# Fractals and Multifractals

- Institution: IIT Guwahati
- Department: Mathematics
- Teacher: Prof. M. Guru Prem Prasad Prof. A. Gowrisankar
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111103001)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111103001)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

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

1. [Lecture 1: Cantor set](https://invisiblenetwork.ai/learn/nptel/111103001/4030d7932c2b) (YouTube)
2. [Lecture 2: Sierpinski Triangle, Von Koch Curve](https://invisiblenetwork.ai/learn/nptel/111103001/84b93a921d1e) (YouTube)
3. [Lecture 3: Fractals and Examples](https://invisiblenetwork.ai/learn/nptel/111103001/414a7cbd0220) (YouTube)
4. [Lecture 4: Metric Space, Compact Metric Space](https://invisiblenetwork.ai/learn/nptel/111103001/3eb316c4e1d1) (YouTube)
5. [Lecture 5: Sequence, Equivalent metrics](https://invisiblenetwork.ai/learn/nptel/111103001/fbec1c5c5bec) (YouTube)
6. [Lecture 6: Continuous and Uniform Continuous Function, Homeomorphism](https://invisiblenetwork.ai/learn/nptel/111103001/e613dc7ebf60) (YouTube)
7. [Lecture 7: Cauchy Sequence, Complete Metric Space, Contraction Mapping](https://invisiblenetwork.ai/learn/nptel/111103001/144d545bcf70) (YouTube)
8. [Lecture 8: Fixed Point, Banach Contraction Mapping Theorem](https://invisiblenetwork.ai/learn/nptel/111103001/4665e4153fec) (YouTube)
9. [Lecture 9: Hyperspace H(X), Hausdorff Metric](https://invisiblenetwork.ai/learn/nptel/111103001/c801f052b232) (YouTube)
10. [Lecture 10: Hausdorff distance of two closed interval in R](https://invisiblenetwork.ai/learn/nptel/111103001/06d02407f7c3) (YouTube)
11. [Lecture 11: Completeness of the Space of Fractals](https://invisiblenetwork.ai/learn/nptel/111103001/13d91d3c4fca) (YouTube)
12. [Lecture 12: Dilation of a Set and its Associated Properties](https://invisiblenetwork.ai/learn/nptel/111103001/742b743ade8c) (YouTube)
13. [Lecture 13: Iterated Function System (IFS), Contraction on H(X)](https://invisiblenetwork.ai/learn/nptel/111103001/bf64e490d1e7) (YouTube)
14. [Lecture 14: Hutchinson-Barnsley Mapping and Its Properties](https://invisiblenetwork.ai/learn/nptel/111103001/ff28e5bc016b) (YouTube)
15. [Lecture 15: Deterministic Fractals and Examples](https://invisiblenetwork.ai/learn/nptel/111103001/2a2074e56d99) (YouTube)
16. [Lecture 16: Condensation Set, Collage Theorem, Random Iteration Algorithm](https://invisiblenetwork.ai/learn/nptel/111103001/090cd86058da) (YouTube)
17. [Lecture 17: Deterministic Algorithm](https://invisiblenetwork.ai/learn/nptel/111103001/0436c6119fc7) (YouTube)
18. [Lecture 18: Classical Interpolation](https://invisiblenetwork.ai/learn/nptel/111103001/0fe2dd6860d4) (YouTube)
19. [Lecture 19: IFS for Fractal Interpolation Function](https://invisiblenetwork.ai/learn/nptel/111103001/3b0af2af4490) (YouTube)
20. [Lecture 20: Existence of Fractal Interpolation Function](https://invisiblenetwork.ai/learn/nptel/111103001/dfd176d2c00b) (YouTube)
21. [Lecture 21: Types of Fractal Interpolation Functions, Vertical Scaling Factor](https://invisiblenetwork.ai/learn/nptel/111103001/7dd307932c74) (YouTube)
22. [Lecture 22: Estimation of Fractal Interpolation Parameters](https://invisiblenetwork.ai/learn/nptel/111103001/0fef4f7fa8ab) (YouTube)
23. [Lecture 23: Variable Vertical Scaling Factors](https://invisiblenetwork.ai/learn/nptel/111103001/71970f248d30) (YouTube)
24. [Lecture 24: Iteration in Fractal Interpolation](https://invisiblenetwork.ai/learn/nptel/111103001/1e86c1522178) (YouTube)
