Fibonacci word fractal
The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word. Definition This curve is built iteratively by applying the Odd–Even Drawing rule to the Fibonacci word 0100101001001...: For each digit at position k: Draw a segment forward If the digit is 0: Turn 90° to the left if k is even Turn 90° to the right if k is odd To a Fibonacci word of length (the nth Fibonacci number) is associated a curve made of segments. The curve displays three different aspects whether n is in the form 3k, 3k + 1, or 3k + 2. Properties Some of the Fibonacci word fractal's properties include: The curve contains segments, right angles and flat angles. The curve never self-intersects and does not contain double points. At the limit, it contains an infinity of points asymptotically close. The curve presents self-similarities at all scales. The reduction ratio is . This number, also called the silver ratio, is present in a great number of properties listed below. The number of self-similarities at level n is a Fibonacci number \ −1. (more precisely: ). The curve encloses an infinity of square structures of decreasing sizes in a