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Sierpinski arrowhead curve

Sierpinski arrowhead curve

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The Sierpiński arrowhead curve is a fractal curve similar in appearance and identical in limit to the Sierpiński triangle
Sierpinski triangle
The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...

.


Representation as Lindenmayer system


The Sierpiński arrowhead curve can be expressed by a rewrite system
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. What is considered are rewriting systems...

 (L-system
L-system
An L-system or Lindenmayer system is a parallel rewriting system, namely a variant of a formal grammar, most famously used to model the growth processes of plant development, but also able to model the morphology of a variety of organisms...

).
Alphabet: X, Y
Constants: F, +, −
Axiom: XF
Production rules:
X → YF + XF + Y
Y → XF − YF − X


Here, F means “draw forward”, + means “turn left 60°”, and − means “turn right 60°” (see turtle graphics
Turtle graphics
Turtle graphics is a term in computer graphics for a method of programming vector graphics using a relative cursor upon a Cartesian plane...

).

Like many two-dimensional fractal curves, the Sierpiński arrowhead curve can be extended to three dimensions:


Literature

  • Peitgen et al., Chaos and Fractals, Springer-Verlag, 1992.
  • Roger T. Stevens, Fractal Programming in C, M&T Books, 1989.

See also

  • List of fractals by Hausdorff dimension
  • Sierpiński curve
    Sierpinski curve
    Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \rightarrow \infty completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling...