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)
L0 =3→this is the original triangle
L1 =1
L2 =1/3
Ln ={3, 1, 1/3, 1/9, 1/27…}
The ratio of this series is (1/3)
We get the formula:
Ln= 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:
Nn ={3, 12, 48…}
The common ratio then is four, and since we started with
three sides, the series begins with three.
The formula is thenNn=(3)(4)n
n= the iteration number
N0 =3 N1 =12 N2=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:
Nn=(3)(4)n
Ln= 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:
Pn= Nn Ln
Pn=3*S(4/3)n
Again, if the side length is equal to 3, for P0 :
P0= 3*3 (4/3)0=9
P1= 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!
2nd 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
Ln= S(1/3)n
Nn=(3)(4)n
Pn= Nn Ln
Pn=3*S(4/3)n