An Exploration of Symmetry (Eligible for NTU Credit)

Nanyang Technological University, Singapore

[IF YOU ARE LOOKING FOR THE NO-CREDIT ON-DEMAND VERSION OF THIS COURSE PLEASE GO TO]. Learn how to identify symmetrical forms and appreciate their importance in nature, art, architecture, crystallography and technology. Understand symmetry quantitatively, recognize its role in beauty and design, and appreciate its function in our everyday life. The level of difficulty is intermediate-to-hard with a workload is 7-10 hrs/week. This MOOC is for credit, and students who obtain a Verified Certificate by submitting and authenticating at least 8 out of the 9 assignments with an overall mark of more than 60%, earn 3 Academic Units (AU) that can be directed towards either an Unrestricted Elective (UE) or General Elective (GE-STS) subject at NTU. The usual NTU examination procedure requires student identification through webcam plus keystroke patterning for each assignment submission with a no-tolerance policy towards cheating ( and Students who are found to cheat, tamper with or falsify grades, or collude on assignments will be denied credits. It is not necessary to be enrolled at NTU to be awarded 3 AU and once received the AU remain valid 3 years. Because this is a for-credit course students must join the Signature Track stream within the first 2 week add-drop period (up to 22 January 2016, 2359 hrs) in accordance with NTU rules.

Nanyang Technological University and Coursera are proud to offer Beauty, Form & Function: An Exploration of Symmetry that caters to both non-specialists via a non-academic credit on-demand version ( and specialists through this academic credit fixed-session version.  Below can be found the:

Syllabus • Grading Scheme • Recommended Readings •

Required Background Essential Resources Student Responsibility • FAQs

Symmetry is everywhere. In the grand scheme of things it is the blueprint by which the universe operates. We see symmetry in nature, art, architecture, science and engineering. This course explores the Beauty, Form and Function of Symmetry in common objects, then progresses to investigate tiling and tessellation, gems and crystals, and finally its indispensable role in technology.  Symmetry and asymmetry are also key factors that ensure the successful reproduction of plants and animals. For example, the bilateral symmetry of orchids is widely admired and this geometry is essential for insects to pollinate these flowers.

Duration: 8 weeks

Time Commitment: 7-10 hours per week

Target Group: Scientists and Engineers (current and prospective NTU students)

Recognition: Verified Certificate through Signature Track enrollment

NTU Academic Units: Verified Certificate Students earn 3 Academic Units

NTU Course Type:  Unrestricted Elective or General Elective

This course gives you the means to address several fundamental questions.

  • What tags do we recognize in decorative arrangements?
  • What guidelines shape the patterns in the world around us?
  • What functions do these symmetrical patterns serve?
  • How are drug and materials design engineered through atomic scale symmetry?
  • How is the internal arrangement of atoms reflected by crystal faceting?
  • How does symmetry manipulation create polytypes, polymorphs and polysomes?

To amplify these ideas, you will undertake field-exercises and be introduced to specialists - botanists, artists, geomancers, historians, scientists and engineers - that work with symmetry, and who will provide their personal insights into its ‘magic’ and impact on their disciplines.

The option of earning AUs is open to all students. It is not necessary to be already enrolled at NTU.  Once earned, the AUs remain valid for 3 years.

Course Difficulty

This is a challenging course with 65% of students rating the material as Difficult to Very Difficult, but many find the subject matter stimulating.


The course content is released in two week blocks and consists of approximately 1 hr of lectures and tutorials per week. There are specific windows for uploading homework and undertaking peer-to-peer marking. These are detailed in the downloadable course timetable.

Part I: Basic Concepts of Symmetry

  • Week 1: ‘A World of Symmetry’ will refresh your intuitive appreciation of symmetry in furniture, botany and common objects and introduce the mathematics of point symmetry.

  • Week 2: ‘Tiling and the Asymmetric Unit’ begins by examining tessellations in architecture that will illustrate key concepts in plane symmetry including the asymmetric unit.  ‘Chirality’ discusses this key concept in symmetry and its impact on drug design and delivery.

Part II: Plane (2D) and Space (3D) Symmetry

  • Week 3: ‘Escher and Graphic Design’ introduces symmetry operations in 2 dimensions and the difference between ‘primitive’ and ‘centered’ unit cells are illustrated using the art of Escher. ‘Nets’ examines the role of symmetry in Islamic architecture and history in the context of regular networks.

  • Week 4: ‘Space Symmetry’ expands the symmetry operations to include 3 dimensional motion.  The methodology for reading and using the symmetry diagrams of the 17 plane groups and 230 space groups is introduced.

