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. 

Examination Material or equipment

None.

Required Resources

Reading and suggested resources will be given on Canvas over the course of the term. There are no required resources.

Recommended Resources

There are a variety of online platforms you will use to participate in your study program at ANU, across all of your courses. These could include videos 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/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

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). Feedback can also be provided to Course Conveners and teachers via the Student Experience of Learning & Teaching (SELT) feedback program. SELT surveys are confidential and also provide the Colleges and ANU Executive with opportunities to recognise excellent teaching, and opportunities for improvement.

Other Information

MATH6116 students will be required to give more advanced proofs, beyond what MATH3062 students will be expected to know, throughout the course. These more advanced proofs will appear on the in-class assessments and final exams, but not during the workshops. However, the distribution of points for the various assessment categories is identical to that for the MATH3062 version of the course.

Use of Artificial Intelligence (AI): invigilated assessment tasks prohibit the use of electronic devices, and no AI usage is allowed in such tasks. Because all assessment is in person, there will not be AI use permitted during any assessment. AI use is permitted for the non-assessed homework.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Weeks 1-3: Metric spaces: the analysis needed to study fractals.	Workshops start in Week 2: there will be weekly workshops with assessment to be done during the workshop. There will be weekly homework assigned but not assessed.
2	Weeks 4-6: Group actions: the algebra needed to study fractals.	There will be one in class assessment (based on the homework) at the beginning of week 4.Workshops continue. Weekly homework (not assessed) continues.
3	Weeks 7-12: Fractals and dynamical systems - a study via examples.	There will be a second in class assessment (based on the homework) during week 8.Workshops continue.There will be a final exam in the standard university exam period, after Week 12.

Tutorial Registration

Students are to register for the workshops through MyTimetable. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Learning Outcomes
In class assessment (based on homework)	40 %	1,2,3,4,5,6,7
Workshop assessment	10 %	1,2,3,4,5,6,7
Final Exam	50 %	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 Integrity Rule before the commencement of their course. Other key policies and guidelines include:

• Academic Integrity Policy and Procedure
• Student Assessment (Coursework) Policy and Procedure
• Extenuating Circumstances Application
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure
• Code of practice for teaching and learning

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 Skills website. 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 ‘Canvas’ 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)

Students should consult the course Canvas site and the ANU final examination timetable to confirm the date, time and venue of the exams and the in-class assessments.

Assessment Task 1

In class assessment (based on homework)

There are two such assessments, for a total of 40 percent of the final mark. See the course Canvas site for assignment dates and deadlines. It is intended that these will be returned as quickly as possible after submission. To help prepare for these assessments, regular homework will be set (but the homework does not directly contribute to the course mark). It is students' responsibility to work through the homework problems that have been set and ensure they know how to solve them (they are not turned in for marking). If students wish to discuss solutions to homework problems they are encouraged to attend consultation hours. More advanced questions will be assigned to MATH6116 students only.

To be held in weeks 4 and 8.

Assessment Task 2

Workshop assessment

Every workshop will have an assessment item to be completed jointly with other students but written up and turned in individually by the students during the workshop.

These assessment items will be worth a total of 10% of the final mark, and students will not have to attend every workshop in order to obtain full marks. Details will be given in class and on the course Canvas site.

Assessment Task 3

Final Exam

There will be an in-person final exam, worth 50% of the final mark. More advanced proofs will be assigned to MATH6116 students only.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. The University’s students are an integral part of that community. The academic integrity principle commits all students to engage in academic work in ways that are consistent with, and actively support, academic integrity, and to uphold this commitment by behaving honestly, responsibly and ethically, and with respect and fairness, in scholarly practice.

The University expects all staff and students to be familiar with the academic integrity principle, the Academic Integrity Rule 2021, the Policy: Student Academic Integrity and Procedure: Student Academic Integrity, and to uphold high standards of academic integrity to ensure the quality and value of our qualifications.

The Academic Integrity Rule 2021 is a legal document that the University uses to promote academic integrity, and manage breaches of the academic integrity principle. The Policy and Procedure support the Rule by outlining overarching principles, responsibilities and processes. The Academic Integrity Rule 2021 commences on 1 December 2021 and applies to courses commencing on or after that date, as well as to research conduct occurring on or after that date. Prior to this, the Academic Misconduct Rule 2015 applies.

 

The University commits to assisting all students to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. All coursework students must complete the online Academic Integrity Module (Epigeum), and Higher Degree Research (HDR) students are required to complete research integrity training. The Academic Integrity website provides information about services available to assist students with their assignments, examinations and other learning activities, as well as understanding and upholding academic integrity.

Online Submission

All assessment is in-person; if some rare exception is made to allow online submission in specific circumstances, then you will be required to agree to a declaration as part of the submission of, that will record your understanding of ANU academic integrity principles. Please keep a copy of any online submission for your records. MATH6116 does not use Turnitin, having been granted an exemption.

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. Hardcopy assignments will be scanned and uploaded to Gradescope.

Late Submission

Late submission of assessment tasks without an extension will not be permitted.

Referencing Requirements

The Academic Skills website has information to assist you with your writing and assessments. The website includes information about Academic Integrity including referencing requirements for different disciplines. There is also information on Plagiarism and different ways to use source material. Any use of artificial intelligence must be properly referenced. Failure to properly cite use of Generative AI will be considered a breach of academic integrity.

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.

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 Accessibility 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 supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents all ANU students