Cantor set

Fractals: theory and applications Didier Gonze Unité de Chronobiologie Théorique Service de Chimie Physique - CP 231 Université Libre de Bruxelles Belgium

Overview  Theory of fractals Definition and properties Dimension

 Examples of fractals Cantor set & devil's staircase Mandelbrot set Julia set Sierpinsky triangle

 Appications in biology Physiology Biological time series Kinetics

Definition A fractal is generally "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole" (B. Mandlebrot). This property is called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured". Benoît Mandelbrot (1924-2010)

Source: wikipedia

Britain coast How long is the coast of Bretagne?

Britain coast Lewis Fry Richardson (1881-1953) While studying the causes of war between two countries, Richardson decided to search for a relation between the probability of two countries going to war and the length of their common border. While collecting data, he realised that there was considerable variation in the various gazetted lengths of international borders. For example, that between Spain and Portugal was variously quoted as 987 or 1214 km while that between The Netherlands and Belgium as 380 or 449 km. As part of his research, Richardson investigated how the measured length of a border changes as the unit of measurement is changed. He published empirical statistics which led to a conjectured relationship. This research was quoted by mathematician Benoît Mandelbrot in his 1967 paper "How Long Is the Coast of Britain"? http://fr.wikipedia.org/wiki/Lewis_Fry_Richardson

Britain coast

This hypothetic map illustrates the notion of self-similarity. Note however that in reality, coasts are not purely self-similar!

Properties A fractal often has the following features:  It has a fine structure at arbitrarily small scales.  It is self-similar (at least approximately).  It is too irregular to be easily described in traditional Euclidean geometric language.  It has a dimension which is non-integer and greater than its topological dimension (i.e. the dimension of the space required to "draw" the fractal).  It has a simple and recursive definition.

Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals — for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics.

Examples of natural fractals Brocoli

Lung

Mountain landscape Fern

Geographic map

Examples of mathematical fractals Julia set

Mandelbrot set

Koch snowflake

Dragon curve

Sierpinsky triangle

Cantor set The Cantor set is obtained by deleting recursively the 1/3 middle part of a set of line segments: S0 S1 S2 S3 S4 S5 S6 ... S∞

Properties of the Cantor set C (obtained after an infinite number of iterations): • C has a structure at arbitrarily small scale (like worlds within worlds). • C is self-similar: it contains smaller copies of itself at all scales. • The dimension of C is not an integer (see later). Source: Strogatz

Cantor set The Cantor set is obtained by deleting recursively the 1/3 middle part of a set of line segments: S0 S1 S2 S3

How long in the Cantor set?

S4 S5 S6 ... S∞

Each set Sn completely covers all the sets that come after it in the construction. Hence the Cantor set C=S∞ is covered by each of the set Sn. So the total length of the Cantor set must be less than the total length of Sn, for any n. Let Ln denote the length of Sn. Then from the construction, we see that L0=1, L1=2/3, L2=2/3*2/3=(2/3)2, and in general Ln=(2/3)n. Since Ln -> ∞ as n -> ∞, the Cantor set as a length = limn->∞((2/3)n)=0. This suggest that the Cantor set is small. However, the Cantor set contains an infinite (uncountable) number of points.... Source: Strogatz

Dimension Any point on a 1D curve can be represented by one number, the distance d from the start point.

Any point on a 2D surface can be represented by two numbers. One possible method is to grid the surface and to measure two distance along the grid lines

Any point in 3D can be represented by three numbers. Typically these numbers are the coordinated of the point using an orthogonal corrdinate system.

Euclidian geometry

Dimension Any point on a 1D curve can be represented by one number, the distance d from the start point.

Euclidian geometry

reference point

0 distance

Koch curve The Koch curve is obtained as follows: start with a line segment S0. To generate S1, delete the middle 1/3 part of S0 and replace it with two other 2-sides of an equilateral triangle. Subsequent stages are generated recursively by the same rule. The limit K=S∞ is the von Koch curve.

