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Topic: A constrained fractal
Replies: 17   Last Post: Jul 15, 2006 3:28 PM

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cuthbert

Posts: 10
Registered: 6/26/06
A constrained fractal
Posted: Jun 28, 2006 7:25 AM
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Iteration 0: an equilateral parallelogram. Imagine this as two isosceles triangles meeting base-to-base. Internal angles are deg 60,60,120,120.

Iteration 1: Each side of the parallelogram becomes the base of a new isosceles triangle, deg 120,30,30. So there are 4 new triangles at iteration 1.

Iteration 2: Each side of the new triangles becomes the base of further similar triangles. 8 new triangles at this iteration.

And so on.

Now there's a constraint. At iteration 4, the shorter sides of two of the newly constructed triangles co-incide. At subsequent iterations, do not construct on these sides. That is, develop the figure without internal construction.

At subsequent iterations, further co-incidences of lines occur, and these lines again become 'dead ends', on which no further construction will be made.

Continue ad infinitum.

Questions:

1. How many triangles will be constructed at the n-th iteration?
2. What are the area and perimeter of the resulting figure?



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