimension is clearly an estimate. The higher the dimension the more jagged and wiggly the coastline is. The lower the dimension, the straighter and smoother the coastline. So according to this analysis, the fractal dimension of Britain, In this case, we can measure the slope and find it to be approximately 1.21. The fractal dimension of the coastline, D, is simply the slope of the line in the graph above. If the Y-values change at the same rate as the X-values, then the slope = 1, and the graphed line has an angle of 45 degrees. If the Y values stay the same, then the slope = 0/X = 0, and the line is flat. Slope is simply defined as the change in the Y-value divided by the change in the X-value. The slope of this line tells us how quickly the perimeter changes versus the magnification factor. We can use a simple process to find a line (in blue) that best fits the datapoints. Now there is a very useful reason we take the logarithm of the values before we graph them: It means the points we plot on the graph come out more or less on a To be consistent, we're actually going to define r to be the magnification factor, which is the Toĭo this, we'll need to measure the perimeter using rulers of different lengths. To derive the dimension, we'll use a similar method as for the geometric fractals, and we'll see how the measured length changes as a function of scale. This is why the fractal dimension is a very useful concept to describe a coastline. More closely, and it approaches infinity. The answer is: the coastline gets longer and longer as you measure it AndĪt a microscopic level, sand is a fractal as well, and cannot be easily measured. To get a more accurate value, you'd have to measure the length around every boulder, and every rock, every pebble, and even every grain of sand. But if you were to walk around the beaches of Britain and add up all your steps, you'd arriveĪt a very different (and larger) perimeter. If you measure the coastline by taking a map and placing a rulerĪround the edge you can get a certain value for the perimeter. The answer is that it depends on how closely you look at it, or how long your measuring stick is. If we ask "How long is the coast of Britain," Recall from the chapter on Fractal Geography that a coastline - such as that of Britain - shows more and more detail the closer we zoom into it.
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