Part III: Symmetry in Crystals

  • Week 5: ‘Crystal Forms and the Beauty of Minerals’ looks the external shape of mineral crystals and we learn about crystal power by visiting a geomancer. The manner in which crystal faceting reflects the internal atomic arrangements is described. ‘Planes, Directions and Unit Cell’ introduces the mathematics for describing perfect crystal structures.

  • Week 6: ‘Platonic Solids and Atomic Bonding’ shows that regular geometrical shapes can be derived from folding plane nets into polyhedra that often describe atomic clustering and also the appearance of defects or asymmetry that are important aesthetically and technologically.  A laboratory tour will show the experiments used to extract crystallographic information.

Part IV: Symmetry and Technology

  • Week 7: ‘Tailoring Crystals for Technology’ brings together chemistry and crystallography, shows how they are inter-related, and how they are adapted to engineer designer technological materials.  A clean room fabrication facility provides real world context for exploiting chemically- and physically-driven symmetry.

  • Week 8:  ‘Looking for Broken Symmetry - Atomic Scale Imaging’ looks beyond 3 dimensional symmetry to higher dimensional repetitions found in incommensurate crystal structures. ‘Deceptive Perception - Symmetry & Furniture Design’ returns to the everyday appearance of symmetry that can now be explored analytically.

At the conclusion of this study, you will possess the tools and the motivation to look at symmetry with even greater delight, wonder and appreciation!

Recommended Background

Mathematics at GCE Advanced Level or International Baccalaureate Diploma or equivalent to junior high school. In addition, high school or undergraduate (freshman) chemistry is very helpful.


Essential Resources Required

This is an undiluted university course and resources equivalent to an undergraduate student at NTU are essential to complete these studies.  These include:

  • Access to a Windows or Apple personal computer:  While lectures can be viewed on tablets and smartphones it is not possible to run the specialized programs on these architectures.  There can be no troubleshooting of compatibility problems.

  • Access to a smartphone with camera or a digital camera:  The field-work exercises involve photographing objects in your environment and analysing their symmetry.  Submission of these graded exercises is essential to obtain course recognition.

  • Access to Microsoft Office or Open Office: Homework must be prepared in PowerPoint or Word using the pptx and docx templates provided, converted to a pdf and uploaded. If you have have an older version of Powerpoint and Word you can convert the pptx and docx files to doc and ppt files at and

Suggested Readings

In a topic as broad as Symmetry no single text can do justice to the field.  Therefore, the notes and supplementary materials provided in this course strive to be self-contained.  There are also many web resources to access.

Video book reviews are provided to get a sense of the breadth of publications available.

Quite academic treatments can be found in the research papers of the Open Access journal Symmetry.

For those who wish to study conventionally the following texts, that are widely available in public libraries, are recommended.

Visual Symmetry by Magdolna Hargittai and István Hargittai
ISBN 978-981-283-531-4

Symmetry is as simple or as complicated as we are ready to absorb it in everything around us. From flowers to bridges, buildings, coke machines, and snowflakes; from molecules to walnuts, fences, pine cones, and sunflowers; from music to children's drawings; from hubcaps to bank logos, propellers, wallpaper decorations, and pavements, we recognize it if we walk around with open eyes and an open mind. This book provides aesthetic pleasure and covert education, immersing the reader in both the familiar and the unknown and leading always to unexpected discoveries.  The authors, world-renowned scientists, have already produced a dozen books on symmetry for professionals as well as lay persons, for grownups as well as children, in English, Russian, German, Hungarian, and Swedish languages. They provide this attractive account of symmetry in few words and many, as many as 650, images in full colour from the most diverse corners of our globe. An encounter with this book will open up a whole new experience for the reader, who will never look at the world with the same eyes as before.

Crystals and Crystal Structures by Richard J. D. Tilley
ISBN: 978-0-470-01820-0

 Crystals and Crystal Structures is an introductory text for students and others who need to understand the subject without necessarily becoming crystallographers. Using the book will enable students to read scientific papers and articles describing a crystal structure or use crystallographic databases with confidence and understanding.  Reflecting the interdisciplinary nature of the subject the book includes a variety of applications as diverse as the relationship between physical properties and symmetry, and molecular and protein crystallography. As well as covering the basics the book contains an introduction to areas of crystallography, such as modulated structures and quasicrystals, and protein crystallography, which are the subject of important and active research.

