1

Dr. Scott Schaefer

Fractals and Iterated Affine Transformations

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What are Fractals?

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What are Fractals?

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What are Fractals?

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What are Fractals?

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What are Fractals?

Recursion made visible

A self-similar shape created from a set of contractive transformations

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Contractive Transformations

A transformation F(X) is contractive if, for all compact sets

Transformations on sets? Distance between sets?

),())(),(( 2121 XXDXFXFD HH ,21 XX

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Transformations on Sets

Given a transformation F(x),

In other words, F(X) means apply the transformation to each point in the set X

}|)({)( XxxFXF

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Hausdorff Distance

1X 2X

?),( 21 XXDH

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Hausdorff Distance

1X 2X

?),( 21 XXDH

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Hausdorff Distance

21 XX

0),( 21 XXDH

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Hausdorff Distance

1X 2X

),(),( 1221 XXDXXD HH

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Hausdorff Distance

1X 2X

|)|min(max 212211

21xxd

XxXxXX

1x

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Hausdorff Distance

1X 2X

|)|min(max 212211

21xxd

XxXxXX

1x

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Hausdorff Distance

1X 2X

|)|min(max 212211

21xxd

XxXxXX

1x

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Hausdorff Distance

1X 2X

|)|min(max 212211

21xxd

XxXxXX

1x

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Hausdorff Distance

1X 2X

|)|min(max 212211

21xxd

XxXxXX

1x

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Hausdorff Distance

1X 2X

),max(),(122121 XXXXH ddXXD

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Contractive Transformations

A transformation F(X) is contractive if, for all compact sets

),())(),(( 2121 XXDXFXFD HH ,21 XX

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Iterated Affine Transformations

Special class of fractals where each transformation is an affine transformation

,...})(,)(,)({ 332211 xMxFxMxFxMxF

100232221

131211

mmm

mmm

M i

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Rendering Fractals

Given starting set X0

Attractor is

j

iji XFX )(1

X

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Rendering Fractals

Given starting set X0

Attractor is

j

iji XFX )(1

X

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Rendering Fractals

Given starting set X0

Attractor is

j

iji XFX )(1

X

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Rendering Fractals

Given starting set X0

Attractor is

j

iji XFX )(1

X

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Rendering Fractals

Given starting set X0

Attractor is

j

iji XFX )(1

X

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Contractive Transformations

A transformation F(X) is contractive if, for all compact sets

A set of transformations has a unique attractor if all transformations are contractiveThat attractor is independent of the starting

shape!!!

),())(),(( 2121 XXDXFXFD HH ,21 XX

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

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Rendering Fractals

1T3T

2T

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Rendering Fractals

1T3T

2T

1T2T 3T

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Rendering Fractals

1T3T

2T

1T2T 3T

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Rendering Fractals

1T3T

2T

1T2T 3T

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Rendering Fractals

1T3T

2T

1T2T 3T

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Finding Transformations

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Finding Transformations

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Finding Transformations

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Finding Transformations

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Finding Transformations

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Fractal Tennis

Start with any point x

For ( i=1; i<100; i++ )

x = Mrandom*x

For ( i=1; i<100000; i++ )

draw(x)

x = Mrandom*x

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Fractal Tennis

Start with any point x

For ( i=1; i<100; i++ )

x = Mrandom*x

For ( i=1; i<100000; i++ )

draw(x)

x = Mrandom*x

Gets a point on the fractal

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Fractal Tennis

Start with any point x

For ( i=1; i<100; i++ )

x = Mrandom*x

For ( i=1; i<100000; i++ )

draw(x)

x = Mrandom*x

Creates new points on

the fractal

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Fractal Tennis – Example

25,000 Points

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Fractal Tennis – Example

50,000 Points

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Fractal Tennis – Example

75,000 Points

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Fractal Tennis – Example

100,000 Points

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Fractal Tennis – Example

125,000 Points

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Fractal Tennis – Example

150,000 Points

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Fractal Tennis – Example

175,000 Points

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Fractal Tennis – Example

200,000 Points

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Modified Fractal Tennis

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Modified Fractal Tennis

25,000 Points

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Modified Fractal Tennis

50,000 Points

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Modified Fractal Tennis

75,000 Points

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Modified Fractal Tennis

100,000 Points

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Modified Fractal Tennis

125,000 Points

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Modified Fractal Tennis

150,000 Points

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Modified Fractal Tennis

175,000 Points

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Modified Fractal Tennis

200,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

200,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

25,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

50,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

75,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

100,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

125,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

150,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

175,000 Points

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Modified Fractal Tennis

Weight probabilities based on area

200,000 Points

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Condensation Sets

Given a condensation set C

CX 0

CXFXj

iji

)(1

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Condensation Set Example

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Condensation Sets – Example

Condensation Set

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Condensation Sets – Example

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Rendering Fractals

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Rendering Fractals

1T2T

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Rendering Fractals

1T2T

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Rendering Fractals

1T2T

1T2T

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Rendering Fractals

1T2T

1T2T

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Rendering Fractals

1T2T

1T2T

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Rendering Fractals

1T2T

1T2T

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Rendering Fractals

1T2T

1T2T

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Condensation Set Example

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

1)log(

)2log(

21

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

2)log(

)4log(

21

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

26.1)log(

)4log(

31

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

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Fractal Dimension

Fractal dimension = )log(

)log(#

factorscale

tionstransforma

2)log(

)2log(

22

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Weird Fractal Properties

Fractal curves can have infinite length!!!

40 len

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Weird Fractal Properties

Fractal curves can have infinite length!!!

?1 len

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Weird Fractal Properties

Fractal curves can have infinite length!!!

43

41 len

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Weird Fractal Properties

Fractal curves can have infinite length!!!

43

42

2

len

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Weird Fractal Properties

Fractal curves can have infinite length!!!

43

43

3

len

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Weird Fractal Properties

Fractal curves can have infinite length!!!

43

4i

ilen

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Weird Fractal Properties

Fractal curves can have infinite length!!!

43

4lim

i

ilen

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

4

30 area

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

?1 area

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

4

3

9

411

area

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

4

3

81

16

9

412

area

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

4

3

729

64

81

16

9

413

area

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

i

j

j

iarea0 9

4

4

3

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Weird Fractal Properties

Fractal curves can have infinite length!!! But only enclose finite area?

78.1

1

4

3

9

4

4

3lim

94

0

i

j

j

iarea

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Weird Fractal Properties