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 ratio (see figure). The number of those square structures is a Fibonacci number. The curve can also be constructed in different ways (see gallery below): Iterated function system of 4 and 1 homothety of ratio and By joining together the curves and Lindenmayer system By an iterated construction of 8 square patterns around each square pattern. By an iterated construction of octagons The Hausdorff dimension of the Fibonacci word fractal is , with the golden ratio. Generalizing to an angle between 0 and , its Hausdorff dimension is , with . The Hausdorff dimension of its frontier is . Exchanging the roles of "0" and "1" in the Fibonacci word, or in the drawing rule yields a similar curve, but oriented 45°. From the Fibonacci word, one can define the «dense Fibonacci word», on an alphabet of 3 letters: 102210221102110211022102211021102110221022102211021... . The usage, on this word, of a more simple drawing rule, defines an infinite set of variants of the curve, among which: a "diagonal variant" a "svastika variant" a "compact variant" It is conjectured that the Fibonacci word fractal appears for every sturmian word for which the slope, written in continued fraction expansion, ends with an infinite sequence of "1"s. Gallery The Fibonacci tile The juxtaposition of four curves allows the construction of a closed curve enclosing a surface whose area is not null. This curve is called a "Fibonacci tile". The Fibonacci tile almost tiles the plane. The juxtaposition of 4 tiles (see illustration) leaves at the center a free square whose area tends to zero as k tends to infinity. At the limit, the infinite Fibonacci tile tiles the plane. If the tile is enclosed in a square of side 1, then its area tends to . Fibonacci snowflake The Fibonacci snowflake is a Fibonacci tile defined