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

Definition

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 ${\displaystyle F_{n))$ (the n^th Fibonacci number) is associated a curve ${\displaystyle {\mathcal {F))_{n))$ made of ${\displaystyle F_{n))$ 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:^[2][3]

• The curve ${\displaystyle {\mathcal {F_{n))))$ 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 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 ${\displaystyle 1+{\sqrt {2))}$ (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 of octagons
• The Hausdorff dimension of the Fibonacci word fractal is ${\displaystyle 3\,{\frac {\log \varphi }{\log(1+{\sqrt {2)))))\approx 1.6379}$, with ${\displaystyle \varphi ={\frac {1+{\sqrt {5))}{2))}$ the golden ratio.
• Generalizing to an angle ${\displaystyle \alpha }$ between 0 and ${\displaystyle \pi /2}$, its Hausdorff dimension is ${\displaystyle 3\,{\frac {\log \varphi }{\log(1+a+{\sqrt {(1+a)^{2}+1)))))}$, with ${\displaystyle a=\cos \alpha }$.
• The Hausdorff dimension of its frontier is ${\displaystyle {\frac {\log 3}((\log(1+{\sqrt {2))})))\approx 1.2465}$.
• 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

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 in a square of side 1, then its area tends to ${\displaystyle 2-{\sqrt {2))=0.5857}$.

Perfect tiling by the Fibonacci snowflake

Fibonacci snowflake

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

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

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

with ${\displaystyle q_{0}=\epsilon }$ 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.
• its perimeter at order n equals ${\displaystyle 4F(3n+1)}$, where ${\displaystyle F(n)}$ is the n^th Fibonacci number.
• its area at order n follows the successive indexes of odd row of the Pell sequence (defined by ${\displaystyle P(n)=2P(n-1)+P(n-2)}$).