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Golden Section And Fractals Nature & Astronomy Seray ARSLAN and Yoana Dineva Golden Section The ratio has a special relationship with Fibonacci Numbers(1,1,2,3,5,8,13,21…). Each number is the sum of the two numbers before it. If you take any two successive Fibonacci Numbers their ratio is very very close to Golden Section. Nature&Astronomy The golden section is a ratio which can be found in many places in nature we can’t even guess. 233 / 144 =1.61805555555555 55555555555555556 Golden Ratio:1.618 Fibonacci Numbers= 1,1,2,3,5,8,13,21,34, 55,89,144,233,377… Adolf Zeising Adolf Zeising, whose main interest is mathematics and philosophy,discovered the golden section in the arrangement of brances along the stems of plants and of veins in leaves. He extented his research to the skeleton of animals and the branchings of their veins and nerves to the proportions of chemical compounds and geometry of crystals. Golden Rectangle b/a=1.618 Golden Spiral Spirals in nature Approximate logarithmic spirals can occur in nature. It is sometimes stated that nautilus shells get wider in the pattern of a golden spiral. In truth, nautilus shells exhibit logarithmic spiral growth, but at an angle distinctly different from that of the golden spiral. This pattern allows the organism to grow without changing shape. Spirals are common features in nature; golden spirals are one special case of these. To be continued By Yoana Dineva Fractals In modern mathematics, the golden ratio occurs in the description of fractals, figures that exhibit self-similarity and play an important role in the study of chaos and dynamical systems. A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity. Roots of the idea of fractals go back to the 17th century, while mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass, Georg Cantor and Felix Hausdorff a century later in studying functions that were continuous but not differentiable; however, the term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." triangle Serpinski Koch snowflake 2nd Century Cathar Mosaic in the form of a Sierpenski Triangle Stunning and Intricate Fractal Arabesque on a ceiling panel (4’ x 4’) at Delware Jain Temple, Mt. Abu, India, 1031AD. There are several examples of fractals, which are defined as portraying exact selfsimilarity, quasi selfsimilarity, or statistical self-similarity. While fractals are a mathematical construct, they are found in nature, which has led to their inclusion in artwork. They are useful in medicine, soil mechanics, seismology, and technical analysis. The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Gottfried Leibniz considered recursive self-similarity It was not until 1872 that a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the nonintuitive property of being everywhere continuous but nowhere differentiable. The graph of such a function would be called today a fractal. In 1904, Helge von Koch, dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve. Wacław Sierpiński constructed his triangle in 1915 . In the 1960s, Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff– Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computerconstructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". In nature Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves is currently being used to determine how much carbon is contained in trees Fractal Rivers in Savannah, Georgia Patterns of Visual Math - Fractal Technology, Art & History Weierstrass function - Wikipedia, the free encyclopedia Thank you to the team 11 SOU “ Zheleznik” Stara Zagora Bulgaria Thank you for listening me