Definitions of noninteger dimensions and mathematical formalisms

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Definitions of non-integer dimensions and
mathematical formalisms

• Edith Perrier, IRD France,
• Visiting scientist at UCT,
• Applied Maths Dept., room 417.1, Ext. 3205,
• E-mail edith_at_maths.uct.ac.za
• Web site where to download previous lectures
  lect.ppt (or lect.zip if links included)
  http//www.mth.uct.ac.za/Affiliations/BioMaths/Ind
  ex.html

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The Von Koch Curve
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The middle third Cantor Set
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• A fractal is by definition a set for which the
  Hausdorff-Besicovitch dimension strictly exceeds
  the topological dimension (Mandelbrot,1975)

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Fractal Geometry. Mathematical foundations and
applications (Kenneth Falconer, 1990, Ed. Wiley,
UK)

• My personal feeling is that the definition of a
  fractal should be regarded in the same way as
  the biologist regards the definition of life .
  There is no hard and fast definition, but just a
  list of properties characteristic of a living
  thing, such as the ability to reproduce or to
  move or to exist to some extent independently of
  the environment. Most living beings have most of
  the characteristics on the list, though there are
  living objects that are exceptions to each of
  them. In the same way, it seems best to regard a
  fractal as a set that has properties such as
  those listed below, rather than to look for a
  precise definition which will almost certainly
  exclude some interesting cases. From the
  mathematicians point of view, this approach is
  no bad thing. It is difficult to avoid developing
  properties of dimension other than in a way that
  applies to fractal and non-fractal sets
  alike. For non-fractals, however, such properties
  are of little interest they are generally
  almost obvious and could be obtained more easily
  by other methods.
• When we refer to a set F as a fractal, therefore,
  we will typically have the following in mind.
• (i) F has a fine structure, i.e. detail on
  arbitrarily small scales.
• (ii) F is too irregular to be described in
  traditional geometrical language both locally and
  globally
• (iii) Often F has some form of self-similarity,
  perhaps approximate or statistical
• (iv) Usually, the fractal dimension of F
  (defined in some way) is greater than its
  topological dimension
• (v) In most cases of interest F is defined in a
  very simple way, perhaps recursively

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Fractal dimension (1)

• Topological dimension DT
• Hausdorff dimension DH
• Exercise 1 Assuming that the geometrically
  defined middle cantor fractal set is a fractal of
  finite dimension, proof that this dimension is

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Solutions Exercice 1

• Easy assuming that at sdimHF, Hs(F) is finite
  and non-zero
• Hs(F) Hs(FL) Hs(FR) (1/3)s Hs(F) (1/3)s
  Hs(F)
• 12 (1/3)s and sLog2/Log3
• Rigorous calculation Falconer p.31-32

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An illustration from number theory

• Almost all numbers (in the sense of Lebesgue
  measure) are normal to all bases . That is they
  have base-m expansions containing equal
  proportions of the digits 0,1,..,m-1 for all m.
  If F(p0,p1,pm) is the set of numbers x in 0,1
  with base-m expansions containing the digits
  0,1,..,m-1 in proportions (p0,p1,pm) ,
  F(1/m,1/m,.1/m) has Lebesgue measure 1 thus
  dimension 1 for all m
• If the pj are not equal one can show that
• Application to another definition for the Cantor
  set no digit 1 in base 3 expansions

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• Fractal if and only if?
• Cantor set
• Von Koch curve
• The fractal dimension may be a non
  integer whereas (The Hausdorff
  dimension of the space filling Peano Curve
  equals 2)

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• On the wide variety of fractal dimension in
  use, the definition of Haussdorff (1919) based on
  a construction of Carathéodory (1914), is the
  oldest and probably the more important. Hausdorff
  dimension has the advantage of being defined for
  any set, and is mathematically convenient, as it
  is based on measures, which are relatively easy
  to manipulate. A major disadvantage is that in
  many cases it is hard to calculate or to estimate
  by computational methods.
• Properties of Hausdorff measures have been
  developped during the XXth century, largely by
  Besicovitch and his students.

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Fractal dimensions (2)

• Box Counting dimension (Kolmogorov entropy,
  entropy dimension, logarithmic density,
  information dimension, Minkowski variant)
• Examples

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Fractal dimensions

• discussion .

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Numerical calculations

• (Example from the Web at http//www.msci.memphi
  s.edu/giri/BIP/)
• Since there is no consensus on a definition of
  the Fractal Dimension of a plane curve, our
  program computes it in eleven different ways. The
  merits and demerits of the different methods of
  computing the fractal dimension were studied.
  There are no widely accepted definitions of FD.
  The software BIP computes it in 11 different
  ways. The methods include Euclidean Distance Map
  (EDM), Minkowski Sausage Method (Dilation), Box
  Counting Method, Corner Method (Counting and
  Perimeter), Fast Method (Regular and Hybrid),
  Parallel Lines Method, Cumulative Intersection
  Method, and Mass Radius (Short and Long).
• On the whole, (biofilm) images are about as
  fractal as can be expected within the
  variabilities in nature. The most stable and
  reliable methods are EDM, Minkowski Sausage
  Method, and Box Counting Method

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Words

• The term "fractal" was coined by Benoit
  Mandelbrot in 1975 in his book Les Objets
  Fractals . It comes from the Latin fractus,
  meaning an irregular surface like that of a
  broken stone. Fractals are non-regular geometric
  shapes that have the same degree of
  non-regularity on all scales. Just as a stone at
  the base of a foothill can resemble in miniature
  the mountain from which it originally tumbled
  down, so are fractals self-similar whether you
  view them from close up or very far away.
  Fractals are the kind of shapes we see in nature.
  We can describe a right triangle by the
  Pythagorean theorem, but finding a right triangle
  in nature is a different matter altogether. We
  find trees, mountains, rocks and cloud formations
  in nature, but what is the geometrical formula
  for a cloud? How can we determine the shape of a
  dollup of cream in a cup of coffee? Fractal
  geometry, chaos theory, and complex mathematics
  attempt to answer questions like these. Science
  continues to discover an amazingly consistent
  order behind the universe's most seemingly
  chaotic phenomena. .

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Lecture series

• Lecture 1. March. 3rd. Introduction to fractal
  geometry . Measures and power laws.
• Lecture 2. March 5th. Definitions of non-integer
  dimensions and mathematical formalisms
• Lecture 3. March 10th. Iteration of functions and
  fractal patterns
• Lecture 4. March 12th. Extensions Self-similar
  and self-affine sets . Multifractals.
• Lecture 5. March 17th. Fractals / Geostatistics
  / Time series analysis
• Lecture 6. March 19th. Dynamical processes
  Fractal and random walks
• Lecture 7. March 24th. Dynamical processes
  Fractal and Percolation
• Lecture 8. March 26th. Dynamical processes
  Fractal and Chaos