#### Transcript THE KOCH SNOWFLAKE

**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