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Home , Packing dimension

Talk Wu Min Department of South China University of Technology Main content

一. What are ? 二. What are Fractal ? 三. What are Multifractals analysis ? 一. What are Fractals ?

1. A classics example ——Koch curve 2. The basic features of fractal set 3. The other examples Koch Curve_ The mathematician H·V·Koch constructed this curve in 1904 The construction of Koch curve

Begin with a straight line E0, divide the line segment into three segments of equal length, draw an equilateral triangle that has the middle segment from step 1 as its base and points outward to get E1:

E0 E1 The construction of Koch curve

Now repeat, taking each of the four resulting segments, dividing them into three equal parts and replacing each of the middle segments by two sides of an equilateral triangle :

E3 E2 The construction of Koch curve

Continue this construction ， the Koch curve is the limiting curve obtained by applying this construction an infinite number of times. Koch snowflake___ Three copies of the Koch curve placed around the three sides of an equilateral triangle

n steps The features of Koch Curve

1. F has a fine structure at arbitrarily small scales；

2. It is too irregular to be easily described in traditional Euclidean geometric language；

3. the length be infinite，the area be zero；

4. It is self-similar ；

5. It has a simple and recursive definition . What are Fractals ?

1), 2) and 3) above imply irregularity, while 4) and 5) imply regularity.

In general, the fractals we studied are sets or geometric object with above properties. A fractal has the following features:

• It has a fine structure at arbitrarily small scales. • It is too irregular to be easily described in traditional Euclidean geometric language. • It has a fractal which is greater than its topological dimension . • It is self-similar (at least approximately or stochastically). • It has a simple and recursive definition. The other examples • Weierstrass function • • Peano curves • Sierpinski triangle carpet • • • Weierstrass function _ introduced by German mathematician Karl Weierstrass in 1872 ：

+∞ f (x) = ∑ λ(s−2)k sin(λk x) , λ > 1, 1 < s < 2. f :[0,1] → R. k =1 Weierstrass function Cantor set ___ Introduced by German mathematician in 1883 Peano curves _ in 1890,G.Peano (Italy,1858–1932)

A space-filling• curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an N-dimensional hypercube). Sierpinski Triangle __ first described by Wacław Sierpiński in 1916

A model for superconductivity and amorphous solid The construction of Sierpinsk carpet

n steps Menger sponge _ It was first described by Austrian mathematician Karl Menger in 1926

This structure with infinite surface area and zero volume. A perfect model for catalyst in chemical reaction Menger sponge

• The sponge has a of (log 20) / (log 3) The Menger sponge is a compact and has Lebesgue measure 0. Julia set ___ In 1918-1920，the French mathematicians G.Julia (1893-1978) constructed this set

n ， k ． f : C → C is a complex polynomial f (z) = ∑ ak z k =0 The closure of the repelling periodic points of f is called the Julia set of f．

If f (z) = z 2 , then the Julia set of f is the unite circle．

If f (z) = z 2 + C and C ≠ 0， the Julia set of f is very complicatied． Figure of Julia set

C = -1 C = -0.5+0.5i

C=-0.2+0.75 i C=0.64 i Figure of Julia set Mandelbrot set_ a famous example of fractal

2 Let Pc (z) = z + c

∞ M = {c{Pn (0)} bounded } is called a Mandelbrot set． c n=1 Mandelbrot set _ a famous example of fractal

• The Mandelbrot set shows more intricate detail by the closer one looks or magnifies the image, usually called "zooming in". • Its border would show an inconceivable number of different fractal structures. • The Mandelbrot set in general is not strictly self-similar but it is quasi-self-similar. Mandelbrot set

A closer view of the Mandelbrot set magnifies the local image of Mandelbrot set

• magnifies the local image of Mandelbrot set random fractal —— random Cantor set

A model for the distribution of the noise in comunication line, important in the study of nonlinear dynamics. random fractal— random Koch Curve

A model for coastline random fractal —

the irregular motion of tiny particles from the pollen grains of flowers 二. What are Fractals dimension ?

1. How long is the coast of Britain?

2. What are Fractals dimension ? How long is the coast of Britain?

《 How Long Is the Coast of Britain ? Statistical Self- Similarity and Fractional Dimension 》------by B.Mandelbrot published in in 1967. question: How Long Is the Coast of Britain ?

answer: The measured length of the Coast of Britain is not exact ！The length of a stretch of coastline depends on the scale of measurement. How long is the coast of Britain? How long is the coast of Britain?

The measured length L(G) increases without limit as the measurement scale G decreases towards zero. Richardson conjectured that L(G) = MG1−D Mandelbrot interprets this result as showing that coastlines can have a property of statistical self- similarity, with the exponent D measuring the of the coastline. Random Koch Curve -- a model for coastline

The dimension of the coastline of South Africa is 1.02， of the West coast of Britain is 1.25. What are Fractals dimension ?

Classic dimension

– point---of dimension 0

– line---of dimension 1

– plane---of dimension 2

– body---of dimension 3 What are Fractals dimension ?

Fractal dimensions • Hausdorff dimension (Hausdorff 1919)

• Box-counting dimension (1928-1932)

• Packing dimension (Tricot 1982) What are Fractals dimension ? —— Hausdorff measure

E ⊆ Rn , s ≥ 0, ∀δ > 0, ∞ Η s (E) = inf{ |U |s : U ⊃ E, 0 <|U |≤ δ}, δ ∑ i ∪i=1 i i s s Η (E) = limΗδ (E) δ →0 Ηs is a regular metric outer measure satisfying: n n H (E) = cn L (E) when n is an integer and Ηs (λE) = λsΗs (E). What are Fractals dimension ? ——Hausdouff dimension

If 0≤ s 0⇒Ηs (E) =∞.

s s dimH E =sup{s:Η (E) >0}=sup{s:Η (E) =∞} =inf{s:Ηs(E) =0}=inf{s:Ηs(E) <∞} What are Fractals dimension ? ——Box-counting dimension

n E ⊆ R ,δ > 0, let Nδ (E) be the smallest number of sets of diameter at most δ which can cover E.

log Nδ (E) log Nδ (E) dimBE = lim , dimBE = lim . δ →0 − logδ δ →0 − logδ

Define dimBE = dimBE = dimBE if dimBE = dimBE. What are Fractals dimension ? ——Packing dimension

E ⊆ Rn , s ≥ 0, ∀δ > 0, s s Pδ (E) = sup{∑| Bi | : Bi are disjoint balls of radii at most δ with center in E} s s P0 (E) = lim Pδ (E) δ →0 ∞ Ps (E) = inf{ Ps (F ) : F ⊂ F } ∑ 0 i ∪i=1 i s s dimP E = sup{s : P (E) = ∞} = inf{s : P (E) = 0} 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? 三. What are Multifractals analysis ? • 分形理论的创始人

分形理论的创始人 IBM研究员，耶鲁大学数学教授 •Benoit B．Mandelbrot）1924年11月20日生于波兰华沙