Source: Strogatz

Koch curve What is the length of the Koch cuve? Following the same procedure as for the Cantor set, we find: L0=1 L1=4/3 L2=(4/3)2 Ln=(4/3)n L∞=limn->∞(4/3)n = ∞ The length is thus infinite. In fact the same infinite limit is obtained for any value of L0. Thus, the distance between any pair of points on the Koch curve is infinite. This suggests that K is more than 1D, but would we say that it is 2D? It has no "area"... Source: Strogatz

Fractal dimension The simplest fractals (such as the Cantor set and the Koch curve) are self-similar: they are made of scale-down copies of themselves. The dimension of such fractals can be defined by extending an elementary observation about "classical" self-similar objects. A mathematical description of dimension is based on how the "size" of an object behaves as the linear dimension increases. Example:

number of squares = m = 1 scale factor = r = 1

m=4 r=2

m=9 r=3

If we shrink the square by a factor of 2 in each direction, it takes 4 of the small squares to equal the whole. If we scale the originl square by a factor of 3, we need 9 small squares. In general, if we reduce the side dimensions of the square by a factor r, it takes r2 smaller squares to equal the original. If we play the same game with cubes, we notice that we need 8 cubes of scale 2, 27 cube of scale 3, ... i.e. r3 cubes of scale r. Source: Strogatz

Fractal dimension In summary: In 1D (line segment), we need r segments of scale r to equal the original segment. In 2D (squares) we need r2 squares of scale r to equal the original square. In 3D (cube) we need r3 cubes of scale r to equal the original cube. The exponents 1,2,3 are not accident. They reflect the dimentionality d of these "classical" objects. This relationship between the dimension d, the scaling factor r and the number m of rescaled copies required to cover the original object is thus:

m=r

d

Rearranging the above equation gives an expression for dimension depending on how the size changes as a function of linear scaling:

!

ln m d= ln r

Source: Strogatz

Fractal dimension Application to the Cantor set S0 S1 S2 S3 S4 S5 S6 ... S∞

Each set is composed of m = 2 copies of itself. Each copy is scaled by a factor r = 3. Thus the dimension of the Cantor set is

ln m ln2 d= = " 0.63 ln r ln 3 Source: Strogatz

Fractal dimension Application to the Koch curve set Each set is composed of m = 4 copies of itself (from one segment, we get 4 segments). Each copy is scaled by a factor r = 3 (each segment has a length=1/3 of the length of its parent). Thus the dimension of the Koch curve is:

ln m ln 4 d= = " 1.26 ln r ln 3

! Source: Strogatz

Fractal dimension The box dimension To deal with fractals which are not self-similar, we need to further generalize the notion of dimension. Various definition have been proposed. All the definitions share the idea of "measurement at a scale ε" and how the measurement varies as ε->0. One kind of measurement involved covering the set with boxes of size ε.

The minimum number of segments of size ε needed to cover the original segment of length L is: N(ε) = L / ε

The minimum number of squares of size ε needed to cover the grey aera A is: N(ε) = A / ε2 Source: Strogatz

Fractal dimension The box dimension In general, for "classical" objects, we have a power-low relationship: N(ε) ~ 1 / εd We interpret d as a dimension, usually called the box dimension. An equivalent definition is:

ln N(") " #0 ln(1/" )

d = lim

Application to the Cantor set by each of the set Sn used in its !Recall that the Cantor set is covered n n

construction. Each Sn consists of 2 intervals of length (1/3) , so if we pick ε =(1/3)n, we need 2n of these intervals to cover the Cantor set. Hence, N=2n when ε =(1/3)n. Since ε ->0 as n->∞, we find: ln N(") ln2 n n ln2 ln2 d = lim = = = n " #0 ln(1/" ) ln 3 n ln 3 ln 3

Note that this is consistent with the similarity measurement introduced previously

Source: Strogatz

Fractal dimension Application to a fractal that is not self-similar Let's consider the following fractal: A square region is divided into 9 equal squares. One of them is selected at random and discarded. Then the process is repeated on each of the 8 remaining squares and so on.