Course Format

The class is divided into 4 major Parts and will consist of:

  • Lecture videos about 9 - 15 minutes in length. Most lectures contain 1-2 in video quiz questions for self-assessment. These are not graded.
  • Multiple Choice Question (MCQ) Quizzes at the end of each Part which will be graded.  Model MCQ and detailed answers are available for revision.
  • 3 open - ended assignments with peer-to-peer assessment.  To guide you in this homework, student focus-group recordings show how undergraduates tackled these assignments.
  • 2 learning logs.  These logs are designed to help you reflect on what you have learned and share with fellow students.
  • 4 MCQ quizzes  (20% of total grade)
  • 3 peer-graded assignments (72%)
  • 2 peer-graded learning logs (8%)
The pass mark is >= 60% of the maximum possible score averaged across all graded assignments.

At least 8 out of the 9 assignments must be submitted and authenticated to gain 3 Academic Units (AU).

Student Responsibility

It is the responsibility of students to ensure assignments are submitted on time. Note that times are for Singapore and we are unable to entertain late submission.

  • Use the international clock provided near the top of the navigation menu to check the time difference with your locality.
  • Download the course timetable and print it out to ensure you do not miss homework deadlines.
  • Do not leave your homework until the last minute.
  • For the graded multiple choice question (MCQ) sets either 1 or 2 attempts are allowed. When 2 attempts are made the highest mark is taken towards grading.
  • For peer-to-peer assignments submit 1-2 days early in case of internet failures.
  • While homework submission is not compulsory it is very difficult to pass this course without attempting every assignment.
  • For SIGNATURE TRACK STUDENTS at least 8 out of the 9 assignments must be submitted and authenticated to gain 3 Academic Units (AU).
  • For SIGNATURE TRACK STUDENTS authentication through a webcam plus keystroke recognition is always required for the assignment to be recognized.  ALWAYS  USE A COMPUTER WITH WEBCAM.
  • For SIGNATURE TRACK STUDENTS a live portrait captured by webcam is compulsory for all submitted assignments.  Capturing student ID cards, passport photos, drivers licences etc is not acceptable.
  • All NTU MATRICULATED SIGNATURE TRACK STUDENTS must enrol using their NTU email to avoid paying the course fee and allow transfer of academic credits. NTU students that enrol with Coursera using a non-NTU email must change it to their university email account. Please follow the steps at to change the email address.
  • SIGNATURE TRACK AUTHENTICATION IS  NOT SUPPORTED ON MOBILE DEVICES such as iPhone and iPads. Always submit your homework from a desktop or laptop computer to avoid any problems with authentication.
  • CHEATING AND UNETHICAL BEHAVIOUR ARE UNACCEPTABLE and students must comply with the Coursera Honour Code ( and NTU Examination Policy (  Failure to abide by these requirements will, at minimum, lead to loss of Academic Units and exclusion from all NTU MOOC, with egregious cases resulting in suspension or expulsion from NTU.
  • For peer-to-peer evaluated assignments PLAGIARISM WILL NOT BE TOLERATED and whether detected by students or the course manager will automatically lead to a score of zero for that assignment.
  • PEER MARKERS HAVE A DUTY-OF-CARE to mark according to the rubrics and follow all instructions.  Giving marks for obvious non-compliance with rubric instructions or clear plagiarism will lead to an automatic loss of academic units.
  • If you are sharing a computer with other students, make sure you log out to avoid your homework and grades being over-written on the Coursera server by another student's uploads and answers.  Once lost, your work cannot be restored.
  • Matriculated NTU students must enrol using the secure website and their NTU email account to ensure seamless transfer of Academic Units to their degree transcript.
  • Peer-to-peer marked assignments will gain a score that is the mean taken from either 2 or 5 student assessors.  It is essential that students mark carefully against the simple rubrics provided.


 1. Is this a hard course?
  • The course  is definitely challenging.  The content and assessments replicate a usual NTU undergraduate course.
  • Parts III and IV contain lectures of general interest, but there is a step change in difficulty compared to Parts I and II, and the second half of the course is appropriate for science and engineering students.

2. How do I gain a Statement of Accomplishment?
  • Complete Parts I to IV and obtain a score of >= 60%.

3. How do I gain NTU Academic Units (AUs)?
  • Enrol in the Signature Track and earn a Verified Certificate.
  • Complete Parts I, II, III and IV and obtain a score of >= 60%.
  • All SUBMITTED ASSIGNMENTS MUST BE AUTHENTICATED using keystroke recognition and a live webcam image.  Images of students IDs, passports, drivers licenses etc. are not acceptable.
  •  SIGNATURE TRACK AUTHENTICATION IS NOT SUPPORTED ON MOBILE DEVICES such as iPhone and iPads. You can watch video lectures, participate in the forums, and save drafts of your quiz and assignment answers from a mobile device, but it is ESSENTIAL THAT HOMEWORK IS SUBMITTED FROM A DESKTOP OR LAPTOP COMPUTER to avoid problems with authentication.

