This isn't really a purely philiosophical question, but I didn't know where else to post it.
Its a mathematical question (but a rather
philiosphical mathematical question
)
Anyway, I have just finsihed a book on chaos theory, and one thing that really interested me is fractals. I have a question regarding the Koch curve.
Looking at a koch curve, it is easy to see that there is a definite region which is enclosed by the curve. There is also a definate region which is outside of the curve. My question involves the area of "fuzzyness" where the two regions meet. If we were to choose a point inside this region of less then obvious certainty, is there a way to prove without doubt that a point is or is not enclosed by the curve?
It would be possible to prove if a point
is inside the curve, it is simply a matter of going through the various iteration until one of the new triangles overlaps the point. However, assuming that after, say, 100 iterations, the point has not been covered, how can we tell that after further iteratons the point will not be covered?
Is there a method to prove that a point lies outside the curve? If so what is it?
BTW, the book I have just finished is Chaos: Making a New Science by James Gleick. If anyone has suggestions for further reading I would be grateful. Preferably nothing too heavy, as you can see I am new to the subject, and I am not a mathematician.