The first one is related to the circle limits of M.C.Escher ( van Dusen et. Below, two special tilings with differently sized Koch tiles are shown, one with small and one with a large snowflake in the center. ![]() Examples can be found on the Koch Snowflake Wikipedia page. Periodic tilings of the plane may be achieved by using two or more Koch tiles. Note, that the inside decorated hexagon is identical to a next generation outside decorated triangle. To the right the curve is decorating a hexagon. To the left the curve is decorating the edge of a triangle either at the outside or at the inside. Below four snowflake tiles are shown using the same edge. Koch curves can be connected to form so called Koch Snowflakes or Koch Islands. Inside you see white Koch snowflakes of different sizes. The grey area surrounding the hexagon is the outside of an inverted Koch snowflake. But it is more interesting to look at the space in between and outside. The black pixels approximate the fractal hexagon. K3 Tile shapes for edge sequence, rule b) Fractal hexagon: An fractal storm of Koch’s snowflakes. ![]() K2 Edge shapes of a number of generations of the 2×2 rhomb tiles with edge sequence (1/2, -1/2) and n=3 (or equivalently, edge sequence (1, -1)), and type b) substitution rule.īy applying substitution rule b) repeatedly, the circumference of our rhomb tiles also gets a fractal appearance (Fig. ![]() The edges for even n are identical to the Koch curves. At each new iteration n, all the line segments are replaced by a dent or a dimple with a connecting angle of 120 degrees. The main difference is that our building instruction is different. ![]() The edges of our type b) rhomb tilings are reminiscent of these Koch curves as shown in Fig. K1) Fig.K1 Four iterates of the Koch curve. The mathematician Niels Fabian Helge von Koch (1870-1924) invented a famous curve by iteratively breaking up a line segment into three line segments of equal length and by replacing the central one by an equilateral triangle and removing the base line of the triangle.
0 Comments
Leave a Reply. |