25. [Lecture 25: Integration in Fractal Interpolation](https://invisiblenetwork.ai/learn/nptel/111103001/b1041a6dd9a3) (YouTube)
26. [Lecture 26: alpha Fractal Interpolation Function](https://invisiblenetwork.ai/learn/nptel/111103001/f3ce0dfafcdd) (YouTube)
27. [Lecture 27: Fractal Dimension](https://invisiblenetwork.ai/learn/nptel/111103001/79e2a585e12e) (YouTube)
28. [Lecture 28: Fractal Dimension](https://invisiblenetwork.ai/learn/nptel/111103001/3f8abde751f6) (YouTube)
29. [Lecture 29: Estimation of Fractal Dimension](https://invisiblenetwork.ai/learn/nptel/111103001/db9824af866b) (YouTube)
30. [Lecture 30: Box-counting Theorem](https://invisiblenetwork.ai/learn/nptel/111103001/257d2dc0efe2) (YouTube)
31. [Lecture 31: Box-counting dimension of Sierpinski Triangle, von Koch Curve and Cantor set](https://invisiblenetwork.ai/learn/nptel/111103001/79e1a6b3863d) (YouTube)
32. [Lecture 32: Invariance of Fractal Dimension](https://invisiblenetwork.ai/learn/nptel/111103001/a4edd29fa6f4) (YouTube)
33. [Lecture 33: Properties of Fractal Dimension](https://invisiblenetwork.ai/learn/nptel/111103001/888c76d1df47) (YouTube)
34. [Lecture 34: Fractal Dimension of an IFS Attractor](https://invisiblenetwork.ai/learn/nptel/111103001/bb1a44136b8c) (YouTube)
35. [Lecture 35: Fractal Dimension of a Fractal Interpolation Function](https://invisiblenetwork.ai/learn/nptel/111103001/3a115aacd16f) (YouTube)
36. [Lecture 36: Topological Dimension and Examples](https://invisiblenetwork.ai/learn/nptel/111103001/6653f7cdf25b) (YouTube)
37. [Lecture 37: Measuring Sets, Physical Interpretation of Hausdorff-Besicovitch Dimension, Hausdorff Dimension of von Koch curve](https://invisiblenetwork.ai/learn/nptel/111103001/d8b7b961983f) (YouTube)
38. [Lecture 38: Diameter of a Set, δ-cover of a Set, s-dimensional Hausdorff measure](https://invisiblenetwork.ai/learn/nptel/111103001/e82a6c249523) (YouTube)
39. [Lecture 39: Scaling property, Holder condition, Hausdorff Dimension, Properties of Hausdorff dimension](https://invisiblenetwork.ai/learn/nptel/111103001/dfe125b87c66) (YouTube)
40. [Lecture 40: Calculation of Hausdorff dimension, Equivalent definition of Hausdorff dimension](https://invisiblenetwork.ai/learn/nptel/111103001/780feeeb1035) (YouTube)
41. [Lecture 41: Estimation of Hausdorff measure, Hausdorff Dimension and Examples](https://invisiblenetwork.ai/learn/nptel/111103001/377d50b76cc2) (YouTube)
42. [Lecture 42: Dimensions of Self-Similar Sets](https://invisiblenetwork.ai/learn/nptel/111103001/99831eb9907f) (YouTube)
43. [Lecture 43: Conserved Quantity in the Triadic Cantor Set](https://invisiblenetwork.ai/learn/nptel/111103001/340cd88e163f) (YouTube)
44. [Lecture 44: Legendre transformation, Theory of multifractality](https://invisiblenetwork.ai/learn/nptel/111103001/9e080a05f8ba) (YouTube)
45. [Lecture 45: Dyadic Cantor set, Deterministic Multifractal](https://invisiblenetwork.ai/learn/nptel/111103001/d7c9b7a57e48) (YouTube)
46. [Lecture 46: Cut and Paste model on Dyadic Cantor Set](https://invisiblenetwork.ai/learn/nptel/111103001/14ab9e811c8f) (YouTube)
47. [Lecture 47: Dyadic Sierpinski Triangle, Cut and paste model on Sierpinski carpet](https://invisiblenetwork.ai/learn/nptel/111103001/cfeca07d5e71) (YouTube)
48. [Lecture 48: Multifractal Measure on the Koch Curve, Multifractal Measure on the Sierpinski Carpet](https://invisiblenetwork.ai/learn/nptel/111103001/da56f31a0236) (YouTube)
49. [Lecture 49: Curdling and the Devil’s Staircase, Curdling with Several Length Scales, Generalized Fractal Dimensions](https://invisiblenetwork.ai/learn/nptel/111103001/79a770cceb2d) (YouTube)