Sierpiński curve

Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n → ∞ {displaystyle n o infty } completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve. Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n → ∞ {displaystyle n o infty } 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 n → ∞ {displaystyle n o infty } ) is 2 {displaystyle 2} . The Euclidean length of the n {displaystyle n} th iteration curve S n {displaystyle S_{n}} is i.e., it grows exponentially with n {displaystyle n} beyond any limit, whereas the limit for n → ∞ {displaystyle n o infty } of the area enclosed by S n {displaystyle S_{n}} is 5 / 12 {displaystyle 5/12,} that of the square (in Euclidean metric). The Sierpiński curve is useful in several practical applications because it is more symmetrical than other commonly studied space-filling curves. For example, it has been used as a basis for the rapid construction of an approximate solution to the Travelling Salesman Problem (which asks for the shortest sequence of a given set of points): The heuristic is simply to visit the points in the same sequence as they appear on the Sierpiński curve. To do this requires two steps: First compute an inverse image of each point to be visited; then sort the values. This idea has been used to build routing systems for commercial vehicles based only on Rolodex card files.