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Course11110300149 lectures

Fractals and Multifractals

Prof. M. Guru Prem Prasad Prof. A. Gowrisankar · IIT Guwahati

Start with lecture 1 At NPTEL

About

Fractals has been an interesting subject in mathematics since its emergence in the mid-1970s. Although fractal theory is relatively modern, the foundational mathematical tools required to study fractals are drawn primarily from basic concepts in real analysis. Multifractals represent a more advanced extension of the fractal theory. The Euclidean geometry effectively discusses regular forms; however, many real-world phenomena exhibit irregular structures. The course of fractals and multifractals provides a powerful mathematical framework for analyzing such irregular sets and functions through the notion of non-integer (fractal) dimensions. This course is designed for undergraduate, postgraduate students of both Engineering and Mathematical Sciences.

Intended audience: Final year UG students, PG students of Science and Engineering Disciplines

Prerequisites: Basic Calculus or Real Analysis

Lectures

Week 1: History of Fractals

  1. Lecture 1: Cantor set YouTube
  2. Lecture 2: Sierpinski Triangle, Von Koch Curve YouTube
  3. Lecture 3: Fractals and Examples YouTube

Week 2: Compact and Complete Metric Space

  1. Lecture 4: Metric Space, Compact Metric Space YouTube
  2. Lecture 5: Sequence, Equivalent metrics YouTube
  3. Lecture 6: Continuous and Uniform Continuous Function, Homeomorphism YouTube
  4. Lecture 7: Cauchy Sequence, Complete Metric Space, Contraction Mapping YouTube

Week 3: Fractal Space and Iterated Function System

  1. Lecture 8: Fixed Point, Banach Contraction Mapping Theorem YouTube
  2. Lecture 9: Hyperspace H(X), Hausdorff Metric YouTube
  3. Lecture 10: Hausdorff distance of two closed interval in R YouTube
  4. Lecture 11: Completeness of the Space of Fractals YouTube
  5. Lecture 12: Dilation of a Set and its Associated Properties YouTube
  6. Lecture 13: Iterated Function System (IFS), Contraction on H(X) YouTube

Week 4: IFS Theory

  1. Lecture 14: Hutchinson-Barnsley Mapping and Its Properties YouTube
  2. Lecture 15: Deterministic Fractals and Examples YouTube
  3. Lecture 16: Condensation Set, Collage Theorem, Random Iteration Algorithm YouTube
  4. Lecture 17: Deterministic Algorithm YouTube

Week 5: Fractal Interpolation

  1. Lecture 18: Classical Interpolation YouTube
  2. Lecture 19: IFS for Fractal Interpolation Function YouTube
  3. Lecture 20: Existence of Fractal Interpolation Function YouTube
  4. Lecture 21: Types of Fractal Interpolation Functions, Vertical Scaling Factor YouTube
  5. Lecture 22: Estimation of Fractal Interpolation Parameters YouTube

Week 6: Fractal Interpolation

  1. Lecture 23: Variable Vertical Scaling Factors YouTube
  2. Lecture 24: Iteration in Fractal Interpolation YouTube
  3. Lecture 25: Integration in Fractal Interpolation YouTube
  4. Lecture 26: alpha Fractal Interpolation Function YouTube
  5. Lecture 27: Fractal Dimension YouTube

Week 7: Fractal Dimension

  1. Lecture 28: Fractal Dimension YouTube
  2. Lecture 29: Estimation of Fractal Dimension YouTube
  3. Lecture 30: Box-counting Theorem YouTube
  4. Lecture 31: Box-counting dimension of Sierpinski Triangle, von Koch Curve and Cantor set YouTube

Week 8: Fractal Dimension

  1. Lecture 32: Invariance of Fractal Dimension YouTube
  2. Lecture 33: Properties of Fractal Dimension YouTube
  3. Lecture 34: Fractal Dimension of an IFS Attractor YouTube
  4. Lecture 35: Fractal Dimension of a Fractal Interpolation Function YouTube

Week 9: Topological Dimension and Hausdorff Dimension

  1. Lecture 36: Topological Dimension and Examples YouTube
  2. Lecture 37: Measuring Sets, Physical Interpretation of Hausdorff-Besicovitch Dimension, Hausdorff Dimension of von Koch curve YouTube
  3. Lecture 38: Diameter of a Set, δ-cover of a Set, s-dimensional Hausdorff measure YouTube

Week 10: Hausdorff Measure and Hausdorff Dimension

  1. Lecture 39: Scaling property, Holder condition, Hausdorff Dimension, Properties of Hausdorff dimension YouTube
  2. Lecture 40: Calculation of Hausdorff dimension, Equivalent definition of Hausdorff dimension YouTube
  3. Lecture 41: Estimation of Hausdorff measure, Hausdorff Dimension and Examples YouTube
  4. Lecture 42: Dimensions of Self-Similar Sets YouTube

Week 11: Multifractal Measure

  1. Lecture 43: Conserved Quantity in the Triadic Cantor Set YouTube
  2. Lecture 44: Legendre transformation, Theory of multifractality YouTube
  3. Lecture 45: Dyadic Cantor set, Deterministic Multifractal YouTube
  4. Lecture 46: Cut and Paste model on Dyadic Cantor Set YouTube

Week 12: Random Fractals

  1. Lecture 47: Dyadic Sierpinski Triangle, Cut and paste model on Sierpinski carpet YouTube
  2. Lecture 48: Multifractal Measure on the Koch Curve, Multifractal Measure on the Sierpinski Carpet YouTube
  3. Lecture 49: Curdling and the Devil’s Staircase, Curdling with Several Length Scales, Generalized Fractal Dimensions YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Guwahati
Department
Mathematics
Teacher
Prof. M. Guru Prem Prasad Prof. A. Gowrisankar
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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