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## Geometry (all content)

### Course: Geometry (all content)>Unit 8

Lesson 8: Koch snowflake fractal

# Koch snowflake fractal

A shape that has an infinite perimeter but finite area. Created by Sal Khan.

## Want to join the conversation?

• "" is there a way for me to calculate the area of the koch triangle, i mean it is finite right? • So, is infinite, but it is in a limited area, so it's infinite, but at the same time it's finite?
Edit: Wouldn't it eventually get so small that you couldn't make it any bigger?
Edit: So if it can get smaller and smaller forever, doesn't it have a infinite perimeter and an infinite area?
Edit: My brain hurts. •  It's perimeter is infinite. Its area is confinable and approaches a real number which could be rational (but doesn't have to be), but it is a real number.
You can't actually say exactly how big it is, but you can say what it is smaller than and will never be bigger than.
• Would it be true to say that no object of this shape can possibly exist because any object made of matter would find a limit when the size of the triangle at the next layer of triangles is smaller than the size of the elementary particles it is made of? • As Sal says on this video the perimeter of this koch snowflake is infinite.

One really intriguing question popped out of my mind.
Are not all irrational numbers like pi based on some simple recursive formula as fractals do.

So we could be able to make a clear definition to irrational numbers by fractals.
What is the fractal formula for pi?? • Does this Koch Snowflake Fractal goes on forever till infinity? • When will this be used in a job or real life scenario? • It certainly could be used (and is used) in art installations, graphics and design, also fabric design. Because it is used in these fields, it must be used in computer programming, because to represent the shapes, you have to know the mathematics that would allow you to produce the shapes.

On another level, no one will hire you just because you know how to construct a Snowflake Fractal, or calculate its area or perimeter (well, maybe someone who needs to produce a design would), but there are many applications that might benefit from knowledge of Koch snowflakes. Notice that Sal mentioned a little of this late in the video where he was comparing the coasts of real land masses to fractal shapes. I have seen the mathematics of fractals used in biology as a way of modeling very complex body surfaces (say, of lizards covered with clunky scales). Mathematical modeling is an important way of understanding the reason why a condition exists.

I would say, it pays to learn the concepts regardless of whether they seem useful to you, because you will keep your mind sharp and you may find a better way to predict drought, for example, based on some mathematical concept that you happened to learn. It happens all the time.
• In the video, where you first made your two marks on each side of the triangle, can you make the smaller triangles going inward instead of outward, and having it still be considered a fractal? • If we consider the 10000000000th iteration then side of the triangle will be s/(3^1000000000)=0.000000000033s. Which is a very small value. So how does this small value contribute to the perimeter of the Koch triangle?? • does the shape have to be a triangle? • There are thousands of fractals. Some take triangles as base ( like Koch Snowflake or Sierpinski Triangle), but other can use squares ( Vicsek fractal, Sierpinsky carpet), and, in fact, you don't have to be based on any shape, a beautiful example of that are Julia's Sets and Mandelbrot's Set ( which are based on complex numbers but generate mysteriously elaborate shapes ).

Thera are many more examples, but the most fun thing is to try to come up with your own fractal patterns! 