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 the Odd–Even Drawing rule to the Fibonacci word 0100101001001...:

For each digit at position ''k'':
# 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
# 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 $\mathcal_n$ made of $F_n$ segments. The curve displays three different aspects whether ''n'' is in the form 3''k'', 3''k'' + 1, or 3''k'' + 2.

Properties

Some of the Fibonacci word fractal's properties include:
* The curve $\mathcal$ contains $F_n$ segments, $F_$ right angles and $F_$ 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 $1+\sqrt$. 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: $F_-1$).
* The curve encloses an infinity of square structures of decreasing sizes in a ratio $1+\sqrt$ (see figure). The number of those square structures is a Fibonacci number.
* The curve $\mathcal_n$can also be constructed in different ways (see gallery below):
** Iterated function system of 4 and 1 homothety of ratio $1/(1+\sqrt2)$ and $1/(1+\sqrt2)^2$
** By joining together the curves $\mathcal_$ and $\mathcal_$
** 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 $3\,\frac\approx 1.6379$, with $\varphi = \frac$ the golden ratio.
* Generalizing to an angle $\alpha$ between 0 and $\pi/2$, its Hausdorff dimension is $3\,\frac$, with $a=\cos\alpha$.
* The Hausdorff dimension of its frontier is $\frac\approx 1.2465$.
* 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... . 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

File:Fibonacci fractal F23 steps.png, Curve after $\textstyle$ iterations.
File:Fibonacci fractal self-similarities.png, Self-similarities at different scales.
File:FWF Dimensions.png, Dimensions.
File:Fibonacci fractal F21 & F20.png, Construction by juxtaposition (1)
File:Fibonacci Fractal F22 & F21.png, Construction by juxtaposition (2)
File:Fibonacci word fractal, order 18.svg, Order 18, with some sub-rectangles colored.
File:Fibonacci word fractalX.jpg
File:FWF alternative construction.png, Construction by iterated suppression of square patterns.
File:FWF octogons.png, Construction by iterated octagons.
File:Fibonacci word gaskett.png, Construction by iterated collection of 8 square patterns around each square pattern.
File:Fibo 60deg F18.png, With a 60° angle.
File:Inverted Fibonacci fractal.png, Inversion of "0" and "1".
File:Fibonacci word fractal variants.png, Variants generated from the dense Fibonacci word.
File:Fibonacci word fractal compact variant.jpg, The "compact variant"
File:Fibonacci word fractal svastika variant.jpg, The "svastika variant"
File:Fibonacci word fractal diagonal variant.jpg, The "diagonal variant"
File:FWF PI8.png, The "π/8 variant"
File:FWF Samuel Monnier.jpg, Artist creation (Samuel Monnier).

The Fibonacci tile

The juxtaposition of four $F_$ 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 $2-\sqrt = 0.5857$.

Fibonacci snowflake

The Fibonacci snowflake is a Fibonacci tile defined by:Blondin-Massé, Alexandre; Brlek, Srečko; Garon, Ariane; and Labbé, Sébastien (2009).
Christoffel and Fibonacci tiles
, ''Lecture Notes in Computer Science: Discrete Geometry for Computer Imagery'', p.67-8. Springer. .
* $q_n = q_q_$ if $n \equiv 2 \pmod 3$
* $q_n = q_\overline_$ otherwise.
with $q_0=\epsilon$ and $q_1=R$, $L =$ "turn left" and $R =$ "turn right", and $\overline = L$.

Several remarkable properties:A. Blondin-Massé, S. Labbé, S. Brlek, M. Mendès-France (2011).
Fibonacci snowflakes
.
* 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

References

External links

Generate a Fibonacci word fractal
, ''OnlineMathTools.com''.

{{Fractals

Fractals
Fractal curves