#2-2 Chaos theory, infinity, randomness, fractals and african people

A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.
A fractal often has the following features:
•It has a fine structure at arbitrarily small scales.
•It is too irregular to be easily described in traditional Euclidean geometric language.
•It is self-similar (at least approximately or stochastically).
•It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
•It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. 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; for instance, it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, as it is possible to zoom into a region of the image that does not exhibit any fractal properties.
The mathematics behind fractals began to take shape in the 17th century when mathematician and philosopher Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).
It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch curve. (The image at right is three Koch curves put together to form what is commonly called the Koch snowflake.) In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. In 1918, Bertrand Russell recognised a "supreme beauty" within the emerging mathematics of fractals. The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Lévy C curve. Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals.
Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered.
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 computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".

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[bongmannz]

Wow if only Africans had the resources to do things we could be living far beyond that of what we are now.

[echoinfinity100]

"...they have the heritage that's about mathematics, that it's not about singing and dancing..." I think that it is very crucial for humankind as whole to try to realize and understand each other. We have to learn about different cultures and share knowledge. Today this is not happening fast enough and sometimes not at all. Instead of teaching people around the world about everything we know and learn from them everything they know people allow that those with power use and harm those without.

[eggfooyoung24]

Great 2 parter

Got a thumb's up from me

[cavoj]

number 1 = absolute number .......proof

[Helden6621]

could you imagin a planet without power hungry capitalisim :P

[RoyManter]

@bongmannz they had it. Now we, sorry, you have it.

[ausaurwest]

@ 5:20-5:24 "Every digital circuit in the world started in Africa."

This is a profound statement worthy of the utmost consideration.

[ladykillertime]

@bongmannz Have you ever asked why they don't have the resources?

[ladykillertime]

@jmaioran 4:53-5:17 wouldn't be profound if it weren't for 3:23-3:56....3:23-3:56 if profound all on it's own.

[GagnezTV]

Some people just don't get ancient knowledge and technology... Singing and dancing also uses mathematics... Thanks for sharing this video though...

-G