FractalsFractals

Siobhán RaffertySiobhán Rafferty

What Are Fractals?What Are Fractals?

a set of points whose fractal dimension exceeds a set of points whose fractal dimension exceeds its topological dimension its topological dimension

A “self-similar” geometrical shape that includes A “self-similar” geometrical shape that includes the same pattern, scaled down and rotated and the same pattern, scaled down and rotated and repeated.repeated.

The Koch CurveThe Koch Curve The Koch Curve is a The Koch Curve is a

famous example of a famous example of a Fractal published by Niels Fractal published by Niels Fabien Helge von Koch in Fabien Helge von Koch in 19061906

Stage 0 is a straight line Stage 0 is a straight line segmentsegment

Stages 1 - infinity are Stages 1 - infinity are produced by repeating produced by repeating stage1 along every line stage1 along every line segment of the previous segment of the previous stage.stage.

The Koch SnowflakeThe Koch Snowflake The perimeter of each The perimeter of each

stage is 1.33 x the stage is 1.33 x the perimeter of the perimeter of the pervious stage.pervious stage.

When we repeat the When we repeat the stages to infinity the stages to infinity the perimeter is infinite.perimeter is infinite.

Most geometrical shapes Most geometrical shapes have an Area –Perimeter have an Area –Perimeter Relationship. This does Relationship. This does not hold with Fractalsnot hold with Fractals

An infinite perimeter An infinite perimeter encloses a finite area.encloses a finite area.

DimensionsDimensions

Fractals have non-integer dimensions that can Fractals have non-integer dimensions that can be calculated using logarithms.be calculated using logarithms.

If the length of the edges on a cube is If the length of the edges on a cube is multiplied by 2, 8 of the old cubes would fit multiplied by 2, 8 of the old cubes would fit into the new curve. into the new curve.

Log8/Log2 = 3, a cube is 3 dimensional.Log8/Log2 = 3, a cube is 3 dimensional. Similarly for a fractal of size P, made of Similarly for a fractal of size P, made of

smaller units (size p), the number of units (N) smaller units (size p), the number of units (N) that fits into the larger object is equal to the that fits into the larger object is equal to the size ratio (P/p) raised to the power of dsize ratio (P/p) raised to the power of d

D = Log(N)/Log(P/p)D = Log(N)/Log(P/p)

Dimensions, an example.Dimensions, an example. Each line in stage 1 is Each line in stage 1 is

made up of lines 3cm made up of lines 3cm long (P=3)long (P=3)

There are 12 line There are 12 line segmentssegments

Stage 2 has lines of Stage 2 has lines of length 1cm (p=1)length 1cm (p=1)

It has 48 line segments It has 48 line segments (N = 48/12 = 4)(N = 48/12 = 4)

d = log 4/ log 3 = d = log 4/ log 3 = 1.26191.2619

Benoit MandelbrotBenoit Mandelbrot Born in 1924 and currently Born in 1924 and currently

a mathematics professor at a mathematics professor at Yale UniversityYale University

““The Mandelbrot Set” The Mandelbrot Set” xx²+c, where c is a complex ²+c, where c is a complex

number.number. XX1 1 = 0² + c= 0² + c XX22 = c² + c = c² + c XX33 = (c² + c)² + c = (c² + c)² + c

The Mandelbrot Set The Mandelbrot Set If it takes very few If it takes very few

iterations for the iterations for the iterations to become very iterations to become very large and tend to infinity large and tend to infinity then the value “c” is then the value “c” is marked in marked in redred..

Numbers are marked on Numbers are marked on the set following the light the set following the light spectrumspectrum orangeorange, , yellowyellow, , greengreen, , blueblue, , indigoindigo, , violet violet in order of those tending in order of those tending to infinity at different to infinity at different rates.rates.

The values shown in The values shown in black do not escape to black do not escape to infinityinfinity

The result is a fractal!The result is a fractal!