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Koch coastline
by RS  admin@creationpie.com : 1024 x 640


1. Koch coastline

2. Koch coastline development
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A fractal is a self-similar object (i.e., with fractional dimension).

3. Step 0
Koch coastline #0Start with a straight line as the base case.

4. Step 1
Koch coastline #0 #1Divide the line segment in into three parts.

5. Step 2
Koch coastline #0 #1 #2Extend the middle segment. Here are the base and step cases. The fractional dimension in log(4/3).

6. Step 3
Koch coastline #0 #1 #2 #3Now do the same for the step case to get the next step case.

7. Step 4
Koch coastline #0 #1 #2 #3 #4Continue in the same manner.

8. Step 5
Koch coastline #0 #1 #2 #3 #4 #5Continue in the same manner.

9. Step 6
Koch coastline #0 #1 #2 #3 #4 #5 #6Continue in the same manner.

10. Step 7
Koch coastline #0 #1 #2 #3 #4 #5 #6 #7Continue in the same manner.

11. Step 8
Koch coastline #0 #1 #2 #3 #4 #5 #6 #7 #8In 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.

12. 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.

13. Koch coastline
Koch coastlinesDo 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.

14. Randomized coastline
Randomized Koch coastlineImagine 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.

15. Realistic coastline
Randomized and colored Koch coastlineColor shading makes the computer-generated coastline look more realistic.

16. 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. Koch coastline

17. Regular coastline
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18. Randomized coastline
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19. Regular coastline
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20. Randomized coastline
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21. End of page

by RS  admin@creationpie.com : 1024 x 640