Iterative Mathematics Fractal Geometry

Fractal Geometry T. Serino One-, two-, and three-dimensional figures are common. However, many objects, are difficult to categorize as one-, two-, or three-dimensional. Many of these shapes occur in nature: a coastline, the bark on a tree, a mountain, or a path followed by lightning. For a long time, mathematicians assumed that making realistic geometric models of shapes that occur in nature was almost impossible. The development of fractal geometry, has made this possible.

Fractal Geometry The photos below were made by using fractal geometry. T. Serino The photos below were made by using fractal geometry. The discovery and study of fractal geometry has been one of the most popular mathematical topics in recent times.

Fractal Geometry T. Serino The word fractal (from the Latin word fractus, “broken up, fragmented”) was first used in the mid-1970’s by mathematician Benoit Mandelbrot to describe shapes that had several common characteristics, including some form of “self-similarity.” Similar? This fractal is called the Mandelbrot set. Can you see the self similarity?

Fractal Geometry T. Serino Typical fractals are extremely irregular curves or surfaces that “wiggle” enough so that they are not considered one-dimensional. Fractals have dimension between 1 and 2. For example, a fractal may have a dimension of 1.37. Fractals are developed by applying the same rule over and over again, with the end point of each step becoming the starting point for the next step, in a process called recursion.

Fractal Geometry T. Serino Let’s look at the fractal tree. Start with a tree trunk. Draw two branches, each one a bit smaller than the trunk. Draw two branches from each of those branches, and continue. Continue to see this construct. If you take a little piece of any branch and zoom in on it, it will look exactly like the original tree. Fractals are scale independent, which means that you cannot really tell whether you are looking at something very big or something very small because the fractal looks the same whether you are close to it or far from it.

Fractal Geometry T. Serino To illustrate this idea of scale independence, let’s look at Sierpinski’s Triangle. One way to construct the Sierpinski Triangle is by drawing three line segments connecting the midpoints of each side of an equilateral triangle. The line segments form an “upside down triangle.” Shade this new triangle and then continue to do this to each new triangle formed.

Fractal Geometry T. Serino You can see that if you zoom in on a small part of the Sierpinski triangle that it looks exactly the same as a previous iteration of the triangle.

T. Serino Fractal Geometry

Try this. Create the Sierpinski carpet. T. Serino Create the Sierpinski carpet. Starting with a square, cut the square up into 9 equal sized squares as shown. Color in the middle square. Repeat this iteration for each of the white squares that are created. Complete this and one additional iteration in your notes.

Fractal Geometry T. Serino One more iteration of the Sierpinski carpet would look like this. Both of the previous fractals are named after Waclaw Sierpinski, a Polish mathematician who is best known for his work with fractals and space-filling curves.

Fractal Geometry T. Serino Let’s use the recursive process to develop one more famous fractal called the Koch Snowflake. Start with an equilateral triangle (step 1). Whenever you see an edge replace it with (steps 2-4).

Fractal Geometry T. Serino If the initial edge of the snowflake were 3 units long, and each of the four segments of is 1 unit long, calculate the perimeter of each step. Could we calculate the area of each figure?

Fractal Geometry T. Serino The following is a portion of the boundary of the Koch snowflake known as the Koch curve or the snowflake curve.

Fractal Geometry T. Serino The Koch curve consists of infinitely many pieces of the form . Notice that after each step, the perimeter is 4/3 times the perimeter of the previous step. Therefore, the Koch snowflake has an infinite perimeter (The perimeter diverges). It can be shown that the area of the snowflake is 1.6 times the area of the starting equilateral triangle. Thus, the area of the snowflake is finite (not infinite). The Koch snowflake has a finite area enclosed by an infinite boundary! This fact may seem difficult to accept, but it is true. However, the Koch snowflake, like other fractals, is not an everyday run-of-the-mill geometric shape.

Fractal Geometry T. Serino Fractals provide a way to study natural forms such as coastlines, trees, mountains, galaxies, polymers, rivers, weather patterns, brains, lungs, and blood supply. Fractals also help explain that which appears to be chaotic. The branching of arteries and veins appears chaotic, but closer inspection reveals that the same type of branching occurs for smaller and smaller blood vessels, down to capillaries. Thus, fractal geometry provides a geometric structure for chaotic processes in nature. The study of chaotic processes is called chaos theory.

Fractal Geometry T. Serino Fractals nowadays have a potentially important role to play in characterizing weather systems and in providing insight into various physical processes such as the occurrence of earthquakes or the formation of deposits that shorten battery life. Some scientists view fractal statistics as a doorway to unifying theories of medicine, offering a powerful glimpse of what it means to be healthy.

Fractal Geometry T. Serino Fractals lie at the heart of current efforts to understand complex natural phenomena. Unraveling their intricacies could reveal the basic design principles at work in our world. Until recently, there was no way to describe fractals. Today, we are beginning to see such features everywhere. Tomorrow, we may look at the entire universe through a fractal lens.

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