What is the box dimension of this fractal? Pick the unit of length to equal the side of the oringinal square. Then S1 is covered (with no wastage) by N=8 squares of side ε=1/3. Similarly S2 is covered by N=82 squares of side ε=(1/3)2. In general N=8n when ε=(1/3)n. hence: ln N(") ln8 n n ln8 ln8 d = lim = = = n " #0 ln(1/" ) ln 3 n ln 3 ln 3 Source: Strogatz

Fractal dimension Hausdorff dimension When computing the box dimension it is not always easy to find a minimal cover. There is an equivalent way to compute the bow dimension that avoids this problem. We cover the set with a square mesh of boxes of side ε, count the number of occupied boxes N(ε), and compute d as before. Even with this improvement the box dimension is rarely used in practice. Its computation requires too much storage and computer time. the box dimension also suffers from some mathematical drawbacks. One alternative dimension, called the Hausdorff dimension, is also based on covering sets with boxes but uses boxes of varying sizes. It is however hard to compute numerically.

List of fractals objects sorted by their Hausdorff dimension: http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Source: Strogatz

Multifractals Multifractals are a generalization of a fractals which are not characterized by a single dimension, but rather by a continuous spectrum of dimensions.

Fractal vs Euclidian geometry Euclidian geometry

Fractal geometry

traditional

modern invention

based on a characteristic size or scale

no specific size or scale

suits description of man made object

appropriate for geometry in nature

described by a usually simple formula example: x2+y2+z2 = r2 described a sphere

described by an algorithm example zn+1=zn2+z0 describes the Mandelbrot set

Examples of fractals

Cantor set: variants Cantor comb

Cantor collar

Cantor quare

Cantor set: variants Cantor cube (Cantor dust)

Cantor tartan

Devil's staircase

As Pacman eats the dots, he gets heavier. Imagine that his weight after eats all the dots is 1. Let’s graph his weight with time. Obviously, it will not increase gradually, but will instead go up like this: This is called the Devil’s Staircase. In this picture, we colored the area below it to make it more visual. The Devil's stair case is also called the Cantor function.

Source: http://library.thinkquest.org/26242/full/fm/fm7.html

Devil's staircase Suppose that we pick a point at random from the Cantor set. The Cantor function (or Devil's stair case) actually gives the probability P(x) that this point lies to the left of x, where 0 does not belong to the set => 

z0 = i

z1 = z02+z0 = i2 + i = -1+i z2 = z12+z0 = (-1+i)2 + i = - i z3 = z12+z0 = (-i)2 + i = -1+i ... => oscillates between 2 values => bounded =>belongs to the set =>  Source: wikipedia

Mandelbrot set The Mandelbrot set is the set of points z0 in the complex plane for which the iteration zn+1 = zn2 + z0 remains bounded. Algorithm foreach z0 in the complex plane for i=1:N z=z^2+z0 end if |z|>th plot z in while else plot z in black end end

A more colourful picture can be generated by coloring points not in the set according to how quickly the sequence diverges to infinity

Mandelbrot set The Mandelbrot set is the set of points z0 in the complex plane for which the iteration zn+1 = zn2 + z0 remains bounded.

Image: wikipedia

Mandelbrot set The Mandelbrot set is the set of points z0 in the complex plane for which the iteration zn+1 = zn2 + z0 remains bounded.

Image: wikipedia

Mandelbrot set Properties of the Mandelbrot set  The Mandelbrot set has a fractal boundary.  The Mandelbrot set is connected (Hubbard & Douady).  The area of the set obtained by pixel counting is 1.50659177 +/- 0.00000008 (Sloane's; Munafo; Lesmoir-Gordon et al. 2000) and by statistical sampling is 1.506486 +/- 0.000004 with 95% confidence (Mitchell 2001).  The region of the Mandelbrot set centered around -0.75+i is sometimes known as the sea horse valley because the spiral shapes appearing in it resemble sea horse tails. Similarly, the portion of the Mandelbrot set centered around 0.3+0i with size approximately 0.1+0.1i is known as elephant valley.

