Chaos

Chaos : 

Chaos, a new branch of mathematics, began in the 1960s It explains things like the shape of a snowflake Jupiter’s great red spot heartbeats brainwaves disease epidemics traffic jams the pattern of smoke rising water falling from a tap A lot of the background work for chaos was done by people trying to predict the weather Chaos

Edward Lorenz (1917-2008) : 

Edward Lorenz (1917-2008) Lorenz was a meteorologist His work on weather forecasting led him accidentally to discover the maths of chaos theory In 1972 he published a paper “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas?” The chaos theory, quantum physics and relativity are the three big scientific discoveries of the 20th century

Lorenz studied the weather : 

Lorenz studied the weather In 1961, Edward Lorenz put data into programs to predict the weather, then he put the same data into the same programs but rounded a figure to the 4th decimal place. He found the weather would be almost the same for two to three days but then changed dramatically, caused by the effects of chaos. Small changes in the initial conditions of a system can lead to huge differences in the outcome, (the weather a few days later in this case).

Lorenz’s experiment : 

The difference between the starting values of these curves is only 0·000127 Lorenz’s experiment

Lorenz immediately knew that the future of long-range weather forecasting was doomed. : 

Lorenz immediately knew that the future of long-range weather forecasting was doomed.

“Butterfly Effect” : 

“Butterfly Effect” Edward Lorenz coined the phrase “butterfly effect” to describe how a slight change in initial conditions can lead to drastically different outcomes. “The flap of a butterfly’s wings in South America could be responsible for a tornado in Texas.” He chose a butterfly because a 3 dimensional image of the trajectory mapped out by his equations looks like a butterfly

Definition of fractals : 

Mandelbrot’s definition A fractal set is a metric space for which the Hausdorff-Besicovitch dimension D is greater than the topological dimension DT Definition of fractals

Definition of fractals : 

Fractals for the layperson Geometrical shapes that are irregular all over yet are “self-similar” in that the shapes look the same from all distances Definition of fractals

Non fractal : 

Non fractal

Fractal : 

Fractal

Non fractal : 

Non fractal

Fractal : 

Fractal

Slide 13: 

Fractal

Slide 14: 

Fractal

Features of fractals : 

Features of fractals Self similarity (part is similar to the whole) Fractional dimension The result of infinite iterations Too irregular to be described in traditional geometric language

Fractal : 

Fractal

Fractal : 

Fractal

Self Similarity : 

Self Similarity Ferns Sierpenski gasket Koch snowflake Mandelbrot Set Julia set

Slide 19: 

Mandelbrot Set

Julia Set : 

Julia Set

Fractals in nature : 

Fractals in nature

Fractals in nature : 

Fractals in nature

Sierpinski Gasket : 

Sierpinski Gasket

Sierpinski Gasket : 

Sierpinski Gasket

Koch Snowflake : 

Koch Snowflake Helge von Koch (1870 – 1924)

Slide 26: 

Draw a simple equilateral triangle Koch Snowflake On each of the three sides place another equilateral triangle exactly one third and in the middle of the side

Slide 27: 

Repeat the process Koch Snowflake

Slide 28: 

Derive a general formula for the perimeter of the n th curve in this sequence, Pn . P0 = L Koch Snowflake

Slide 29: 

Koch Snowflake P1

Slide 30: 

Koch Snowflake P2

Slide 31: 

P0 = L Koch Snowflake

Slide 32: 

The perimeter of the curve is infinite. Koch Snowflake

Slide 33: 

The area An of the nth curve is finite. This can be seen by constructing the circumscribed circle about the original triangle as shown. Koch Snowflake

Slide 34: 

P0 = L The perimeter of the curve is infinite but the area is finite!! Koch Snowflake

Do you want to end show? : 

Do you want to end show?