Dimension of quasi-2D Fractal
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h |
1/3 |
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N |
6 |
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h |
1/9 |
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h |
1/3n |
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N |
36 |
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N |
3n2n |
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h |
1/27 |
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N |
216 |
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D |
= |
log(N) |
= |
log(3n ) +log(2n ) |
=1+ log(2n ) =1+ DF |
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F ,2 |
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log(1/ h) |
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log(3n ) |
log(3n ) |
x |
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In general, DF ,2 = DFx + DFy
Alam, Principles of Nanobiosensors, 2013 |
9 |
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Same DF, but different geometry
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same dimension, |
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What about this |
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because DF=1+log(m)/log(n) |
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irregular fractal? |
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Alam, Principles of Nanobiosensors, 2013 |
10 |
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Dimension of a irregular fractal surface
h=6416 |
h=8 |
h=32h=4 |
5 |
5 |
4 |
4 |
3 |
(N) |
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3 |
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2 |
log |
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2 |
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1 |
1 |
0 |
2 |
3 |
4 |
5 |
6 |
01 |
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1 |
2 |
3 |
4 |
5 |
6 |
104
102
10100 0
Slope=DF=1.54.
102 |
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104 |
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log (1/h) |
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Alam, Principles of Nanobiosensors, 2013
11
Irregular to regular surfaces
DF,CT= DF,stick |
DF,CT=1+log(m)/log(n)
For DF,stick=1.5
Let m=2, solve for n: log(n)=log(2)/(DF,stick-1) Result: n=4
Generation algorithm: Take a line segment
Remove the fraction (n-2)/n from its centre (result: ½)
repeat …
12