4. When will I receive my Statement of Accomplishment?
  • All Statements of Accomplishment and Verified Certificates are issued approximately 4-6 weeks after completion of the 8 week course.
  • Prior to issuing Verified Certificates checks for cheating, plagiarism and compliance with the Coursera Honour Code and NTU Regulations will be undertaken.
  • NTU has a ZERO TOLERENCE POLICY to breaches of the honour code and students who are found not to comply will, at minimum, be denied Academic Units with more serious cases leading to suspension or expulsion from NTU. 

5. I am already an NTU student - why should I join this MOOC?

  • You will be able to pursue this course without having to fit it into your class schedule in NTU.
  • You will be guaranteed a place in the course as there is no limit to student numbers.
  • You can study when you are on attachment or on overseas exchange.
  • This MOOC is offered every semester and you can attempt multiple times without penalty.

6. Who is eligible to gain NTU Academic Units from this MOOC?
  • Whether already an NTU student, or considering joining NTU, you can earn 3 AU by obtaining a Verified Certificate with Distinction on the Signature Track.
  • You will enjoy learning without worrying about your results as the MOOC is graded Pass/Fail, and there will be no impact on your cumulative grade point average (CGPA) and the classification of your degree.
  • You can discontinue the course at any time.

7. I am an NTU student and unsure whether to take this course for credit - how much time do I have to decide?

  • This MOOC operates in a similar way to normal face-to-face courses with an "Add" period.
  • You can try out the course for up to two weeks after commencement as a normal student before converting to Signature Track to obtain a Verified Certificate.
  • If you miss the two week window to enrol as a Verified Certificate student, then change your mind, you will have to re-enrol the following semester.

8. Is there a course fee?
  • The course may be taken by anyone free-of-charge to earn a Statement of Accomplishment.
  • Those not currently matriculated NTU students and who may wish to transfer credit for the course when they subsequently enrol at NTU must take the Signature Track for which a fee of USD49 is payable to Coursera which will issue the Verified Certificate. 
  •  Matriculated full-time NTU undergraduate students will be able to sign-up for the Signature Track without paying a fee and be able to transfer the credits earned if they passed the course.

10. How does an NTU student enrol to the Signature Track of the course?
  • Please ensure you have either one of the picture IDs (National ID card, Passport, Driver’s license) on hand before enrolling.
  •  Login to Teamsites via with your network account.
  • Select from the line “Registration to Symmetry MOOC's Signature Track” 
  •  For NTU students who have earlier registered for the course on the free track using their NTU email address choose Option 1a
  • For NTU students newly enrolling choose Option 1b
  • Follow the instructions to sign up for the waived-fee Signature Track and complete the online form to indicate your choice of credit transfer.
  • Please ensure that you create your Signature Profile immediately after you have signed up for Signature Track. This is to ensure that your Signature Profile is link to your identity to each piece of graded coursework that you submit for your course. Also, when you successfully complete the course, you’ll earn a Verified Certificate.

11. How do full time NTU students transfer the credits earned?
  • Wait for email from Office of Academic Services (OAS) and the Centre for IT Services (CITS).
  • Follow the instructions in the email to indicate whether you are transferring credits to fulfil your GER-STS General Elective or UE Unrestricted Elective requirement.
  • A reminder will also be sent at the end of the course. 
  • As long as you pass the course, earn a Verified Certificate with Distinction through the Signature Track, and remain a full-time matriculated NTU undergraduate student at the point of completion, your credits will be automatically transferred towards fulfilling the requirements of your NTU degree programme.

12. How do new students transfer the credits earned?
  • You should apply to your School for the credits to be transferred in the first semester of your enrolment to NTU. 
  • You must produce the Verified Certificate with Distinction issued by Coursera for the Signature Track, at the point of application.
  • Only Verified Certificates with Distinction issued within 3 years of their award by Coursera are accepted by NTU for transfer of 3 Academic Units (AUs).
  • The courses transferred will be recorded in your transcript.
  • Please be reminded that you must comply with the following requirements for graduation:
       - Minimum candidature period for the degree       - Minimum AU of letter-graded courses        - Minimum years of study at NTU (i.e. residential period).
  • 11 January 2016, 8 weeks
  • 10 August 2015, 8 weeks
  • 12 January 2015, 8 weeks
  • 11 August 2014, 8 weeks
  • 17 February 2014, 8 weeks
Course properties:
  • Free:
  • Paid:
  • Certificate:
  • MOOC:
  • Video:
  • Audio:
  • Email-course:
  • Language: English Gb


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