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Introduction to FRACTALS

Introduction to FRACTALS. Larry Liebovitch, Ph.D. Florida Atlantic University 2004. Non-Fractal. Fractal. Partial List of Acknowledgments. Florida Atlantic University. Center for Complex Systems. Dr. J. A. S. Kelso. Department of Psychology. Dr. David Wolgin. Continuing Education.

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Introduction to FRACTALS

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  1. Introduction to FRACTALS Larry Liebovitch, Ph.D. Florida Atlantic University 2004

  2. Non-Fractal

  3. Fractal

  4. Partial List of Acknowledgments Florida Atlantic University Center for Complex Systems Dr. J. A. S. Kelso Department of Psychology Dr. David Wolgin Continuing Education Dr. Phyllis Jonas Graphics Adrien Spano Instructional Services W. Douglas Trabert Harvard University Pulmonary Critical Care Unit Dr. Barney Hoop Telecommunications Manager Peg Doyle

  5. Non - Fractal Size of Features 1 cm 1 characteristic scale

  6. Fractal Size of Features 2 cm 1 cm 1/2 cm 1/4 cm many different scales

  7. Self-Similarity Water Water Water Land Land Land

  8. Scaling The value measured for a property depends on the resolution at which it is measured.

  9. Dimension 2 3 1 4

  10. Statistical Properties moments may be zero or non-finite. for example, mean 0 variance 

  11. Self-Similarity Geometrical The magnified piece of an object is an exact copy of the whole object.

  12. Self-Similarity Statistical The value of statistical property Q(r) measured at resolution r, is proportional to the value Q(ar) measured at resolution ar. Q(ar) = kQ(r) d pdf [Q(ar)] = pdf [kQ(r)]

  13. Branching Patterns nerve cells in the retina, and in culture Caserta, Stanley, Eldred, Daccord, Hausman, and Nittmann 1990 Phys. Rev. Lett. 64:95-98 Smith Jr., Marks, Lange, Sheriff Jr., and Neale 1989 J. Neurosci, Meth. 27:173-180

  14. Branching Patterns air ways blood vessels in the retina in the lungs Family, Masters, and Platt 1989 Physica D38:98-103 Mainster 1990 Eye 4:235-241 West and Goldberger 1987 Am. Sci. 75:354-365

  15. TISSUE CULTURE MEDIUM (ELECTROLYTE) CELLS SMALL GOLD ELECTRODE (10-1 CM2) LARGE GOLD COUNTER ELECTRODE (101 CM2) 4000 Hz AC SIGNAL 1 VOLT LOCK-IN AMPLIFIER PC DATA ACQUISITION AND PROCESSING V t Variations in Time cell height above a substrate Giaever and Keese 1989 Physica D38:128-133

  16. Variations in Time cell height above a substrate Giaever and Keese 1989 Physica D38:128-133 5.4 5.8 5.0 5.7 5.6 5.6 5.2 5.5 4.8 mV (a measure of the average height) 0 20 40 60 80 100 0 200 400 600 800 1000 5.78 5.77 5.74 5.76 0 2 4 6 8 10 0.0 0.4 0.8 min min

  17. 1 2 3 4 Variations in Time voltage across the cell membrane Churilla, Gottschalke, Liebovitch, Selector, Todorov, and Yeandle 1996 Ann. Biomed. Engr. 24:99-108 -5 mV -12 0 5 sec

  18. Currents Through Ion Channels ATP sensitive potassium channel in cell from the pancreas Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591:375-391) FC = 10 Hz 5 sec 5 pA FC = 1k Hz 5 msec

  19. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  20. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  21. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record Closed Time in ms

  22. Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Number of closed Times per Time Bin in the Record A/D =170 Hz Closed Time in ms

  23. How is the body formed? 1,000,000 capillaries 100,000 genes ? Heart 100,000,000,000 nerve cells Brain DNA

  24. How is the body formed? Self-Similar Structures Repeated Application of these Rules Heart Rules Brain DNA

  25. Scaling Self-Similarity Q (ar) = k Q(r) Q (r) = B rb Self-Similarity can be satisfied by the power law scaling: Q (r) = B rb Proof: using the scaling relationship to evaluate Q(a) and Q(ar), Q (r) = B rb Q (ar) = B ab rb if k = ab then Q (ar) = k Q (r)

  26. log ar log a log ar + log r log a Self-Similarity Scaling Q (r) = B rb f(Log[r]/Log[a]) Q (ar) = k Q(r) Self-Similarity can be satisfied by the more complex scaling: log r log a Q (r) = B rbf ( ) where f(1+x) = f(x) Proof: using the scaling relationship to evaluate Q(a) and Q(ar), log r log a Q (r) = B rbf ( ) Q (ar) = B ab rbf ( ) = Babrbf ( ) log ar log a log r log a = B ab rbf ( ) = Babrbf ( ) 1+ if k = ab then Q (ar) = k Q (r)

