Creating a Koch Curve Fractal

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

Draw Stage 1 figure below the Stage 0 figure 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

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

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

Draw Stage 2 and 3 for the fractal 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

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

(Number of segments times length of segments) Make a chart to collect data on each stage 0- 2. Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 2 3

(Number of segments times length of segments) Make a chart to collect data on each stage. Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 4/3 1.33 2 16 1/9 16/9 1.78 How do the lengths change from stage to stage?

(Number of segments times length of segments) Predict the total length at Stage 3. Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 4/3 1.33 2 16 1/9 16/9 1.78

(Number of segments times length of segments) Predict the total length at Stage 3. Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 4/3 1.33 2 16 1/9 16/9 1.78 3 64 1/27 64/27 2.37

(Number of segments times length of segments) Use exponents to rewrite your numbers in the columns labeled “Total Length, Fraction Form” for Stages 0-3 Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 4/3 1.33 2 16 1/9 16/9 1.78 3 64 1/27 64/27 2.37

(Number of segments times length of segments) Predict the Stage 4 lengths. Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 41/31 1.33 2 16 1/9 16/9=42/32 1.78 3 64 1/27 64/27=43/33 2.37

(Number of segments times length of segments) How many segments will Stage 4 contain? How long will each segment be? Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 41/31 1.33 2 16 1/9 16/9=42/32 1.78 3 64 1/27 64/27=43/33 2.37 44/34=256/81

(Number of segments times length of segments) How many segments will Stage 4 contain? How long will each segment be? Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 4 1/3 41/31 1.33 2 16 1/9 16/9=42/32 1.78 3 64 1/27 64/27=43/33 2.37 256 1/81 44/34=256/81 3.16

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

Example 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.

(Number of segments times length of segments) Total Length (Number of segments times length of segments) Stage Number Number of segments Length of each segment Fraction Form Decimal Form 1 1=1 1.00 1(51)=51 1(1/3)=(1/3)1 51(1/3)1=(5/3)1 1.67 2 5(5)=52 (1/3)(1/3)=(1/3)2 52(1/3)2=(5/3)2 2.78 17 517 (1/3)17 (5/3)17 5907.84