To a Fibonacci word of length 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..
The curve encloses an infinity of square structures of decreasing sizes in a ratio . The number of those square structures is a Fibonacci number.
The curve can also be constructed by different ways :
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 :
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 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 un a square of side 1, then its area tends to.
Fibonacci snowflake
The Fibonacci snowflake is a Fibonacci tile defined by:
if
otherwise.
with and, "turn left" et "turn right", and, Several remarkable properties ::
It is the Fibonacci tile associated to the "diagonal variant" previously defined.
It tiles the plane at any order.
It tiles the plane by translation in two different ways.
its perimeter, at order n, equals. is the nth Fibonacci number.
its area, at order n, follows the successive indexes of odd row of the Pell sequence.
