Mathematical Investigations IV
Name
Mathematical Investigations IV
Snowflake Project
In this project, you'll work with a partner to study the Koch Snowflake curve, and then you'll
create and analyze your own fractal. You'll use your understanding of sequences and series, you'll
draw a little, and you'll explain what you've found.
Step One:
Complete the worksheet on the Koch Snowflake curve.
Step Two:
Use the sliders in the Mathematica notebook Koch, etc.nb. You may find this notebook on the
student server > public > math > Mathematica notebooks > MI 4 > S & S. Play a little to see
what happens to different patterns. Some are pretty, and some are just bizarre.
Step Three:
Create your own pattern! Begin with one segment. Your next step should have at least three
segments. You'll need to think carefully about how you're subdividing each segment and what
angles you're using. Draw at least four stages, where a straight line segment is stage one.
Next, consider an enclosed region as was done with the Snowflake Curve. You may begin with
either an equilateral triangle or a square. Your choice! (Or perhaps you want to start with a
pentagon or hexagon if you're feeling adventurous!)
Step Four:
Study what happens to the total length of your fractal as your initial triangle or square goes
through more and more iterations.
Step Five: Study what happens to the total area of your shape as your initial triangle or square
goes through more and more iterations.
Snowflake Project.1
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Mathematical Investigations IV
Name
You and your partner will hand in the snowflake worksheet and one project with the components
listed below. This must be typed. Use a reasonable font, 12 pt, double-spaced, and use
reasonable margins.
To hand in:
(1)
Koch Snowflake worksheet – one copy per student.
(2)
Your project – one copy per pair of students. This should include the following:
(a)
An introductory paragraph. What are you planning to tell us?
Referring to one line segment only:
(b)
Describe very clearly how one iteration of your pattern is created from a line
segment. Be careful with segment lengths and angles.
(c)
Graphics. Create the first iteration (beyond the straight line segment) using
technology. For this, you may use the drawing program in Word or you may use any
other software and copy the drawing into your Word document. Show two more
iterations. These may be drawn by hand or using technology. Make sure these graphics
are sufficiently large so they are clear to the reader.
For an enclosed figure:
(d)
Explain how you're calculating the perimeter of you enclosed "snowflake" and
show your calculations/equations. Find the perimeter, if it exists, as n  , and interpret
this result.
(e)
Explain what happens to the area of your enclosed "snowflake." Explain and
show your calculations and the area, if it exists, as n  .
Generally:
(f)
Write a concluding paragraph.
Snowflake Project.2
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