Introduction to fractals
and
boundary value problems in irregular domains
Part 1 - Introduction to fractals
Welcome! This module presents an introduction to fractals and their mathematical foundations. Motivated by examples and applications, we introduce box counting dimensions, Hausdorff measure and dimension and some methods for their computation. In the second part of the module we focus on particular self-similar fractal structures: we introduce the Iterated Function Systems, we investigate the dimension of the related self-similar sets and we present a dynamical system perspective for this framework. A part of the module is devoted to the implementation, with Wolfram Mathematica software, of simple visualization and fractal dimension algorithms.
Schedule
wed. (theory) 14.30 - 16.00 and fri. (mathematica lab) 14.30 - 16.00
Logistics
The course will be held via Google Meet, here's the link to the next appointment
Friday April 17 h. 14.30
https://meet.google.com/tey-aiqu-ocr?hs=122&authuser=1
Before the lesson, please download these files
An image
https://drive.google.com/file/d/1D90W0A7gxoy3VRviU4h2c9QVEiml88G9/view?usp=sharing
and a notebook
https://drive.google.com/file/d/1KEDCRO2FV9RV3idetjTbsMQwAn0wiEpZ/view?usp=sharing
Registration
Interested students and teachers from Sapienza University are invited to register (with their institutional email @uniroma1.it) at the Google Classroom page.
The code of the course is
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Programme & Materials
March 25. Introduction to fractal sets and classical examples. Hausdorff measure: definition, relation with Lebesgue measure, scaling properties, examples. Hausdorff measure of the Cantor set. Hausdorff dimension: definition and first examples. Notes download
March 27. Introduction to Wolphram Mathematica: execution of instructions, variable assignment, lists. Simple list operations. Definition of functions. Recursion. Project: recursive definition of a Cantor prefractal. Mathematica nb download
April 1. Recalls on Hausdorff measure and Hausdorff dimension. Hausdorff dimension of fractal sets: some examples. Properties of Hausdorff dimension. Equivalent definitions of Hausdorff dimensions. Box counting dimensions: definition, examples, properties. Comparison between Hausdorff and box counting dimensions. Notes download
April 3. Timing function and recursion. List manipulation: Tuples and Flatten. Construction of Cantor Dust (as product of Cantor sets and as attractor of an IFS). The Map function. Graphics with Mathematica: graphic primitives, Graphics and Show function, some options. Manipulate functions. Example: displaying Middle Third Cantor set. Exporting options: SetDirectory and Export functions. Mathematica .nb download
April 8. Upper and lower estimates for Hausdorff dimension. Mass distribution principle. Examples. Preliminaries on iterated function schemes: contractions, similarity, iterated function systems, Hutchinson operator, invariant and self-similar sets. Existence and uniqueness of invariant sets and their upper approximations. Notes download
April 10 Graphics primitives and directives: color, thickness, line style. Displaying graphics and setting parameters with Manipulate. Examples based on KochCurve function. Expression manipulation: Part and ReplacePart functions. List manipulation: Partition function. Examples: arrowed KochCurve. Mathematica nb download
April 15 Attractors of Iterated Function Systems: Hausdorff distance and Contraction Lemma. Attractors approximation, trees and addresses. Hausdorff dimension of self-similar attractors. Examples. Notes download
April 17 Nest and NestList function, RandomChoice and applications to the Random Iteration Algorithm for fractal generation. Image import and manipulation, list manipulation: a simple algorithm for the approximation of the Box Counting Dimension. Mathematica notebook (Random Iteration Algorithm) Mathematica notebook (Box counting dimension) Test image: koch curve
... That's all folks! (at least for the first module)
Contacts
Module 1 - Introduction to fractals:
Anna Chiara Lai
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma
anna.lai_at_sbai.uniroma1.it
Module 2 - Boundary value problems in irregular domains
Maria Rosaria Lancia
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma
maria.lancia_at_sbai.uniroma1.it