Fibonacci word fractal

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

Definition

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 $F_{n}$ (the n^th Fibonacci number) is associated a curve $\mathcal{F}_n$ made of $F_{n}$ segments. The curve displays three different aspects whether n is in the form 3k, 3k + 1, or 3k + 2.

Properties^[1][2]

The Fibonacci numbers in the Fibonacci word fractal.

• The curve ${\displaystyle {\mathcal {F_{n}}}}$, contains $F_{n}$ segments, ${\displaystyle F_{n-1}}$ right angles and ${\displaystyle F_{n-2}}$ 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 ${\displaystyle \scriptstyle {1+{\sqrt {2}}}}$. 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 : ${\displaystyle F_{3n+3}-1}$).
• The curve encloses an infinity of square structures of decreasing sizes in a ratio ${\displaystyle \scriptstyle {1+{\sqrt {2}}}}$. (see figure) The number of those square structures is a Fibonacci number.
• The curve $\mathcal{F}_n$can also be constructed by different ways (see gallery below):
  • Iterated function system of 4 and 1 homothety of ratio ${\displaystyle \scriptstyle {1/(1+{\sqrt {2}})}}$ and ${\displaystyle \scriptstyle {1/(1+{\sqrt {2}})^{2}}}$
  • By joining together the curves ${\displaystyle {\mathcal {F}}_{n-1}}$ and ${\displaystyle {\mathcal {F}}_{n-2}}$
  • 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 ${\displaystyle \scriptstyle {3{\frac {\log \varphi }{\log(1+{\sqrt {2}})}}=1,6379}}$, with ${\displaystyle \scriptstyle {\varphi ={\frac {1+{\sqrt {5}}}{2}}}}$, the golden ratio.
• Generalizing to an angle $\alpha$ between 0 and $\pi /2$, its Hausdorff dimension is ${\displaystyle \scriptstyle {3{\frac {\log \varphi }{\log(1+a+{\sqrt {(1+a)^{2}+1}}}})}}$, with ${\displaystyle a=\cos \alpha }$.
• The Hausdorff dimension of its frontier is ${\displaystyle \scriptstyle {{\frac {\log 3}{{\log(1+{\sqrt {2}}})}}=1,2465}}$.
• 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 varaint»
  • 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

• 

  Curve after ${\displaystyle \textstyle {F_{23}}}$ iterations.

• 

  Self-similarities at different scales.

• 

  Dimensions.

• 

  Construction by juxtaposition (1)

• 

  Construction by juxtaposition (2)

• 
• 

  Construction by iterated suppression of square patterns.

• 

  Construction by iterated octagons.

• 

  Construction by iterated collection of 8 square patterns around each square pattern.

• 

  With a 60° angle.

• 

  Inversion of "0" and "1".

• 

  Variants generated from the dense Fibonacci word.

• 

  The "compact variant"

• 

  The "svastika variant"

• 

  The "diagonal variant"

• 

  The "pi/8 variant"

• 

  Artist creation (Samuel Monnier).

The Fibonacci tile

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

The juxtaposition of four ${\displaystyle F_{3k}}$ 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 ${\displaystyle \scriptstyle {2-{\sqrt {2}}=0.5857}}$.

Perfect tiling by the Fibonacci snowflake

Fibonacci snowflake

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

• ${\displaystyle \scriptstyle {q_{n}=q_{n-1}q_{n-2}}}$ if ${\displaystyle \scriptstyle {n\equiv 2{\pmod {3}}}}$
• ${\displaystyle \scriptstyle {q_{n}=q_{n-1}{\overline {q}}_{n-2}}}$ otherwise.

with ${\displaystyle q_{0}=\epsilon }$ and ${\displaystyle q_{1}=R}$, ${\displaystyle L=}$"turn left" et ${\displaystyle R=}$"turn right", and ${\displaystyle \scriptstyle {{\overline {R}}=L}}$,

Several remarquable 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 ${\displaystyle 4F(3n+1)}$. $F(n)$ is the n^th Fibonacci number.
• its area, at order n, follows the successive indexes of odd row of the Pell sequence (defined by ${\displaystyle P(n)=2P(n-1)+P(n-2)}$).

References

1. The Fibonacci word fractal
2. "Hausdorff Dimension of Generalized Fibonacci Word Fractals".
3. Christoffel and Fibonacci tiles
4. Fibonacci snowflakes

See also

• Golden ratio
• Fibonacci number
• Fibonacci word
• List of fractals by Hausdorff dimension

External links

• "The Fibonacci word fractal", Alexis Monnerot-Dumaine, february 2009
• "Christoffel and Fibonacci tiles", A. Blondin-Massé, S. Labbé, S. Brlek, september 2009
• "Fibonacci snowfalkes", A. Blondin-Massé, S. Labbé, S. Brlek, M. Mendès-France 2010
• "Hausdorff dimension of generalized Fibonacci word fractals", Tyler Hoffman, Benjamin Steinhurst 2016