The idea of "recursive self-similarity" was originally developed by the philosopher Leibniz and he even worked out many of the details. The Dürer's Pentagon largely resembled the Sierpinski carpet, but based on pentagons instead of squares. In 1525, the German Artist Albrecht Dürer published The Painter's Manual, in which one section is on "Tile Patterns formed by Pentagons". ![]() Ethnomathematics like Ron Eglash's African Fractals ( ISBN 0-8135-2613-2) describes pervasive fractal geometry in indigenous African craft work. Objects that are now described as fractals were discovered and described centuries ago. For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves." The length of the Koch snowflake's boundary is therefore infinite, while its area remains finite. Each time new triangles are added (an iteration), the perimeter of this shape grows by a factor of 4/3 and thus diverges to infinity with the number of iterations. However, not all self-similar objects are fractals - for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.Ī Koch snowflake is the limit of an infinite construction that starts with a triangle and recursively replaces each line segment with a series of four line segments that form a triangular "bump". Obvious examples include clouds, mountain ranges and lightning bolts. has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve)ĭue to them appearing similar at all levels of magnification, fractals are often considered to be 'infinitely complex'. ![]() ![]()
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