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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:

  1. If the digit is 0:
  2. * Draw a line segment then turn 90° to the left if k is even
  3. * Draw a line segment then turn 90° to the right if k is odd
  4. If the digit is 1:
  5. * Draw a line segment and stay straight

To a Fibonacci word of length (the n<sup>th</sup> Fibonacci number) is associated a curve made of segments. The curve displays three different aspects whether n is in the form 3k, 3k&nbsp;+&thinsp;1, or 3k&nbsp;+&nbsp;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, is present in a great number of properties listed below.
  • The number of self-similarities at level n is a Fibonacci number \&nbsp;−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.
  • The curve can also be constructed in different ways (see gallery below):
  • Iterated function system of 4 and 1 homothety of ratio and
  • By joining the curves and
  • 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 , with the golden ratio.
  • 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 sequence of&nbsp;"1"s.

Gallery

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 in a square of side 1, then its area tends to .

Fibonacci snowflake

The Fibonacci snowflake is a Fibonacci tile defined by:

  • if
  • otherwise.

with and , "turn left" and "turn right", and .

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 , where is the n<sup>th</sup> Fibonacci number.
  • its area at order n follows the successive indexes of odd row of the Pell sequence (defined by ).

See also

References

External links