page_title: Sierpiński curve
section_title: Arrowhead curve
hierarchical_section_title: Sierpiński curve / Arrowhead curve
caption_attribution_description: English: First six stages of fractal Sierpinski arrowhead curve
context_page_description: Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.
  Because the Sierpiński curve is space-filling, its Hausdorff dimension is.
  The Euclidean length of the ᵗʰ iteration curve is
  i.e., it grows exponentially with beyond any limit, whereas the limit for of the area enclosed by is that of the square.
context_section_description: The Sierpiński arrowhead curve is a fractal curve similar in appearance and identical in limit to the Sierpiński triangle.
  The Sierpiński 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 Sierpiński 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.
caption: Evolution of Sierpiński arrowhead curve
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