The Fibonacci word fractal is a
fractal curve
A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectif ...
defined on the plane from the
Fibonacci word
A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
It is a para ...
.
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'' :
# Draw a segment forward
# 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
(the ''n''
th Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
) is associated a curve
made of
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
, contains
segments,
right angles and
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
. This number, also called the
silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice t ...
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 :
).
* The curve encloses an infinity of square structures of decreasing sizes in a ratio
. (see figure) The number of those square structures is a
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
.
* The curve
can also be constructed by different ways (see gallery below):
**
Iterated function system of 4 and 1 homothety of ratio
and
** By joining together the curves
and
**
Lindenmayer system
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into so ...
** By an iterated construction of 8 square patterns around each square pattern.
** By an iterated construction of
octagons
* The
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
of the Fibonacci word fractal is
, with
, the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
.
* Generalizing to an angle
between 0 and
, its Hausdorff dimension is
, with
.
* The Hausdorff dimension of its frontier is
.
* 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 series of "1".
Gallery
File:Fibonacci fractal F23 steps.png, Curve after 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 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 "pi/8 variant"
File:FWF Samuel Monnier.jpg, Artist creation (Samuel Monnier).
The Fibonacci tile
The juxtaposition of four
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 a square of side 1, then its area tends to
.
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. .
*
if
*
otherwise.
with
and
,
"turn left" et
"turn right", and
,
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
.
is the n
th Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
.
* its area, at order ''n'', follows the successive indexes of odd row of the
Pell sequence
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , and ...
(defined by
).
See also
*
Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
*
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
*
Fibonacci word
A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
It is a para ...
*
List of fractals by Hausdorff dimension
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illus ...
References
External links
Generate a Fibonacci word fractal, ''OnlineMathTools.com''.
{{Fractals
Fractals
Fractal curves