 # FRACTALS

Understanding the Fibonacci Sequence and Golden Ratio. The Fibonacci Sequence The Fibonacci sequence is possibly the most simple recurrence relation occurring in nature. It is 0,1,1,2,3,5,8,13,21,34,55,89, 144… each number equals the sum of the two numbers before it, and the difference of the two numbers succeeding it. It is an infinite sequence which goes on forever as it develops. The Golden Ratio/Divine Ratio or Golden Mean - The quotient of any Fibonacci number and it’s predecessor approaches Phi, represented as ϕ (1.618), the Golden ratio. This iteration can continue both ways, infinitely. The Golden Ratio can be seen from a Chambered Nautilus to a Spiraling Galaxy The Golden Ratio can be applied to any number of geometric forms including circles, triangles, pyramids, prisms, and polygons. Sunflowers have a Golden Spiral seed arrangement. If you graph any number system, eventually patterns appear.

Our universe and the numbers not only go on infinitely linear, but even it’s short segments have infinite points. Image source Phi Golden Ratio. Nature, The Golden Ratio and Fibonacci Numbers. Plants can grow new cells in spirals, such as the pattern of seeds in this beautiful sunflower. The spiral happens naturally because each new cell is formed after a turn. "New cell, then turn, then another cell, then turn, ...

" How Far to Turn? So, if you were a plant, how much of a turn would you have in between new cells? Why not try to find the best value for yourself? Try different values, like 0.75, 0.9, 3.1416, 0.62, etc. Remember, you are trying to make a pattern with no gaps from start to end: (By the way, it doesn't matter about the whole number part, like 1. or 5. because they are full revolutions that point us back in the same direction.) What Did You Get? If you got something that ends like 0.618 (or 0.382, which is 1 − 0.618) then "Congratulations, you are a successful member of the plant kingdom! " Why? Any number that is a simple fraction (example: 0.75 is 3/4, and 0.95 is 19/20, etc) will, after a while, make a pattern of lines stacking up, which makes gaps.

Fibonacci Numbers Why? Golden Ratio. The Idea Behind It Have a try yourself (use the slider): Beauty This rectangle has been made using the Golden Ratio, Looks like a typical frame for a painting, doesn't it? Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape. Do you think it is the "most pleasing rectangle"? Maybe you do or don't, that is up to you! Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece, but it is not really known if it was designed that way. The Actual Value The Golden Ratio is equal to: 1.61803398874989484820...

The digits just keep on going, with no pattern. Calculating It You can calculate it yourself by starting with any number and following these steps: A) divide 1 by your number (=1/number) B) add 1 C) that is your new number, start again at A With a calculator, just keep pressing "1/x", "+", "1", "=", around and around. It is getting closer and closer! Drawing It Here is one way to draw a rectangle with the Golden Ratio: Pentagram. Fibonacci Sequence. The Fibonacci Sequence is the series of numbers: The next number is found by adding up the two numbers before it. The 2 is found by adding the two numbers before it (1+1) Similarly, the 3 is found by adding the two numbers before it (1+2), And the 5 is (2+3), and so on! Example: the next number in the sequence above is 21+34 = 55 It is that simple! Here is a longer list: Can you figure out the next few numbers?

Makes A Spiral When we make squares with those widths, we get a nice spiral: Do you see how the squares fit neatly together? The Rule The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series). First, the terms are numbered from 0 onwards like this: So term number 6 is called x6 (which equals 8). So we can write the rule: The Rule is xn = xn-1 + xn-2 where: xn is term number "n" xn-1 is the previous term (n-1) xn-2 is the term before that (n-2) Example: term 9 is calculated like this: Golden Ratio And here is a surprise.

Using The Golden Ratio to Calculate Fibonacci Numbers. Fractal Foundation Online Course - Chapter 11 - FIBONACCI FRACTALS. The Golden Ratio The Fibonacci Sequence appears in many seemingly unrelated areas. In this section we'll see how the Fibonacci Sequence generates the Golden Ratio, a relationship so special it has even been called "the Divine Proportion. " Remember that the next number in the Fibonacci Sequence is just made by adding the current number in the sequence to the number before it. Let's see how the numbers in the Fibonacci Sequence compare to each other. In particular, how much bigger is any Fiboncci number than the one before it? If we call the n'th Fibonacci number Fn, then the next number in the sequence Fn+1 = Fn + Fn-1 Let's look now at the ratio of successive Fibonacci Numbers, or The ratio of the successive Fibonacci Numbers gets closer and closer to a certain value as n gets larger and larger.

What is the ratio of F11 / F10: [ ] (Use 8 decimals of precision for your answers.) What is the ratio of F12 / F11: [ ] What is the error for F11 / F10 : [ ] What is the error for F12 / F11 : [ ] Golden ratio. Line segments in the golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, or ) represents the golden ratio. The golden ratio also is called the golden mean or golden section (Latin: sectio aurea). Other names include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, and golden number. Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing (see Applications and observations below).

Calculation Therefore, Golden ratio. List of fractals by Hausdorff dimension. Deterministic fractals Random and natural fractals See also Notes and references Further reading Benoît Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Co; ISBN 0-7167-1186-9 (September 1982).Heinz-Otto Peitgen, The Science of Fractal Images, Dietmar Saupe (editor), Springer Verlag, ISBN 0-387-96608-0 (August 1988)Michael F. External links Julia set. Three-dimensional slices through the (four-dimensional) Julia set of a function on the quaternions. The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.

Formal definition Let f(z) be a complex rational function from the plane into itself, that is, , where p(z) and q(z) are complex polynomials. Then there is a finite number of open sets F1, ..., Fr, that are left invariant by f(z) and are such that: the union of the Fi's is dense in the plane andf(z) behaves in a regular and equal way on each of the sets Fi. The last statement means that the termini of the sequences of iterations generated by the points of Fi are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically.

Examples For . Koch snowflake. The first seven iterations in animation Zooming into the Koch curve The Koch snowflake (also known as the Koch star and Koch island) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire) by the Swedish mathematician Helge von Koch. Construction The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: divide the line segment into three segments of equal length.draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.remove the line segment that is the base of the triangle from step 2.

After one iteration of this process, the resulting shape is the outline of a hexagram. En.wikipedia. Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The Julia set and Mandelbrot sets can be considered as icons of fractal art. Fractal art (especially in the western world) is not drawn or painted by hand.

It is usually created indirectly with the assistance of fractal-generating software, iterating through three phases: setting parameters of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other graphics programs are used to further modify the images produced. Types A 3D Mandelbulb fractal generated using Visions of Chaos There are many different kinds of fractal images and can be subdivided into several groups. Techniques Landscapes . Artists Exhibits