World of Fractals (02:40)
Benot Mandelbrot explains fractal geometry as the language that describes complex shapes in nature, such as clouds, that are not adequately addressed by traditional geometry.
Self-similarity is made obvious in the head of a cauliflower by breaking off a piece and comparing it to the whole. Fractal geometry is the science of the structure inherent even in the unpredictable.
Discovery of the Lorenz Attractor (04:09)
Lorenz’s experiment in 1956 to forecast weather reveals that small deviations make any long term prediction inaccurate which throws the current theory of predicting dynamic systems into question.
Chaos Experiment (01:36)
An experiment involving the movement of an iron ball pendulum, the force of gravity, and three differently colored magnets demonstrates the interplay between chaos and fractals.
Computer Simulation Tracks Pattern (02:40)
Computer simulation tracks the pattern created by the pendulum from different starting points to its resting point, creating a systematic overview and a highly complex structure.
Chaos and the Cantor Set (03:32)
The structure of the pattern changes when any of the parameters change, such as gravitational force. The fact that the third color lies between the meeting of any other two colors is observed.
Fractal Dimension (04:45)
Mandelbrot poses how to measure the convoluted coastline of Britain resulting with a ratio he calls fractal dimension.
Infinity of Fractal Structure (05:05)
Studying structures that are self-similar in the strict scientific sense reveals an infinite number of surfaces as each structure is viewed in greater detail.
Geometry Versus Algebra (04:52)
Mandelbrot and Lorenz candidly reflect on the differing views between mathematicians and the influence of important mentors in their lives.
Julia Sets (04:34)
Gaston Julia's 1918 treatise is only expanded upon by Mandelbrot's conjectures and simulations in the late 1970s. The Julia set model that stirs excitement operates with two competing forces.
Mandelbrot Set (04:48)
The dramatic graphic demonstration of the Mandelbrot set illustrates the principles of fractal geometry along with the first images produced by Mandelbrot himself.
Advancing Knowledge of Fractals (03:39)
Progress in the last ten years shows the similarity of the Mandelbrot and Julia sets on a small scale but current limits in our understanding of the Mandelbrot set are met where they deviate.
Science and Art Connection (03:57)
Both science and art search for symmetry with the qualities of balance, harmony, and beauty. In nature, one always sees something new but is comforted by similarity.
Mandelbrot Set Appears as a Constant (05:58)
The former notion that a complex form requires a complex formula is repudiated by the Mandelbrot set which produces an extremely complex form from a simple construction rule.
Voyage of Discovery (03:12)
The beauty of fractal geometry is playing a role in making more people interested in mathematics.
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