Koch Snowflake

Good Afternoon all!
It snowed last night! Yip Yip!! And it's very cold too, the bunnies are all covered in blankets by the heater. In honor of the first snow of the season (granted, only a few flakes hit the ground, but it still counts), today I will show you the koch snowflake. This is a mathematical shape, named after its likeness to a single snowflake. First, a few terms to work with:

Segment: pretty simple, a single line of unit length 1.
Iteration: the act of repeating a process with the aim of approaching a desired goal. each repetition of the process is also called an iteration; thus 3 iterations means repeating the original process 3 times. got it? good.
Area: the quantity that expresses the extent of a two-dimensional figure or shape. i.e, the area of a rectangle is width * height.
Perimeter: the length of the path that surrounds a two-dimensional figure or shape. i.e, the perimeter of a rectangle is width + width + height + height.

All set? Okay, here we go!

The Koch snowflake is named after Helge von Koch, a Swedish mathematician who specialized in pure mathematics, (read number theory). The Koch snowflake is a fractal shape, similar to the fractal tree I described a post or two ago. Fractals are simply shapes that keep on dividing, and splitting. I'll demonstrate:




These are the first 4 iterations of the koch snowflake. In the first iteration, we see an equilateral triangle. To create the 2 iteration, draw a new equilateral triangle in the exact middle of each line segment, proportional to one third the total length of the segment. Simple, right? Now, to create the 3rd iteration, notice we now have 12 smaller lines, instead of 3. No matter, we will continue to draw an equilateral triangle in the exact middle of each of the 12 line segments. Now we have 48 very small line segments. You may notice each time we create a new iteration, the total number of line segments increases drastically. The 4rth iteration will have 192 line segments. Have you figured out the pattern?
Each new iteration multiplies the total line segments of the previous iteration by 4.

Now it is beginning to look like a snowflake!!
Now lets look at some very interesting properties with this snowflake we have created.
Perimeter:
Assume each original side length of our equilateral triangle has a length of 1 unit.
The first iteration has a perimeter of 3 units.
The second iteration has 12 side lengths, but the length of each is only 1/3 as long: .333*12 = 4.
The third iteration has 48 side lengths, but the length of each is only 1/9 as long: 1/9*48=5.333.
The fourth iteration has 192 side lengths, but the length of each is only 1/27 as long: 1/27*192= 7.111
...And you get the idea.
But wait, are we saying that the perimeter of this snowflake gets infinitely large? Yes! If we extend the number of iterations to say, 1 millionth iteration, the perimeter will continue to grow with it. Remember that a line has no width, it can be thinner than a human hair. while our computer screens may not show every detail, in theory the perimeter can be 1 billion units long! wow!

Now lets look at the area. Will the area also be infinitely large? Sadly, no, here's why. In a nutshell, think about this: no matter how complicated the perimeter gets, can we still draw the same size box around it? lets look at a GIF showing the first 7 iterations:





While our snowflake is continuing to grow, it looks like it will never be taller or wider than it appears in the 3rd iteration, that is the rest of the points will grow around the snowflake, but it will still fit inside a circle! So, what is the area of the koch snowflake? The formula is a little complicated, but if we assume the area of our original triangle is about .4333 units squared. Therefore, using the associated formula for a koch snowflake, the total area of an infinite koch snowflake would be about .6928 units squared.

Say, is it still snowing outside? No? That's too bad. Calculating raindrops isn't nearly as much fun. Ah well, it may snow again tonight! Farewell for now!!