Activity
Use the applet to create the standard Koch snowflake curve. The SNOWFLAKE
button will reset the applet for this exploration.
- Explain the iterative process that creates the fractal image.
- Is this process continued infinitely? Explain.
- Does the fractal image exhibit self-similarity? Provide evidence
to support your answer.
Use the fractal image, the number of segments, and total length of each
iteration to answer the following questions. You may find it useful to
make a table like the one below to organize the information.
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Number of Iterations
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Number of sides
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Perimeter
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0
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1
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2
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3
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4
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5
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- As you increase the number of iterations, what happens to the number
of sides? Write a recursive equation to represent this relationship.
Justify your equation by referring to the fractal image.
- What do you think the fractal image would look like at the hundredth
iteration? Explain using the fractal image, the table, and your recursive
equation in question 4.
- As you increase the number of iterations, what happens to the length
of the sides? Write a recursive equation to represent this relationship.
Justify your equation with the fractal image.
- As you increase the number of iterations, what happens to the perimeter
of the fractal image? Explain your answer in relationship to the number
of sides and side length that you discussed in questions 4 and 6. Write
a recursive equation to represent this relationship. Justify your equation
with the fractal image.
- Do you think the perimeter of the Koch snowflake curve is finite
or infinite? Support your answer algebraically.
Extend
- Conjecture whether the area enclosed by the Koch snowflake curve
is finite or infinite. Discuss how this could be verified algebraically.
How does this conjecture compare to your conjecture about whether the
perimeter is finite or infinite?
- What would happen to your conjectures if you changed the fundamental
shape, iteration procedure, or initial stage?
Credits
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