Concept Byte: Fractals

Use With Lesson 7-2

EXTENSION

Fractals are objects that have three important properties:

• You can form fractals by repeating steps. This process is called iteration.
• They require infinitely many iterations. In practice, you can continue until the objects become too small to draw. Even then the steps could continue in your mind.
• At each stage, a portion of the object is a reduced copy of the entire object at the previous stage. This property is called self-similarity.

Example 1

The segment below of length 1 unit is Stage 0 of a fractal tree. Draw Stage 1 and Stage 2 of the tree. For each stage, draw two branches from the top third of each segment.

• To draw Stage 1, find the point that is 1 3 1 third  unit from the top of the segment. From this point, draw two segments of length 1 3 1 third  unit.

  

• To draw Stage 2, find the point that is 1 3 1 third  unit from the top of each branch of Stage 1. From each of these points, draw two segments of length 1 9 1 ninth  unit. The length of each new branch is 1 3 1 third  of 1 3 1 third  unit which is 1 9 1 ninth  unit.

  

Amazingly, some fractals are used to describe natural formations such as mountain ranges and clouds. In 1904, Swedish mathematician Helge von Koch made the Koch Curve, a fractal that is used to model coastlines.

Example 2

The segment below of length 1 unit is Stage 0 of a Koch Curve. Draw Stages 1–4 of the curve. For each stage, replace the middle third of each segment with two segments, both equal in length to the middle third.

Stage  0 1  unit _ cap stage , 0 . modified 1 , unit with under bar below

• For Stage 1, replace the middle third with two segments that are each 1 3 1 third  unit long.

  

• For Stage 2, replace the middle third of each segment of Stage 1 with two segments that are each 1 9 1 ninth  unit long.

  

• Continue with a third and fourth iteration.

  

  

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