Fractal dimension which itself is a class of different dimensions (such as the Hausdorff dimension, similarity dimension, box-counting dimension and the divider dimension) which are all equal for exactly self-similar fractals like the Koch curve. Many variations of fractal dimension have been defined, and for some sets, these different dimensions yield different values. These numbers attempt to quantify how densely the fractal occupies the space in which it lies. They are generally referred to as fractal dimensions.
There are various numbers, associated with fractals, which provide an objective means for comparing fractals. He chose to measure a coastline length by walking a divider of a specific length along the coastline to see how many dividers were needed to cover the entire coastline and calculated the length of the coastline by multiplying the number of dividers by the length of each divider. He examined the coastlines of several countries including Great Britain, South Africa, and Australia as well as the border of Spain and Portugal 35. The variation in the lengths of coastlines at different scales intrigued the scientist Lewis Fry Richardson in the 1920s. There are several possibilities for this failure such as sloppy length measurements, use of different data sets by different researchers (some of which may be erroneous), and use of different measurement methods, leading to differences in measured lengths etc. However, this naive application of Euclidean geometry to natural curves fails because the limit may not exist. As the segment size approaches zero, the sum of the lengths of the segments will approach the length of the curve.
For more accurate measurements, smaller segments can be used. Measuring the length of smooth curves is a simple process of successive approximations by line segments. The irregularities and variations in coastlines have defied attempts of characterization using methods which are based on Euclidean geometry. Their construction is random as compared to deterministic fractals such as the Mandelbrot set (which is formed through repeated iterations of a simple mathematical equation). Man made fractals include the Cantor set, Sierpinski triangle, and the Mandelbrot set etc.Ĭoastlines (the boundary between land and water) and other natural boundaries have been a subject of human fascination since long. The application of fractals can be seen in fractal antennas, digital imaging, computer graphics, computational geometry, geology and many other fields. From clouds to mountains, snowflakes to river networks, broccoli to blood vessels, fractals can be noticed everywhere in nature.
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