Fibonacci word fractal

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

Definition

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The first iterations
 
L-system representation[1]

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

For each digit at position k:

  1. Draw a segment forward
  2. 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

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The Fibonacci numbers in the Fibonacci word fractal.

Some of the Fibonacci word fractal's properties include:[2][3]

  • 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... (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 sequence of "1"s.
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The Fibonacci tile

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

Fibonacci snowflake

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Fibonacci snowflakes for i = 2 for n = 1 through 4:  ,  ,  ,  [4]

The Fibonacci snowflake is a Fibonacci tile defined by:[5]

  •   if  
  •   otherwise.

with   and  ,   "turn left" and   "turn right", and  .

Several remarkable properties:[5][6]

  • 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  , where   is the nth Fibonacci number.
  • its area at order n follows the successive indexes of odd row of the Pell sequence (defined by  ).

See also

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References

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  1. ^ Ramírez, José L.; Rubiano, Gustavo N. (2014). "Properties and Generalizations of the Fibonacci Word Fractal", The Mathematical Journal, Vol. 16.
  2. ^ Monnerot-Dumaine, Alexis (February 2009). "The Fibonacci word fractal", independent (hal.archives-ouvertes.fr).
  3. ^ Hoffman, Tyler; Steinhurst, Benjamin (2016). "Hausdorff Dimension of Generalized Fibonacci Word Fractals". arXiv:1601.04786 [math.MG].
  4. ^ Ramírez, Rubiano, and De Castro (2014). "A generalization of the Fibonacci word fractal and the Fibonacci snowflake", Theoretical Computer Science, Vol. 528, p.40-56. [1]
  5. ^ a b Blondin-Massé, Alexandre; Brlek, Srečko; Garon, Ariane; and Labbé, Sébastien (2009). "Christoffel and Fibonacci tiles", Lecture Notes in Computer Science: Discrete Geometry for Computer Imagery, p.67-8. Springer. ISBN 9783642043963.
  6. ^ A. Blondin-Massé, S. Labbé, S. Brlek, M. Mendès-France (2011). "Fibonacci snowflakes".
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