Fibonacci word fractal

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The Fibonacci word fractal is a fractal curve defined on the plane from the Fibonacci word.

Definition[edit]

The first iterations

This curve is built iteratively by applying, to the Fibonacci word 0100101001001...etc., the Odd–Even Drawing rule:

For each digit at position k :

  • if the digit is 0 : draw a segment in the current direction
  • if the digit is 1 : draw a segment after a 90° angle turn:
    • to the right if k is even
    • to the left 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[1][2][edit]

The Fibonacci numbers in the Fibonacci word fractal.
  • 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 by different ways (see gallery below):
    • Iterated function system of 4 and 1 homothety of ratio and
    • By joining together the curves and
    • Lindermayer 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... ((sequence A143667 in the OEIS)). 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 series of "1".

Gallery[edit]

The Fibonacci tile[edit]

Imperfect tiling by the Fibonacci tile. The area of the central square tends to infinity.

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 un a square of side 1, then its area tends to .
Perfect tiling by the Fibonacci snowflake

Fibonacci snowflake[edit]

The Fibonacci snowflake is a Fibonacci tile defined by:[3]

  • if
  • otherwise.

with and , "turn left" et "turn right", and ,

Several remarkable properties :[3] · :[4]

  • 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 (defined by ).

References[edit]

  1. ^ The Fibonacci word fractal
  2. ^ Hoffman, Tyler; Steinhurst, Benjamin (2016). "Hausdorff Dimension of Generalized Fibonacci Word Fractals". arXiv:1601.04786 [math.MG].
  3. ^ a b Christoffel and Fibonacci tiles
  4. ^ Fibonacci snowflakes

See also[edit]

External links[edit]