  27. Scaling Relationships most common form: Power Law Q (r) = B rb Q (r) Log Q (r) Logarithm of the measuremnt measurement Log r r resolution used to make the measurement Logarithm of the resolution used to make the measurement

  28. log r log a Scaling Relationships less common, more general form: Q (r) = B rb f ( ) Log Q (r) Q (r) Logarithm of the measuremnt measurement r Log r resolution used to make the measurement Logarithm of the resolution used to make the measurement

  29. How Long is the Coastline of Britain? Richardson 1961 The problem of contiguity: An Appendix to Statistics of Deadly Quarrels General Systems Yearbook 6:139-187 AUSTRIALIAN COAST 4.0 CIRCLE SOUTH AFRICAN COAST 3.5 GERMAN LAND-FRONTIER, 1900 Log10 (Total Length in Km) WEST COAST OF BRITIAN 3.0 LAND-FRONTIER OF PORTUGAL 3.5 1.0 2.0 3.0 1.5 2.5 LOG10 (Length of Line Segments in Km)

  30. Scaling of Membrane Area Paumgartner, Losa, and Weibel 1981 J. Microscopy 121:51 - 63

  31. imi inner mitochondrial membrane omi outer mitochondrial membrane er endoplasmic reticulum 30 [1 - D] [1 - D] -0.54 10 imi -0.09 omi BOUNDARY LENGTH DENSITY IN M-1 -0.72 er 1 x10-6 40 60 4 6 20 8 10 RESOLUTION SCALE M-1

  32. Scaling of Ion Channel Kinetics Liebovitch et al. 1987 Math. Biosci. 84:37-68 70 pS Channel, on cell, Corneal Endothelium 1000 100 keffin Hz effective kinetic rate constant 10 1 1 1000 10 100 teffin msec effective time scale

  33. Two Interpretations of the Fractal Scalings Structural The scaling relationship reflects the distribution of the activation energy barriers between the open and closed sets of conformational substates. closed open Energy

  34. Two Interpretations of the Fractal Scalings Dynamical The scaling relationship reflects the time dependence of the activation energy barrier between the open and closed states. t3 t2 t1 closed time Energy open

  35. Biological Examples of Scaling Relationships spatial • area of endoplasmic reticulum • membrane • area of inner mitochondrial membrane • area of outer mitochondrial membrane • diameter of airways in the lung • size of spaces between endothelial • cells in the lung • surface area of proteins

  36. Biological Examples of Scaling Relationships temporal • kinetics of ion channels • reaction rates of chemical • reactions limited by diffusion • washout kinetics of substances • in the blood

  37. Scaling Resolution 1 cm Perimeter = 8 cm Perimeter = 12 cm Resolution 0.5 cm

  38. Scaling scaling relationship: much more interesting one measurement: not so interesting one value slope Logarithm of the measuremnt Logarithm of the measuremnt Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement

  39. DIMENSION A quantitative measure of self-similarity and scaling The dimension tells us how many new pieces we see when we look at a finer resolution.

  40. Fractal Dimension Space filling properties of an object e.g. • Self-similarity dimension • Capacity dimension • Hausodorff-Besicovitch dimension

  41. Topological Dimension how points within an object are connected e.g. • covering dimension • iterative dimension

  42. Embedding Dimension the space that contains an object

  43. Fractal Dimensions Self-similarity Dimension N new pieces when each line segment is divided by M. N = Md 3 = 31 d = 1 1 2 3

  44. Fractal Dimensions Self-similarity Dimension N new pieces when each line segment is divided by M. N = Md 1 2 3 9 = 32 d = 2 6 5 4 7 8 9

  45. Fractal Dimensions Self-similarity Dimension N new pieces when each line segment is divided by M. N = Md 27 = 33 d = 3 1 2 3 4 5 6 7 8 9

  46. Fractal Dimensions Capacity Dimension N (r) balls of radius r needed to cover the object

  47. Log N( ) 1 r Fractal Dimensions Capacity Dimension d = lim Log N(r) r 0 Relationship to self-similarity dimension: M = 1/r, then N = Md

  48. Fractal Dimensions Hausdorff-Besicovitch Dimension Ai = covering sets

  49. H(S,r) = inf (diameter Ai)S i lim r 0 lim r 0 Fractal Dimensions Capacity Dimension H(s,r) = for all s < d for all s > d H(s,r) = 0

  50. Fractal Dimension of the Perimeter of the Koch Curve 1 1 1 1 1 1 1

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