Elegant Mathematics (03:36)
Arthur C. Clarke introduces the Mandelbrot set, or M-set, as one of the most beautiful and remarkable discoveries in the history of mathematics and one that will change the way we view the universe.
Exploring the Mandelbrot Set (03:45)
No matter how much the Mandelbrot set is magnified, infinite patterns emerge. Although the math is simple, the age of computers allows the multiple operations needed for the set's discovery.
Mandelbrot's Discovery (02:52)
Benoit Mandelbrot relates the process of his 1980 discovery of the M-set, which is based on the mathematical sets derived by Gaston Julia in 1917.
Mandelbrot Equation (03:17)
Simple formulas often lead to complicated results, like Einstein's relativity equation. The letters in the M-set equation stand for numbers that flow in two directions, one into infinity and the other into zero.
Colors in Fractals (02:58)
The colors do not represent movement but show the different areas of calculation, just as contour maps show varying elevation, and display the consistency of patterns within the complexity.
Birth of Fractal Geometry (03:58)
Mandelbrot coins the word "fractal" to describe the bifurcation of the pattern that has detail on all scales of magnification. Fractal geometry is born as an extension of classical geometry.
Nature Deals in Fractals (02:40)
Clarke generates a fractal on his computer to illustrate that nature deals in fractals. The shape forms from the repetition of a few instructions just like a butterfly’s wing is programmed in its DNA.
Seeing Fractals Everywhere (04:26)
The organic but patterned structure of fractals is similar to patterns in nature. Fractal geometry changes the kind of patterns we can look for in nature and provides a new way of describing data.
Fractal Geometry in Space (04:04)
Observing space through a telescope reveals fractal-like structures. Stephen Hawking discusses the limit of the universe in terms of the Planck length but, like the M-set, it may go on forever.
Practical Value of Fractals (04:02)
The first application of fractal geometry is a better description of the physical world that is available to all scientists. This new language will translate into practical use for centuries to come.
Barnsley's Collage Theorem (04:06)
In 1991 Michael Barnsley develops a fractal image compression system. A dream gives him the collage theorem, a way to tell a computer to convert digital images into fractal formulas.
Data Compression and Expansion (02:36)
Clarke gives a pixilated image detail by using Barnsley’s fractal analyzer. He discusses the use of data compression and expansion by weather and reconnaissance satellites.
Familiar Patterns (03:17)
Systems of the body, particularly the brain, echo fractals. Clarke and Mandelbrot comment on the familiarity of fractals in such things as nature, mandalas, stained glass windows, and art.
Thought Revolution (04:13)
The mind finds resonance in the Mandelbrot set, paralleling Carl Jung's theory of the collective unconscious. Fractal geometry offers new insights and new questions into the way the universe works.
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