Fibonacci word fractal

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

Definition

The first iterations

L-system representation

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

For each digit at position k:

1. If the digit is 0:
   • Draw a line segment then turn 90° to the left if k is even
   • Draw a line segment then Turn 90° to the right if k is odd
2. If the digit is 1:
   • Draw a line segment and stay straight

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

The Fibonacci numbers in the Fibonacci word fractal.

Some of the Fibonacci word fractal's properties include:

• The curve ${\mathcal {F_{n}}}$ contains $F_{n}$ segments, $F_{n-1}$ right angles and $F_{n-2}$ 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 $1+{\sqrt {2}}$. 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: $F_{3n+3}-1$).
• The curve encloses an infinity of square structures of decreasing sizes in a ratio $1+{\sqrt {2}}$ (see figure). The number of those square structures is a Fibonacci number.
• The curve ${\mathcal {F}}_{n}$can also be constructed in different ways (see gallery below):
  • Iterated function system of 4 and 1 homothety of ratio $1/(1+{\sqrt {2}})$ and $1/(1+{\sqrt {2}})^{2}$
  • By joining together the curves ${\mathcal {F}}_{n-1}$ and ${\mathcal {F}}_{n-2}$
  • 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 $3\,{\frac {\log \varphi }{\log(1+{\sqrt {2}})}}\approx 1.6379$, with $\varphi ={\frac {1+{\sqrt {5}}}{2}}$ the golden ratio.
• Generalizing to an angle $\alpha$ between 0 and $\pi /2$, its Hausdorff dimension is $3\,{\frac {\log \varphi }{\log(1+a+{\sqrt {(1+a)^{2}+1}})}}$, with $a=\cos \alpha$.
• The Hausdorff dimension of its frontier is ${\frac {\log 3}{{\log(1+{\sqrt {2}}})}}\approx 1.2465$.
• 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... (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.

Gallery

• 
  Curve after $\textstyle {F_{23}}$ 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 Fibonacci tile

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

The juxtaposition of four $F_{3k}$ 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 in a square of side 1, then its area tends to $2-{\sqrt {2}}=0.5857$.

Perfect tiling by the Fibonacci snowflake

Fibonacci snowflake

Fibonacci snowflakes for i = 2 for n = 1 through 4: $\sideset {}{_{1}^{\left[2\right]}\quad }\prod$, $\sideset {}{_{2}^{\left[2\right]}\quad }\prod$, $\sideset {}{_{3}^{\left[2\right]}\quad }\prod$, $\sideset {}{_{4}^{\left[2\right]}\quad }\prod$

The Fibonacci snowflake is a Fibonacci tile defined by:

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

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

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

See also

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

External links

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