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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
