Fractal Dimension Examples

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FRACTALS AND FRACTIONAL DIMENSION f m Roughly speaking, a fractal is an object that looks the same, regardless of the degree o agnication or scaling. Examples include coastlines, mountains, clouds, and nancial time series s recorded in continuous time. The key aspect of a fractal, then, is that it is either exactly or statistically

The concept of quotfractal dimensionquot is attributed to a 20th Century mathematician, Benoit Mandelbrot. His fractal theory was developed in order to try to more precisely quantify the immense complexity of nature in relatively simple equations. His favourite example of such complexity was the craggy coast of Britain, which, seen from far above

In fact, all fractals have dimensions that are fractions, not whole numbers. We can make some sense out of the dimension, by comparing it to the simple, whole number dimensions. If a line is 1-Dimensional, and a plane is 2-Dimensional, then a fractional dimension of 1.26 falls somewhere in between a line and a plane. And this describes the Koch

One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means rectifiable the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles.

Example. Use the scaling-dimension relation to determine the dimension of the Sierpinski gasket. Suppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2. Determine the fractal dimension of the fractal produced using the initiator and

Example 5. Use the scaling-dimension relation to determine the dimension of the Sierpinski gasket. Solution. Suppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2. Determine the fractal dimension of the fractal produced using the initiator

We can use non-spatial dimensions--time, color and perspective--to add dimension to otherwise static objects.For example, you may have seen an image on a -dimensional computer screen that appears -dimensional because of perspective.Using computer-generated perspective, architects can visually walk through a entire building's design before construction even begins see Figure 4.2 and Color Plate 22.

a set which is itself a fractal. This portion of the lecture will cover the denition of a fractal and a few examples of such. Denition A fractal is a subset of Rn that is selfsimilar and whose fractal dimension exceeds its topological dimension. Now we need to discuss the concepts that are involved in this denition.

In general, the quotrougher' the line, the steeper the slope, the larger the fractal dimension. Examples of geometric objects with non-integer dimensions Koch Curve. We begin with a straight line of length 1, called the initiator. We then remove the middle third of the line, and replace it with two lines that each have the same length 13 as the

Examples. The concept of fractal dimension described in this article is a basic view of a complicated construct. The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. In practise, however, fractal dimensions can be determined using techniques that approximate scaling and detail from