Creating a Koch Snowflake Fractal K. White

Creating a Koch Snowflake Fractal K. White
K. White
Creating a Koch Snowflake Fractal
By using the iterate command, you can repeat an action again and again on the same figure. Repeatedly taking the
result of an action and applying the action to that result again is called iteration in mathematics and is central to the creation of
fractals. In this activity, you’ll create the first few stages of a fractal that’s called the Koch Snowflake. A true Koch Snowflake,
created in infinitely many stages, has dimension between 1 and 2.
Sketch and Investigate
1.
2.
In a new sketch, construct a horizontal segment ̅̅̅̅ near the bottom of the sketch window.
Mark point A as a center (double click point A), and dilate point B (in Transform menu) by a scale factor of 1/3.
D
A
B'
B'
B
A
B'
B
A
C
E
3.
4.
Also dilate point B by a scale factor of 2/3.
Mark the first (left) point B’ as a center and rotate the other (right) point B’ by 60 degrees.
5.
6.
Hide ̅̅̅̅ .
Connect the five points with segments to form a line with a peak in the middle, as shown
above right. Relabel the points to match the figure (double click label).
Select, in order, point A and point B. Choose Iterate from the transform menu. The iteration
box appears.
Choose Final Iteration Only from the Display pop-up menu. This tells Sketchpad to hide
segments once they get iterated on.
7.
8.
B
After Step 10
9.
10.
11.
12.
13.
14.
Click first on point A in the sketch , then on point C. This tells Sketchpad to “do to segment
AC what was done to segment AB.”
Press the – key twice so that you see just one iteration.
Choose Add New Map from the Structure menu so that you can iterate on a new segment.
Click on point C first, then on point D.
Repeat steps 11 and 12 once for each segment. When you’re done, the Iteration box should
look like the one on the right. Click Iterate.
Hide segments AC, CD, DE, EB. You know have one of the 3 sides you need.
15. Next, highlight the entire structure that you made earlier. Click the Custom Tools menu and
click Create new tool. This makes it to where you can connect any two points with the
structure you created before.
16. Now you need some points to use your new tool on. For the Koch Snowflake this is done on
an equilateral triangle. To create this triangle draw a segment and rotate the segment and
endpoint 60 degrees with the first point being the center. Then rotate that segment 60
degrees to create an equilateral triangle.
After Step 13
After Step 15
17. In order to use your tool you only need the points of the triangle so Hide the segment of
the triangle, but leave the points as they are.
18. Select your fancy Koch tool and connect the dots! TaDa! It’s the Koch Snowflake! Explore
the other iterations by pressing Shift and the + key or pressing the – key.
19. On the back of this page there is a picture of a Koch Snowflake!
Step 16
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