Sierpiński 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.

Evolution of Sierpiński arrowhead curve

The Sierpinski arrowhead curve draws an equilateral triangle with triangular holes at equal intervals. It can be described with two substituting production rules: (A → B-A-B) and (B → A+B+A). A and B recur and at the bottom do the same thing — draw a line. Plus and minus (+ and -) mean turn 60 degrees either left or right. The terminating point of the Sierpinski arrowhead curve is always the same provided you recur an even number of times and you halve the length of the line at each recursion. If you recur to an odd depth (order is odd) then you end up turned 60 degrees, at a different point in the triangle.

In code, given these drawing functions: void draw_line( double distance); void turn( int angle_in_degrees); The code to draw an (approximate) Sierpinski arrowhead curve looks like this.

void sierpinski_arrowhead_curve( unsigned order, double length)
{
    // If order is even we can just draw the curve.
    if ( 0 == (order & 1) ) {
        curve( order, length, +60);
    }
    else /* order is odd */ {
        turn( +60);
        curve( order, length, -60);
    }
}
void curve( unsigned order, double length, int angle)
{
    if ( 0 == order ) {
        draw_line( length);
    } else {
        curve( order - 1, length / 2, - angle);
        turn( + angle);
        curve( order - 1, length / 2, + angle);
        turn( + angle);
        curve( order - 1, length / 2, - angle);
    }
}

Representation as Lindenmayer system[edit]

The Sierpiński arrowhead curve can be expressed by a rewrite system (L-system).

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).

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

Sierpinski arrowhead 3d stage 5.png

References[edit]

Literature[edit]

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

See also[edit]