fractals: properties and applications - applied mathematics · 2016. 3. 29. · • fractals are ....
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Fractals:properties and applications
MATH CO-OP
VERONICA CIOCANEL, BROWN UNIVERSITY
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Fractal ball experiment: DIY!
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How do we thinkof dimension?
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Conclusions: Fractal properties
• Fractals exhibit fractal dimensions: all objects whose dimension is not an integer are fractals.
• Fractals are self-similar.
d = 1.2683
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Fractals - mathematical objects
Mandelbrot set Variation of a Mandelbrot set
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Fractals - around us
Lake Mead coastline The Great Wave off Kanagawa - Hokusai
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1. Fractal antennas
Sierpinskitriangle
Example of fractal antenna
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• Fractal-shape antennas can respond to more frequencies than regular ones.• They can be ¼ the size of the regular ones: use in cellular phones and military
communication hardware.• BUT: Not all fractal shapes are best suited for antennas.
Koch curve fractal antenna
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2. Coastlines
Border length
• Portugal - Spain border
987 km (reported by the Portuguese)
1214 km (reported by the Spanish)
Measurements were using different scales!
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Returning to coastlines…
South Africa Britain
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Approximating a smooth curve using straight lines – guaranteed to get closer to the true value of the curve length
Can we say the same for the UK coastline?
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Scale/ruler length: 𝑙𝑙 = 1𝑟𝑟
Perimeter/length: 𝑁𝑁 ∗ 𝑙𝑙
9 ∗ 11
= 𝟗𝟗 19 ∗ 12
= 𝟗𝟗.𝟓𝟓 48∗ 14
= 𝟏𝟏𝟏𝟏 97 ∗ 18
= 𝟏𝟏𝟏𝟏.𝟏𝟏𝟏𝟏𝟓𝟓
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• Coastlines have fractal-like properties: complexity changes with measurement scale
• A lot like the Koch curve
• This curve has infinite length!
• Length: makes little sense
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But, concept of fractal dimension makes sense!
South Africa: d = 1.02 Britain: d = 1.25
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• This is called the “Coastline paradox”: measured length of a stretch of coastline depends on the measurement scale
• But for practical use, the ruler scale is not that fine: km’s are enough!
• Approximating the coastline with an infinite fractal is thus not so useful in this case.