This course introduces basic mathematical techniques of fractal geometry and dynamical systems, aimed towards understanding and modeling natural shapes and forms from leaves to coastlines. Basic topological and geometrical language to describe and model rough, ("fractal") objects is developed. Relationships between fractal geometry and discrete dynamical systems and chaotic dynamics are emphasized, including symbolic dynamics, stability of attractors, bifurcations and routes to chaos. 

The key ideas are introduced in an intuitive way. The key definitions and theorems are stated but few proofs of theorems are given. However, graduate students will have to attend additional lectures which will provide rigorous mathematical foundations and will be assessed separately from undergraduate students.

In computer laboratory sessions students learn how the mathematical results can be applied in practice by running and modifying simple Python programs. 

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

Upon successful completion of this course, students will have the knowledge and skills to:

1. Be able to construct and analyse a wide range of fractals.

2. Be able to analyse 1-D dynamical systems in terms of attractors, basins and cascades of bifurcations. 

3. Understand how to use fractal geometry to model rough data and natural shapes. 

4. Be familiar with Hutchinson theory of deterministic fractal sets and measures, and be able to prove basic theorems and solve problems in the area. 

5. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from fractal geometry. 

6. Ability to use their deep knowledge and understanding of fractal geometry to formulate responses to complex concrete and abstract problems. 

7. Ability to communicate their understanding and skills in fractal geometry with colleagues and non-experts and apply their knowledge in an occupational situation. 

Recommended Resources

Recommended student system requirements 

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

For more information please see https://www.anu.edu.au/students/systems/recommended-student-system-requirements

Staff Feedback

Students will be given feedback in the following forms in this course:

• written comments
• verbal comments
• feedback to whole class, groups, individuals, focus group etc

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Other Information

The delivery of the course material this year has been adjusted to allow for remote participants. All lecture material will be presented online in the form of short videos, associated notes, and helpful exercises. It will be required that the material be viewed and engaged with by the end of each week. Workshops and Computer labs will be delivered both in-person and online.

Please note, that where there are multiple assessment tasks of the same type, e.g weekly quizzes, a date range is used in the Assessment Summary. The first date is the approximate due date of the first task, the return date is the approximate return date for the final task. Further information is provided in the assessment section of the class summary, and details are provided on the course wattle site.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Lecture material will be delivered online through the edX platform. There will be a series of short videos released each week covering the equivalent of the material that would be delivered in three one-hour in-person lectures. The lecture notes will be published, and small activities provided to test knowledge as you go. Starting on week 3 we will have one workshop per week, and one computer lab every second week. There will be a total of 10 workshops and 5 computer labs. Assignments will be posted on Wattle. Topics covered in week 1: What are fractals? Concepts from analysis: Metric spaces, continuity and completeness, contractions. Affine transformations
2	Topics covered: Topological concepts Iterated function systems (IFSs) Hausdorff metric The completeness of the space of fractals
3	Topics covered: Fixed point theorems Existence of fractal attractors Advanced topics in metric spaces	Assignment 1 due Thursday
4	Topics covered: Code space and mapping to fractals Hutchinson's theorem	Quiz 1 in the workshop session
5	Topics covered: Rigorous definitions of fractal dimension Introduction to dynamical systems	Assignment 2 due Thursday
6	Topics covered: Chaotic dynamics in dynamical systems	Quiz 2 in the workshop session
7	Topics covered: More on dynamical systems Abstract notions of transformation Equivalence and conjugacy	Assignment 3 due Thursday
8	Topics covered: Newton's method	Quiz 3 in the workshop session
9	Topics covered: Fractal interpolation	Assignment 4 due Thursday
10	Topics covered: Julia sets and Mandelbrot sets	Quiz 4 in the workshop session
11	Topics covered: Condensation sets Fractal image compression	Assignment 5 due Thursday
12	Topics covered: Measure theory and fractals	Quiz 5 in the workshop session

Tutorial Registration

Students are to register for the workshops and computer labs through Wattle.

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Homework assignments	30 %	*	*	1,2,3,4,5,6,7
Reading quizzes	7 %	*	*	1,2,3,4,5,6,7
Computer labs	12 %	*	*	1,2,3,4,5,6,7
Midterm exam	16 %	29/03/2021	23/04/2021	1,2,3,4,5,6,7
Final exam	35 %	03/06/2021	01/07/2021	1,2,3,4,5,6,7

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information 
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Integrity . In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Examination(s)

Please note, that where a date range is used in the Assessment Summary in relation to exams, the due date and return date for mid-semester exams indicate the approximate timeframe in which the exam will be held; the due and return date for end of semester exams indicate the approximate timeframe in which the exam will be held and the date official end of Semester results are released on ISIS. Students should consult the course wattle site and the ANU final examination timetable to confirm the date, time and venue of the exam.

Assessment Task 1

Homework assignments

There are five homework assignments due during the session, worth 36 points each (out of 600 total course points). See the Wattle course site for assign dates and deadlines. It is intended that the marked homework will be returned, and solutions posted, within one week after submission.

Homework sets are marked for correctness and clarity of writing. They contain both computational problems and proofs. More advanced proofs will be assigned to HPO/6116 students only.

Due: Thursdays weeks 3, 5, 7, 9 and 11.

Assessment Task 2

Reading quizzes

There are five reading quizzes due during the session, worth 8 points each (out of 600 total course points). See Wattle course site for assign dates and deadlines. It is intended that the marked quizzes will be returned, and solutions posted, within one week after submission.

Due: In workhops weeks 4, 6, 8, 10, 12

Assessment Task 3

Computer labs

There are five computer labs due during the session, worth 14 points each (out of 600 total course points). See Wattle course site for assign dates and deadlines. It is intended that the marked quizzes will be returned, and solutions posted, within one week after submission.

Assessment Task 4

Midterm exam

There is one midterm exam during the session, worth 100 points (out of 600 total course points).

The date range is an general indication of when the mid-semester exam will be held. Please check the course Wattle site and the ANU Examination Timetable to confirm the date, time and location of the end of semester exam.

Assessment Task 5

Final exam

The final exam is worth 210 points (out of 600 total course points). There is a hurdle requirement. You must earn at least 80 points on the final exam to receive a passing mark in the course.

The date range in the Assessment Summary indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the course Wattle site and the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location exam.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. At its heart, academic integrity is about behaving ethically, committing to honest and responsible scholarly practice and upholding these values with respect and fairness.

The ANU commits to assisting all members of our community to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to be familiar with the academic integrity principle and Academic Misconduct Rule, uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with.

The Academic Misconduct Rule is in place to promote academic integrity and manage academic misconduct. Very minor breaches of the academic integrity principle may result in a reduction of marks of up to 10% of the total marks available for the assessment. The ANU offers a number of online and in person services to assist students with their assignments, examinations, and other learning activities. Visit the Academic Skills website for more information about academic integrity, your responsibilities and for assistance with your assignments, writing skills and study.

Online Submission

You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. Unless an exemption has been approved by the Associate Dean (Education) submission must be through Turnitin.

Hardcopy Submission

For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item. Late submission is not accepted for take-home examinations.

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

All assignments are to be returned via Wattle.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Resubmission is permitted up to the deadline.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students