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. 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 [math]\displaystyle{ F_n }[/math] (the n^th Fibonacci number) is associated a curve [math]\displaystyle{ \mathcal{F}_n }[/math] made of [math]\displaystyle{ F_n }[/math] 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 [math]\displaystyle{ \mathcal{F_n} }[/math] contains [math]\displaystyle{ F_n }[/math] segments, [math]\displaystyle{ F_{n-1} }[/math] right angles and [math]\displaystyle{ F_{n-2} }[/math] 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 [math]\displaystyle{ 1+\sqrt{2} }[/math]. 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: [math]\displaystyle{ F_{3n+3}-1 }[/math]).
• The curve encloses an infinity of square structures of decreasing sizes in a ratio [math]\displaystyle{ 1+\sqrt{2} }[/math] (see figure). The number of those square structures is a Fibonacci number.
• The curve [math]\displaystyle{ \mathcal{F}_n }[/math]can also be constructed in different ways (see gallery below):
  • Iterated function system of 4 and 1 homothety of ratio [math]\displaystyle{ 1/(1+\sqrt2) }[/math] and [math]\displaystyle{ 1/(1+\sqrt2)^2 }[/math]
  • By joining together the curves [math]\displaystyle{ \mathcal{F}_{n-1} }[/math] and [math]\displaystyle{ \mathcal{F}_{n-2} }[/math]
  • 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 [math]\displaystyle{ 3\,\frac{\log\varphi}{\log(1+\sqrt 2)}\approx 1.6379 }[/math], with [math]\displaystyle{ \varphi = \frac{1+\sqrt{5}}{2} }[/math] the golden ratio.
• Generalizing to an angle [math]\displaystyle{ \alpha }[/math] between 0 and [math]\displaystyle{ \pi/2 }[/math], its Hausdorff dimension is [math]\displaystyle{ 3\,\frac{\log\varphi}{\log(1+a+\sqrt{(1+a)^2+1})} }[/math], with [math]\displaystyle{ a=\cos\alpha }[/math].
• The Hausdorff dimension of its frontier is [math]\displaystyle{ \frac{\log 3}{{\log(1+\sqrt 2})}\approx 1.2465 }[/math].
• 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 [math]\displaystyle{ \textstyle{F_{23}} }[/math] iterations.

• 

  Self-similarities at different scales.

• 

  Dimensions.

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  Construction by juxtaposition (1)

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  Construction by juxtaposition (2)

• 

  Order 18, with some sub-rectangles colored.

• 
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  Construction by iterated suppression of square patterns.

• 

  Construction by iterated octagons.

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  Construction by iterated collection of 8 square patterns around each square pattern.

• 

  With a 60° angle.

• 

  Inversion of "0" and "1".

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  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 [math]\displaystyle{ F_{3k} }[/math] 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 [math]\displaystyle{ 2-\sqrt{2} = 0.5857 }[/math].

Perfect tiling by the Fibonacci snowflake

Fibonacci snowflake

Fibonacci snowflakes for i = 2 for n = 1 through 4: [math]\displaystyle{ \sideset{}{_1^\left [ 2 \right ] \quad}\prod }[/math], [math]\displaystyle{ \sideset{}{_2^\left [ 2 \right ] \quad}\prod }[/math], [math]\displaystyle{ \sideset{}{_3^\left [ 2 \right ] \quad}\prod }[/math], [math]\displaystyle{ \sideset{}{_4^\left [ 2 \right ] \quad}\prod }[/math]

The Fibonacci snowflake is a Fibonacci tile defined by:

• [math]\displaystyle{ q_n = q_{n-1}q_{n-2} }[/math] if [math]\displaystyle{ n \equiv 2 \pmod 3 }[/math]
• [math]\displaystyle{ q_n = q_{n-1}\overline{q}_{n-2} }[/math] otherwise.

with [math]\displaystyle{ q_0=\epsilon }[/math] and [math]\displaystyle{ q_1=R }[/math], [math]\displaystyle{ L = }[/math] "turn left" and [math]\displaystyle{ R = }[/math] "turn right", and [math]\displaystyle{ \overline{R} = L }[/math].

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

See also

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

External links

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