TheFibonacci word fractal is afractal curve defined on the plane from theFibonacci word.

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

For each digit at positionk:

1. If the digit is 0:
   • Draw a line segment then turn 90° to the left ifk iseven
   • Draw a line segment then Turn 90° to the right ifk isodd
2. If the digit is 1:
   • Draw a line segment and stay straight

To a Fibonacci word of length${\displaystyle F_n}$ (then^thFibonacci number) is associated a curve${\displaystyle {\mathcal{F}}_n}$ made of${\displaystyle F_n}$ segments. The curve displays three different aspects whethern is in the form 3k, 3k + 1, or 3k + 2.

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

• The curve${\displaystyle {\mathcal{F}}_{\mathcal{n}}}$ contains${\displaystyle F_n}$ 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${\displaystyle 1+\sqrt{2}}$. This number, also called thesilver ratio, is present in a great number of properties listed below.
• The number of self-similarities at leveln is aFibonacci number \ −1. (more precisely:${\displaystyle F_{3n+3}-1}$).
• The curve encloses an infinity of square structures of decreasing sizes in a ratio${\displaystyle 1+\sqrt{2}}$ (see figure). The number of those square structures is a Fibonacci number.
• The curve${\displaystyle {\mathcal{F}}_n}$can also be constructed in different ways (see gallery below):
  • Iterated function system of 4 and 1 homothety of ratio${\displaystyle 1/(1+\sqrt{2})}$ and${\displaystyle 1/(1+\sqrt{2}{)}^2}$
  • By joining together the curves${\displaystyle {\mathcal{F}}_{n-1}}$ and${\displaystyle {\mathcal{F}}_{n-2}}$
  • Lindenmayer system
  • By an iterated construction of 8 square patterns around each square pattern.
  • By an iterated construction ofoctagons
• TheHausdorff dimension of the Fibonacci word fractal is${\displaystyle 3\frac{\log \phi }{\log (1+\sqrt{2})}\approx 1.6379}$, with${\displaystyle \phi =\frac{1+\sqrt{5}}{2}}$ thegolden ratio.
• Generalizing to an angle${\displaystyle \alpha }$ between 0 and${\displaystyle \pi /2}$, its Hausdorff dimension is${\displaystyle 3\frac{\log \phi }{\log (1+a+\sqrt{(1+a{)}^2+1})}}$, with${\displaystyle a=\cos \alpha }$.
• The Hausdorff dimension of its frontier is${\displaystyle \frac{\log 3}{\log (1+\sqrt{2})}\approx 1.2465}$.
• 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... (sequenceA143667 in theOEIS). 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 isconjectured that the Fibonacci word fractal appears for everysturmian word for which the slope, written incontinued fraction expansion, ends with an infinite sequence of "1"s.

• 
  Curve after${\displaystyle F_{23}}$ iterations.
• 
  Self-similarities at different scales.
• 
  Dimensions.
• 
  Construction by juxtaposition (1)
• 
  Construction by juxtaposition (2)
• 
  Order 18, with some sub-rectangles colored.
• 
• 
  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 "π/8 variant"
• 
  Artist creation (Samuel Monnier).

The juxtaposition of four${\displaystyle F_{3k}}$ curves allows the construction of a closed curve enclosing a surface whosearea 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 ask 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${\displaystyle 2-\sqrt{2}=0.5857}$.

TheFibonacci snowflake is a Fibonacci tile defined by:^[5]

• ${\displaystyle q_n=q_{n-1}{q}_{n-2}}$ if${\displaystyle n\equiv 2(\mathrm{mod}3)}$
• ${\displaystyle q_n=q_{n-1}{\overline{q}}_{n-2}}$ otherwise.

with${\displaystyle q_0=ϵ}$ and${\displaystyle q_1=R}$,${\displaystyle L=}$ "turn left" and${\displaystyle R=}$ "turn right", and${\displaystyle \overline{R}=L}$.

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.
• itsperimeter at ordern equals${\displaystyle 4F(3n+1)}$, where${\displaystyle F(n)}$ is then^thFibonacci number.
• its area at ordern follows the successive indexes of odd row of thePell sequence (defined by${\displaystyle P(n)=2P(n-1)+P(n-2)}$).

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

• "Generate a Fibonacci word fractal",OnlineMathTools.com.

• Fractals
• Fractal curves