
A constrained fractal
Posted:
Jun 28, 2006 7:25 AM


Iteration 0: an equilateral parallelogram. Imagine this as two isosceles triangles meeting basetobase. 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 coincide. At subsequent iterations, do not construct on these sides. That is, develop the figure without internal construction.
At subsequent iterations, further coincidences 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 nth iteration? 2. What are the area and perimeter of the resulting figure?