Source: Wolfram

Mandelbrot set A striking correspondence between the Mandelbrot set and the logistic map

Source: wikipedia

Julia set A Julia set is a kind of mirror image of the Mandelbrot. Take the same basic family of functions: f(x)=x2+c. But instead of varying c (as done to obtain the Mandelbrot set), keep c fixed, and vary x. The Julia set for c is the set of x-values for which iterating f does not diverge. There's an infinite number of Julia sets - one for every possible c.

c = -0.4+0.6i

c = 0.285+0.01i

c = 0.8+0.156i Figures: wikipedia

Julia set Link with the Mandelbrot set

Figure from: http://mail.colonial.net/~abeckwith/00 6B0D39-70E903AC-006B0D39

Applications in biology

Natural fractals Both, geometrically and dynamically, biological systems are intricate. Concerning geometry, the natural world can not be described in terms of the familiar geometry of lines, triangles, squares, and circles. Instead, mountains clouds, and coastlines are fractals structures that always look the same as portions of them are enlarged.

Quotation from: Agn et al (2000) Cell Biol Int

Fractals in physiology

Fractals in physiology Some of the most visually striking examples of fractal forms are found in physiology: The respiratory, circulatory, and nervous systems are remarkable instances of fractal architecture, branches subdividing and subdividing and subdividing again. Although no clear genetic, enzymatic, or biophysical mechanism yet have been shown to be responsible for this fractal structure, few doubt this. Careful analysis of the lungs reveal fractal scaling, and it has been noted that this fractal structure makes the lungs more fault-tolerant during growth.

Lungs

Heart

Vessels

Source: http://classes.yale.edu/fractals/panorama/Biology/Physiology/Physiology.html

Fractals in physiology Dog

For comparison with human lungs, here are some other mammalian lungs. Note the considerable differences in branching geometry.

Pig

Manatee (lamantin)

Camel

http://classes.yale.edu/fractals/panorama/biology/ physiology/animallungs/animallungs.html

Fractals in biological time series

The fractal concept can be applied not just to irregular geometric forms that lack a characteristic (single) scale of length, but also to certain complex processes that lack a single scale of time. Fractal processes generate irregular fluctuations across multiple time scales, analogous to scale-invariant objects that have a branching or wrinkly structure across multiple length scales. The irregularity seen on different scales is not readily distinguishable, suggesting statistical self-similarity.

Goldberger et al (2002) Fractal dynamics in physiology: Alterations with disease and aging, PNAS 99:2466-72

Fractals in biological time series patient with severe congestive heart failure

healthy patient

patient with severe congestive heart failure

subject with a cardiac arrhythmia, atrial fibrillation, which produces an erratic heart rate.

Goldberger et al (2002) Fractal dynamics in physiology: Alterations with disease and aging, PNAS 99:2466-72

Note that the healthy record (B) is far from a homeostatic constant state (it is characterized by non-stationarity and patchiness). These features are related to fractal and nonlinear properties. Their breakdown in disease may be associated with the emergence of excessive regularity (A) and (C), or uncorrelated randomness (D).

Fractals in the nervous system Werner G (2010) Fractals in the nervous system: conceptual implications for theoretical neuroscience, Frontiers in Physiology

Fractal (enzyme) kinetics Savageau MA. (1995) Michaelis-Menten mechanism reconsidered: implications of fractal kinetics. J Theor Biol 7:115-24.

Fractal (enzyme) kinetics

Savageau MA. (1995) J Theor Biol 7:115-24. Savageau MA. (1998) Biosystems 47:9-36.

Fractal in biological systems Fractal geometry in physiology

? Fractal time series

Fractal kinetics

Some more fractals...

Fractals-generated landscapes

Examples of artistic fractals

Fractal crop circles

Fractals of extra-terrestrial origin ;-)

In the mid 90s some of the found crop circles were based on fractals. For instance the so-called 'Julia set' at Stonehenge in 1996 and the 'Triple Julia set' at Windmill Hill the same year. Or the Koch-fractals in 1997. Source: http://www.cropcircleconnector.com/ Bert/3dfractals.html

Matlab code http://homepages.ulb.ac.be/~dgonze/FRACTALS/fractals.html            

Cantor Koch Koch flake Sierpinsky Menger Menger 3D Ford Julia Mandelbrot Tree Fern Landscape

Further reading

See also Strogatz (1994) "Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering" (see chap. 11)