Symmetry and Tilings: An Exploration
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.
CollapseLearning 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.
- ScienceU-Geometry Center
Tutorial on Symmetry and Tilings; interactive Java symmetry sketching applet
http://www.scienceu.com/geometry/ - MERLOT description of this resource
http://www.merlot.org/artifact/ArtifactDetail.po?oid=1010000000000007722
- ScienceU-Geometry Center
- 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.
Assessment
I grade the assignment questions on the percentage of correct answers; the activity grade represents approximately 1-2% of a semester grade.
References and Resources
- ScienceU-Geometry Center
The Geometry Center contains a variety of appealing material, both textual and visual, including interactive exhibits, simulations, graphics software and a library of reference materials pertaining to geometric tilings and polyhedra.
MERLOT description of "ScienceU-Geometry Center" site.
Direct link to the "ScienceU-Geometry Center" site. - Symmetry and Pattern-The Art of Oriental Carpets
This resource provides materials on the symmetries involved in Oriental rug designs; the Rug Gallery images can be used for further symmetry and tiling analysis.
MERLOT description of "Symmetry and Pattern-The Art of Oriental Carpets" site.
Direct link to the "Symmetry and Pattern-The Art of Oriental Carpets" site. (Access path: Rug Gallery)