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# Symmetry and Tilings: An Exploration

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

#### Summary

Students are directed to read through a Web-based tutorial on Symmetry and Tilings in the form of an short and colorful article entitled Tilings and Tesselations; afterwards, they answer several questions on tilings (tessellations), tiling terminology, types of symmetry (isometries), periodic tilings and Penrose tilings.

In addition, they are given opportunity to use an interactive Java applet in which various types of symmetry can be sketched and explored in the form of wallpaper groups, frieze groups and rosette groups. In a subsequent think-pair-share activity or write-pair-share activity, they analyze some tilings and apply their newly obtained knowledge.

## Learning Goals

To enable students to:
• gain an understanding of the basic terminology involved in symmetry and tilings (tessellations)

• recognize the difference between periodic, non-periodic and Penrose tilings

• gain an understanding of the four types of symmetry (isometries) and their application in tiling construction

• develop their understanding of tilings through the use of an interactive Java sketching applet

• apply their knowledge and understanding in analyzing given tilings

## Context for Use

This activity can be used in a Geometry class as an introduction to the geometric concepts of symmetry and tilings or in a Liberal Arts math class to demonstrate the connections between mathematics and graphic art.

The activity will require approximately 45-60 minutes outside of class depending on how long students spend on the interactive Java applet. The questions are made available via an online quiz or dropbox (WebCT, Angel or other LMS) and students submit their answers prior to the related class discussion. The in-class follow-up activity takes approximately 15-20 minutes.

## Description and Teaching Materials

Activity description
• Students are directed to read through a Web-based tutorial on Symmetry and Tilings outside of class. As they read, they answer several assignment questions (Rich Text File 14kB Jul28 06) concerning the material and submit their answers online via a quiz or dropbox.

• During the following class, the instructor presents some tiling examples and an application (Acrobat (PDF) 112kB Jul28 06) and asks students to analyze the tilings and identify the prototiles, the generating set for each tiling, and fundamental domains. This is done collectively via a think-pair-share activity or write-pair-share activity.

## Teaching Notes and Tips

The ScienceU tutorial is effective and well written but doesn't provide exercises in which students can practice their newly obtained knowledge. When presented with an actual tiling and asked to identify the prototiles and the generating set, many students are somewhat at a loss and need more examples to help them see what is involved. After collectively analyzing several examples of tilings presented by the instructor, students are more able to correctly analyze the application tiling presented.