When you try to measure the length of a coastline, your finding will increase as your measuring device decreases. It means that there is a proportion between these magnitudes. This is why it is possible to find different (even infinite) lengths for a random coastline.
Mathematician Mandelbrot named this proportion as “fractal dimension”.
In the Euclidean geometry a dot has 0, a line has 1, a plane has 2 and a cube has 3 dimensions. But, in the nature shapes of objects are not regular as shown in the Euclidean geometry. In the early 20th century a mathematician named Felix Hausdorff discovered that some shapes have non-integer dimensions. Later on we started calling this non-integer dimension idea as Hausdorff-Besicovitch dimension. This idea was basis for fractal geometry’s development.
In the previous article I showed how one can calculate dimension of a shape in the Euclidean geometry. Same formula can be used in order to calculate objects that don’t have regular shapes. For that, I will be talking about a couple special fractals.
Swedish mathematician Helge von Koch created a geometrical shaped named after him: Koch snowflake.
To create a Koch snowflake, one can start drawing a straight line. Then that line should be divided into thirds as the middle part gets erased:
Draw sides of an equilateral triangle above the removed segment: (In other words, add a peak where there is a gap.)
Continue the same process forever and you will get Koch fractal:
Here are the segments and all of Koch snowflake:
Now let’s use the dimension formula to the Koch snowflake. We only need the number of parts and their lengths in each step of the construction of the Koch snowflake.
In the first step, we had a straight line that was divided into 1/3s:
In the second step we ended up with 4 of those 1/3s:
If we examine each step of the Koch snowflake we will end up with 4 parts that have 1/3 lengths. Therefore fractal dimension of Koch snowflake (which I call d) can be found as follows:
(1/3)d = 4
d ≈ 1,26.
Let’s try a variant of the Koch snowflake, which we call Koch curve. This time we will draw sides of a square instead of an equilateral triangle.
So, we will start with a straight line that is divided into thirds. Then we will remove the middle part and draw sides of a square that has no bottom line:
Next few stages of the Koch curve will look like the following:
Here we see that in each step, we end up with 5 parts that have length 1/3:
Apply this to the dimension formula and this fractal’s dimension will be as follows:
(1/3)d = 5
d ≈ 1,4649.
What does this difference in dimensions mean?
Between the curve and the snowflake, curve has more roughness and it takes up more area than the snowflake. Hence one can conclude that higher dimension means more roughness and more area for Koch fractals:
To be continued…
Another handmade fractal is Sierpinski triangle. This famous fractal was first discovered more than 100 years ago and named after a mathematician named Waclaw Sierpinski.
To construct Sierpinski triangle, one must start with an equilateral triangle:
Then mark middle of each side and connect those points to form a new triangle:
At this point, there are four smaller versions of the original triangle. Cut the middle one out and you will have three equilateral triangles that have half of the side lengths of the original triangle:
Repeat the steps forever and you will get Sierpinski triangle:
- Show that Sierpinski triangle is a fractal.
- Calculate the dimension of the Sierpinski triangle and compare your result with Koch snowflake.
M. Serkan Kalaycıoğlu