Von Koch´s Snowflake Curve

Name: Hannah Kusstatscher

1. Introduction1.1 Sequences

Definition.- A list of numbers where there is a pattern is called a number sequences.

Example: 3, 9, 12, 15, 18, … is a number sequence.

1. Introduction1.2 Type of SequencesThere are two different type of functions:a) Arithmetic Sequences (un+1= un + d)

Ex: let d= 3 and u1=-6 your first term, then the sequence is:

-6,-3, 0, 3, 6, 9, …

a) Geometric Sequences (un = u1.rn-1)Ex: let r= 3 and u1=2 your first term, then the sequence is:

2, 6, 18, 54, …

2.The Koch SnowflakeThe Koch snowflake (also known as the Koch star and Koch island) is a mathematical curve based on the Koch curve, which, according to wikipedia sources it appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry“.

3. Drawing process1. Start with an equilateral triangle.2. Divide each side into 3 equal parts3. On each middle part draw an

equivalent triangle.4. Delete the side of the smaller

triangle wich lies on the original one.

5. Reapeat this procedure for each side.

4. The Von Kock´s Curve

We define the initial triangle as the first curve of our sequence (C1).

The second curve, C2, is the result of «pushing out» equilateral triangles on each edge of the previous curve.

If we repeat this procedure we get a sequence of special curves C1, C2, C3, C4, …

Von Kock´s curve is the limiting case, i.e., when n is infinitely large for this sequence.

4. The Problem

If we define C1 as the perimeter of the initial equilateral curve, the equilateral triangle with sides of one unit, find the perimeter of Von Kock´s curve.

5. Solution5.1 Step 1.- Find the values for C1, C2 and C3

C1=3 units, because we have a 1 unit side

equilateral triagle

C2=3*(*4)

C1C2

13𝑢𝑛𝑖𝑡𝑠 1

9𝑢𝑛𝑖𝑡𝑠

⇐ ⇙L=4∗49=169

→ C3 =3*( )2C3

5.1 Step 2.- Deduce what type of sequence is.

With the previous perimeters we can generalize for Cn as follows:

C1= 3 = 3*()0

C2=3*(*4) = 3*()1

…

→ It´s a Gemetric Sequence!!! =)

Cn = C1.rn-1 , whereC1 = 3r =

5.1 Step 3.- Find the perimeter of Cn when n is infinitely large.

If n is infinitely large then is also infinitely large, because

Thus, the perimeter of a Vom Kock´s curve is infinitely large.

6. Conclusions

We have prooved that the Van Kock´s curve has an infinetely perimeter although it is a closed curve.

7. Bibliography