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Creating a Koch Curve Fractal

Investigate and predict the growth of a Koch Curve fractal through creating stages, recording lengths, and analyzing data on how patterns evolve. Understand recursive rules, identify lengths at various stages, predict future lengths using exponents, and grasp the concept of fractals.

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Creating a Koch Curve Fractal

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  1. Creating a Koch Curve Fractal

  2. Growth of the Koch Curve • In this investigation you will look for patterns in the growth of a fractal. Stage 0

  3. Draw Stage 1 figure below the Stage 0 figure. The first segment is drawn for you on the worksheet. Stage 1 should have four segments. Stage 0 Stage 1

  4. Describe the curve’s recursive rule so that someone can re-create the curve from your description. Stage 0 Stage 1

  5. Using your recursive rule, determine the length of the Stage 1. Stage 0 Stage 1

  6. Draw Stage 2 and 3 for the fractal. Again, the first segment for each stage is drawn for you. Stage 0 Stage 1 Stage 2

  7. Record the total length in the chart. Stage 0 Stage 1 Stage 2

  8. Make a chart to collect data on each stage 0-2.

  9. Make a chart to collect data on each stage. • How do the lengths change from stage to stage?

  10. Predict the total length at Stage 3.

  11. Predict the total length at Stage 3.

  12. Use exponents to rewrite your numbers in the columns labeled “Total Length, Fraction Form” for Stages 0-3

  13. Predict the Stage 4 lengths.

  14. How many segments will Stage 4 contain? • How long will each segment be?

  15. How many segments will Stage 4 contain? • How long will each segment be?

  16. Stage 0 Stage 1 At later stages the Koch curve looks smoother and smoother. But if you magnify a section at a later stage, it is just as jagged as Stage 1. Stage 2 Stage 3 Stage 4 Mendelbrot named these figures fractals. Stage 5

  17. Example Stage 1 • Look at these beginning stages of a fractal: • Describe the fractal’s recursive rule. • Find its length at Stage 2. • Write an expression for its length at Stage 17.

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