Beauty, Form & Function: An Exploration of Symmetry

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Course Date: 11 August 2014 to 06 October 2014 (8 weeks)

Price: free

Course Summary

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 depending on the Track chosen. Workload: 4-6 hrs/week for Track 1 (5 weeks - Statement of Accomplishment); 7-10 hrs/week for Track 2 (8 weeks - Statement of Accomplishment with Distinction). This MOOC is for credit and students who obtain a Verified Certificate with Distinction and complete Track 2 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. For Verified Certificate students the usual NTU examination procedure requires student identification through webcam plus keystroke patterning for each assignment submission. 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 26 January 2015, 2359 hrs) in accordance with NTU rules.

Estimated Workload: 4-10 hours/week

Course Instructors

Tim White

Tim White is a Professor of Materials Science at Nanyang Technological University where he teaches crystallography, electron microscopy and X-ray diffraction to undergraduate and graduate students.  His research encompasses mineralogy, clean energy materials, environmental remediation, nuclear waste disposal and catalysis, where synchrotron X-ray and neutron scattering methods probe atomic-scale symmetry.  In 2006, he introduced a suite of teaching modules for materials scientists called On-line Micro- and Nano-characterisation Instruction (OMNI) that were extended by the Australian Learning and Teaching Council to create MyScope, a national curriculum for blended learning in microscopy and imaging ( ). Tim developed the first comprehensive on-line course at NTU called Symmetry and Crystals from which this MOOC on Beauty, Form and Function: An Exploration of Symmetry evolved.

Course Description

Nanyang Technological University and Coursera are proud to offer Beauty, Form & Function: An Exploration of Symmetry as the first dual-track MOOC that caters to both specialists and non-specialists, and can lead to full academic credits.

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.

Track 1 Overview (PARTS I and II)

Duration: 5 weeks

Time Commitment: 4-6 hours per week

Target Group: Teachers, artists, architects and anyone with a passion for symmetry

Recognition: Certificate of Accomplishment

This Track 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?

Track 2 Overview (PARTS I, II, III and IV)

Duration: 8 weeks

Time Commitment: 7-10 hours per week

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

Recognition: Certificate of Accomplishment with Distinction

NTU Academic Units: Verified Certificate Students earn 3 Academic Units

NTU Course Type:  Unrestricted Elective or General Elective

This Track explores the impact of symmetry in technology through:

  • Describing crystals at the atomic scale to understand drug and materials design
  • Understanding how the internal arrangement of atoms is reflected by crystal faceting
  • Appreciating symmetry modification through polytypism, polymorphism and polysomatism.

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.

Successful completion as a Verified Certificate student earns 3 academic units (AUs) for an Unrestricted Elective or General Elective subject at NTU.

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.


  • Will I get a Statement of Accomplishment after completing this class?
    Yes. Students who successfully complete the class will receive a Statement of Accomplishment from Coursera. 
  • What resources will I need for this class?
    A Smartphone or a digital camera, and the time to read, write, and discuss, with your peers.
  • What is the coolest thing I'll learn if I take this class?
    The world will never look the same again!
    Your eyes will be opened to the wonders and beauty of symmetry in ways you cannot imagine. 


The entire course content is available from the course opening 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.

Part I: Basic Concepts of Symmetry (Track 1 and Track 2)

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

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

Part II: Plane (2D) and Space (3D) Symmetry (Track 1 and Track 2)

  • Week 3: ‘Plane Groups’ describes the symmetry operations in 2 dimensions and the difference between ‘primitive’ and ‘centered’ unit cells illustrated using the art of Escher. ‘Nets’ examines the role of symmetry in Islamic architecture and history in the context of regular and irregular 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 (Track 2)

  • Week 5: ‘Atoms in Crystal Structures’ looks the external shape of mineral crystals by visiting a geomancer. The manner in which crystal faceting reflects the internal atomic arrangements is described. ‘Polyhedral Properties’ shows that regular geometrical shapes can be derived from folding plane nets to create Platonic and Archimedean solids that often describe atomic clustering.

  • Week 6: ‘Planes, Directions and Unit Cell’ introduces the mathematics for describing perfect crystal structures 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 (Track 2)

  • Week 7: ‘Trends and Substitutions’ brings together chemistry and crystallography, shows how they are inter-related and provide the means to engineer designer technological materials.  A clean room fabrication facility provides real world context for exploiting chemically-driven symmetry.

  • Week 8:  ‘Building Complex Structures’ looks beyond 3 dimensional symmetry to higher dimensional repetitions found in incommensurate crystal structures. ‘Technological and Design Implications’ returns to the everyday appearance of symmetry that can now be explored quantitatively.

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!


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

  • Lecture videos about 9 - 15 minutes in length. These contain 1-2 in video quiz questions as part of self assessment. This will not be graded.

  • Multiple Choice Question (MCQ) Quizzes at the end of each Part which will be graded.

  • 3 open - ended assignment which will involve peer to peer assessment.
  • 1 learning log

TRACK 1: Certificate of Accomplishment (PARTS I & II only)
• 2 MCQ quizzes 
• 2 peer-graded assignments
• 1 learning log

TRACK 2: Certificate of Accomplishment with Distinction or Verified Certificate (PARTS I, II, III & IV, equivalent 3AU NTU)
• 4 MCQ quizzes 
• 3 peer-graded assignments
• 2 learning log

For both Tracks, you have to obtain >= 60% of the maximum possible score.

Suggested Reading

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 reseach 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.

Track 1: 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 color 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.

Track 2:
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 Workload

4-10 hours/week

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