Fractals in Math: Definition & Description

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  • 0:01 Definition of Fractal
  • 0:22 Self Similarity
  • 3:23 Fractal Fraction
  • 4:42 Fractals in Nature
  • 5:08 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will know what the defining characteristic of a fractal is. You will see a fractal both in numbers and in an image.

Definition Of Fractal

In math, a fractal is a never-ending pattern. Fractals are built by repeating something over and over again. You can create fractals with numbers, or you can create fractal images. There are even fractals that you can see in the natural world. Yes, we can see math in action in the world around us.

Self Similarity

Because fractals repeat something over and over again, the defining characteristic of fractals is their self similarity. This means that the object is similar or exactly the same to a part of itself. For example, look at this image. Looking at the whole image, you see that you have what we call five petals surrounding a center. Now, look at each petal. Each petal is the whole image in miniature. Each of these petals has five mini-petals surrounding its own center. Each of these mini-petals is again the whole image. But this time, the mini petals are a tinier version of the whole image. This is a fractal, because a pattern is repeated in the petals over and over again. The pattern gets smaller each time. You can say that the pattern in a fractal is repeated internally over and over again. The same thing in the image is replaced with the same image. In this case, the image that is replaced is the five petals surrounding a center. Each petal is replaced with the same image. Yes, we can get infinitely small with these replacements. You can take almost anything and repeat it internally to get a fractal. Look at these other examples of fractals. Interestingly, these fractals have special names. In order from the top, we have the Gosper island, Koch snowflake and box fractal. You can see how they grow the fractal by repeating the same shape over and over.

In the Gosper island, the shape that is repeated is the trapezoid; that is the top half of the hexagon. Look at the hexagon now. Cut the top half off. Now, take this top half and shrink it in size so the bottom is one-third the length of the top of the original hexagon. Now, place this half-hexagon in the middle of the top of the hexagon, and on every side of the original hexagon. Now you have the second image in the series. Next, you take the same half hexagon and you shrink in size again. Now, add this half hexagon to the top line of the second image and to the line to the left of the top line, and you repeat this for every bulge in the second image in the series. Now you have the third image in the series. Now, keep repeating this pattern over and over again and you have the Gosper island.

In the Koch snowflake, it is the triangle that is repeated. It is very similar to the Gosper island, except our whole triangle is what is being repeated. This time our triangle is shrunk in every stage, and put in the middle of each side of the new image in the series. This keeps repeating and you get the Koch snowflake.

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