Mathematical Investigation:
VON KOCH’S SNOWFLAKE CURVE
Ha Yeon Lee 11B
Mathematics HL
 Introduction:
 History of Von Koch’s Snowflake Curve
The Koch snowflake is a mathematical curve, which is believed to be one of the earliest
fractal curves with description. In 1904, a Swedish mathematician, Helge von Koch
introduced the construction of the Koch curve on his paper called, “On a continuous curve
without tangents, constructible from elementary geometry”.
 In this mathematical task, I am going to investigate how the area and perimeter of a
shape/curve changes and find out whether they increase by the same number every time,as
the following process is repeated:
i.
Start with an equilateral triangle.
ii.
Divide each side of the triangle into three equal segments.
iii.
On the middle part of each side, draw an equilateral triangle by connecting lines.
iv.
Now remove the line segment that makes the base of the smaller triangle that was
formed in step 3.
The above process (steps i~iv) can be repeated indefinitely. The shape that emerges is called
“Von Koch’s Snowflake” for obvious reasons. An equilateral triangle, which is the shape used
to start with to draw the Koch Snowflake curve, turns its shape similar to a star or a snowflake
as each side of the previous curve is pushed out.
 Process:
In this investigation, the process of drawing the Koch curve has to repeat in order to generalize
rules for both perimeter and area.
Perimeter: Under the assumption that the equilateral triangle (so-called C0) at the very start
has a perimeter of 3 units, find the perimeter for the next curves (C1, C2, C3, and so on), and
eventually, find the perimeter of Cn.

=
C0
+
C1
+
↑ C0 is composed of three equal line segments.
=
+
+
+
↑ C1 is composed of four equal line segments.
During the second iteration, when extra equilateral triangles are added on the middle part of
16
16
each side of the new curve, C1, the perimeter increases to . The perimeter of of C2 is the
3
3
combination between the previous perimeter of C1 and an extra length of line segment:
C2
=
+
+
4
+
+
+
+
+
3 1
+
3 3
+
=
12
3
4
+3=
𝟏𝟔
𝟑
Through the third iteration, the Koch curve, C3, gains extra line segments, and therefore, the
value of the perimeter continues to increase from that of the previous curve:
1
C3
27
=
× 48 +
144
27
*271 = 13 × 13 × 13
The perimeter of C3 is ended up being
64
9
=
192
27
=
𝟔𝟒
𝟗
16
*144
= 3
27
unit.
Based on my findings in the perimeter of the Koch curve, I created a table that will allow me
to see general patterns much easily:
n (number of iterations)
0
1
2
3
Perimeter (in unit)
3
4
16
3
64
9
↑Table: Investigating the Change in the Perimeter of the Curve after each Iteration
4
If I list all values of the perimeter in a row, I can see that after each iteration, 3 (common ratio)
is constantly multiplied.
-
Number of sides
After each iteration, I found that one side of the curve from the preceding stage has become
four sides in the following stage. The construction of the Koch curve begins with three sides
and therefore the formula for the number of sides can be expressed as below:
Number of sides = 3 × 4n
(for the nth iteration)
The number of sides for the curves created in each iteration (C0, C1, C2, and C3) are 3, 12, 48
and 192, in order.
-
Length of a Side
Another important observation I made, in the process of going through each iteration, is that
1
the length of a side of a new curve is always 3 of the length of a side from the previous stage.
The length of a side of the initial triangle is 1 unit. This can be expressed mathematically as
below:
1
Length of a Side = ( )n
(for the nth iteration)
3
The length of a side for the curves, C0, C1, C2, and C3, are 1,
-
1
1
1
3
9
27
,
, and
, respectively.
Perimeter
Using the mathematical formulae I established in the previous steps, I can now come up with a
general formula for the perimeter of the Koch Snowflake curve:
Perimeter = Number of sides × Length of a Side
1
P = (3 × 4n) × ( )n
3
4
P = 3 × ( )n
(for the nth iteration)
3
All the values until the third iteration are the same with what I have found by drawing
diagrams, involving simple mathematical calculations.
As n (the number of iterations) approaches infinity, the perimeter of each new curve will
continue to increase with no bounds – it means that the value of perimeter after each iteration
will get closer to infinity.
4
n → ∞, P = 3 × ( )∞ = ∞
3
∴ n → ∞, P→ ∞

Area: Under the assumption that the equilateral triangle (so-called C0) at the very start has
an area of 1 unit2, find the areas for the next curves (C1, C2, C3, and so on), and eventually,
find the area of Cn.
⇒ The investigation of the area begins by supposing that C0, the initial curve (an
equilateral triangle) has a total area of 1 unit2.
* s = one side of the initial curve
The area of the initial curve is, therefore,
1
2
×
𝑠 √3
2
×𝑠=
∴ Area =
√3 2
𝑠
4
√3 2
𝑠
4
=1
-First Iteration:
+
C0
C1
=
4
3
=
C2
=
40
27
=
C3
𝐬
𝟑
s
Area:
=
1
1
9
+
*Since s2 =
4 √3
4
3
3
×3
, Area of C1 is units2.
-Second Iteration:
+
C1
4
3
Area:
+
*Since s2 =
4√3
,
3
1
81
Area of C2 is
× 12
40
27
units2.
-Third Iteration:
+
C2
Area:
40
27
+
(× 48)
1
× 48
729
=
376
243
4√3
,
3
*As s2 =
Area of C3 is
376
243
units2.
Based on my findings in the area of the Koch curve, I created a table that will allow me to see
general patterns much easily:
Area (in unit2)
n (number of iterations)
0
1
1
4
3
40
27
1+
1
2
1
1
3
4
1+ [1+9]
1
3
3
4
376
243
4
1+ [1+9+(9)2]
3
↑ Table: Investigating the Change in the Area of the Curve after each Iteration
⇒ The general formula for the area of the Koch curve is,
1
4
4
Area of Von Koch Curve = 1+ [1+5(1 - (9)n−2 ]
3
4
The important point to consider from my findings is that the common ratio is . Therefore, as
9
n gets very large and goes to infinity, the series converges and approaches closer to the
limiting sum of
u1
1−r
(for |r| < 1), which in this case, is
1
3
4
1−9
:
n → ∞, Area of the Koch snowflake curve will get close to 1+
1
3
4
1−9
= 1+159 = 85
 Conclusion:
-
Perimeter
4
Von Koch Snowflake curve has perimeter that grows by 3 (common ratio) in each
iteration. The perimeter for the nth iteration of Von Koch curve, Cn, can be expressed as
the following:
4
P = 3 × ( )n
3
When n is infinitely large, the perimeter of Von Koch curve will be infinitely large as well,
increasing with no limit.
-
Area
The area for the nth iteration of Von Koch curve, Cn, can be expressed as the following:
1
4
4
A = 1+ [1+5(1 - (9)n−2]
3
The area is the total sum of the previous area and extra equilateral triangles (which continue to
be reduced in size) being added after each iterative process.
In conclusion, as the Koch curve endlessly increases its number of sides, its perimeter
8
increases infinitely, while its area reaches to a finite value, of the initial triangle.
5
Works Cited
Brown, Diana. Diagrams of Von Koch Snowflake Curves. October, 12th, 2009
“Fractal”. Wikipedia. 2009. Wikimedia Foundation. October, 18th, 2009
<http://en.wikipedia.org/wiki/fractal>.
“Koch Snowflake”. Wikipedia. 2009. Wikimedia Foundation. October, 18th, 2009
<http://en.wikipedia.org/wiki/Koch_snowflake>.