sequence. If the Thue–Morse sequence members are used in order to select program states: If , move ahead by one unit, If , rotate counterclockwise by an angle of , the resulting curve converges to the Koch snowflake. Representation as Lindenmayer system The Koch curve can be expressed by the following rewrite system (Lindenmayer system): Alphabet : F Constants : +, − Axiom : F Production rules: F → F+F--F+F Here, F means "draw forward", - means "turn right 60°", and + means "turn left 60°". To create the Koch snowflake, one would use F--F--F (an equilateral triangle) as the axiom. Variants of the Koch curve Following von Koch's concept, several variants of the Koch curve were designed, considering right angles (quadratic), other angles (Cesàro), circles and polyhedra and their extensions to higher dimensions (Sphereflake and Kochcube, respectively) Squares can be used to generate similar fractal curves. Starting with a unit square and adding to each side at each iteration a square with dimension one third of the squares in the previous iteration, it can be shown that both the length of the perimeter and the total area are determined by geometric progressions. The progression for the area converges to while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve. The resulting area fills a square with the same center as the original, but twice the area, and rotated by radians, the perimeter touching but never overlapping itself. The total area covered at the th iteration is: while the total length of the perimeter is: which approaches infinity as increases. See also List of fractals by Hausdorff dimension Gabriel's Horn (infinite surface area but encloses a finite volume) Gosper curve (also known as the Peano–Gosper curve or flowsnake) Osgood curve Self-similarity Teragon Weierstrass function Coastline paradox References Further reading External links The Koch Curve poem by Bernt Wahl, Wahl.org. Retrieved 23 September 2019. Application of the Koch curve to an antenna A WebGL animation showing the construction of the Koch surface, tchaumeny.github.io. Retrieved 23 September 2019. De Rham curves L-systems