Сиерпински қисықтары және кеңістікті толтырушы иірімдер

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 (in the limit ) 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 (in Euclidean metric).

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. An alternate constriction is given in the article on the de Rham curve: one uses the same technique as the de Rham curves, but instead of using a binary (base 2) expansion, one uses a ternary (base 3) expansion.