# Fibonacci word fractal

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

## Definition

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

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 $F_{n}$  (the nth 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

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 by 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 series of "1".

## The Fibonacci tile

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

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