The Koch Snowflake
An AMAZING Phenomenon: Infinite Perimeter
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by Cynthia Lanius
A really interesting characteristic of the Koch Snowflake is its perimeter.
Ordinarily, when you increase the perimeter of a geometric figure, you also increase its area. If you have a square with a huge perimeter, it also has a huge area. But wait till you see what happens here!
Remember the process:
- Divide a side of the triangle into three equal parts and remove the middle section.
- Replace the missing section with two pieces the same length as the section you removed.
- Do this to all three sides of the triangle.
Let's investigate the perimeter below.
Question 1: If the perimeter of the equilateral triangle that you start with is 9 units, what is the perimeter of the other figures?
perimeter = 9 units
perimeter = ? units
perimeter = ? units?
Hint: Think of the original triangle with sides of three parts,
and the next figure with sides of four parts.
Question 2: Is there a pattern here? The perimeter of each figure is ___ times the perimeter of the figure before.
Question 3: If the original triangle has a perimeter of 9 units, how many iterations would it take to obtain a perimeter of 100 units? (Or as close to 100 as you can get.)
Question 4: Now think of doing this many, many times. The perimeter gets huge! But does the area? We say the area is bounded by a circle surrounding the original triangle. If you continued the process oh, let's say, infinitely many times, the figure would have an infinite perimeter, but its area would still be bounded by that circle.
An infinite perimeter encloses a finite area... Now that's amazing!!
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Copyright 1996-2007, Cynthia Lanius