THE KOCH SNOWFLAKE

Alison Kiolbasa Heather McClain Cherian Plamoottil

The Koch Snowflake History

• The Koch Snowflake was studied by Niels Fabian

Helge von Koch

• Koch was of Swedish nobility. • From his work he helped to prove the Riemann

hypothesis. This is the rough equivalent to the prime number theory.

•However his most outstanding work was done

on the Koch curve which looks like:

•The Koch Snowflake is similar to this curve except it

starts with an equilateral triangle.

•The snowflake is self-similar. This means that it is

roughly the same on any scale.

•If you were to magnify any part of the snowflake it

would look the same.

The Iterations

•Iterations are essentially functions that are repeated. •Mathematically it can be expressed as: •Assume the function is f(x)=(x+3)/2 •The Second Iteration would be f(f(x))=((x+3)/2 +3)/2 = (x +3+6)/4 = (x+9)/4 •Likewise the third iteration would be: f(f(f(x))) =((x+9)/4)+3)/2 = (x+9+12)/8= (x+21)/8 The iterations for the Koch Snowflake look like this

Length of Sides

Observe the photo: What do you notice about the length of each side by iterations?

Length of Sides

If you said that each iteration is (1/3) of the length before it you are correct!

Lets say that the equilateral triangle starts with a length of S=3. (n=iteration numbers) L 0 =3→this is the original triangle L 1 =1 L n L 2 =1/3 ={3, 1, 1/3, 1/9, 1/27…} The ratio of this series is (1/3) We get the formula: L n = S(1/3) n

Number of Sides

• Like the Length of the sides, the number of sides for the snowflake is also a geometric series.

• The snowflake always starts with three sides. When we make the first iteration, as shown in the picture, we have 12 sides now.

• The second iteration shows 48 sides if we count it.

Number of Sides

We end up with a series: N n ={3, 12, 48…} The common ratio then is four, and since we started with three sides, the series begins with three.

The formula is then N n =(3)(4) n n= the iteration number N 0 =3 N 1 =12 N 2 =48

Common Sense

If we know the length of each side, and the total number of sides, we can also determine the formula for the perimeter of the koch snowflake!

Perimeter of Sides

We already know these two formulas: N n =(3)(4) n L n = S(1/3) n The perimeter of an object that has equal sides, is always the total number of sides times the length. This means: P n = N n L n P n =3*S(4/3) n Again, if the side length is equal to 3, for P 0 : P 0 = 3*3 (4/3) 0 =9 P 1 = 3*3 (4/3) 1 =12

Infinity

Because we can continuously add more and more sides to the snowflake, the perimeter continues to get larger and larger. This means that the snowflake has a perimeter of infinite length!

BUT……

What about area??

What do you notice about the snowflakes as it iterates?

It Never exceeds the box area!

The area of the snowflake is finite…the series converges!

How does this happen?

If we start with the area of a regular equilateral triangle we get this formula….

Area with iterations:

The first iteration: We get the area of the initial triangle, plus the area of three more triangles with side lengths of (1/3) of S.

Tricky Part

We get the 1+(3/9) in there because the sequence alone is not a series, but the partial sums is!

2

nd

Iteration

Formula for Area

Each time you iterate it, we add (3)(4) n triangles of : This means that the following formula is the area. The area converges to:

Any Questions?

Sources: http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/area.htm

http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.ht

m http://en.wikipedia.org/wiki/Koch_snowflake http://mathworld.wolfram.com/KochSnowflake.html

http://math.rice.edu/~lanius/frac/koch.html

http://www.shodor.org/interactivate/activities/koch/index.ht

ml

Formulas to know

L n = S(1/3) n N n =(3)(4) n P n = N n L n P n =3*S(4/3) n