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 nth 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

• Curve after ${\displaystyle \textstyle {F_{23))}$ 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 ${\displaystyle F_{3k))$ 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 nth Fibonacci number.
• its area at order n follows the successive indexes of odd row of the Pell sequence (defined by ${\displaystyle P(n)=2P(n-1)+P(n-2)}$).