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Koch coastline
1. Koch coastline development
1 0: Base
2 0: Base
3 0: Base
4 1: Generator
5 1: Generator
6 1: Generator
7 2: Step 2
8 2: Step 2
9 2: Step 2
2. Step 0
Start with a straight line as the base case.
3. Step 1
Divide the line segment in into three parts.
4. Step 2
Extend the middle segment. Here are the base and step cases. The fractional dimension in log(4/3).
5. Step 3
Now do the same for the step case to get the next step case.
6. Step 4
Continue in the same manner.
7. Step 5
Continue in the same manner.
8. Step 6
Continue in the same manner.
9. Step 7
Continue in the same manner.
10. Step 8
In the limit, the length of the coastline is infinite, continuous (connected), but nowhere differentiable (smooth), what mathematicians call a "
monster curve". Note: A simple recursive procedure can draw the entire curve.
11. Length of fractals
The length of a fractal is infinite. In practice, one stops at a certain point.
The measured length of coastlines has varied depending on measurement scale.
12. Koch coastline
Do you remember the Koch coastline?
What happens if we change the step case, at each step, to add some randomness?
To do so, flip a coin.
13. Randomized coastline
Imagine this as a coastline. More real looking, right. You can see the inlets and protruding islands. Simple to generate. That's the beauty of fractals.
14. Realistic coastline
Color shading makes the computer-generated coastline look more realistic.
15. Koch islands
Here are four variations of a simple Koch island.
This one randomization at each step results in a somewhat realistic island outline.
In practice, many parameters can be varied and randomized.
16. End of page
