SYMMETRY AND TILINGS: AN EXPLORATION

This project involves some reading, learning and interactive experimenting with the mathematical concepts of tilings and symmetry.  Both of these topics have long and illustrious histories; symmetry, in particular, has been used in art and architecture in all major civilizations and cultures.  
 
The purpose of this project is to allow you to gain some knowledge and familiarity with these fundamental aspects of geometry.  To do so, you will need to visit the Web site http://www.scienceu.com/geometry where you will find some nicely written and well presented materials to facilitate your learning.
 
The ScienceU Geometry Center materials originated through a National Science Foundation grant at the University of Minnesota, and the Geometry Center there hosted many workshops, created the Web-based materials you are about to see, and generally promoted geometry to the public-at-large.  ScienceU began hosting some of the original materials after the NSF funding ended.   
 
Assignment:
 
Part 1:
When you arrive at the ScienceU Geometry Center, http://www.scienceu.com/geometry, please click on the link for Symmetry and Tiling and read the pages that follow in the article entitled Tilings and Tesselations.  In the section entitled The Seventeen Wallpaper Tilings, be sure to click on the icon for each type of tiling to see an animation and more detailed description.  As you read, please answer the following questions:
 
1. What is a generating set for a tiling?
2. What is a fundamental domain for a tiling?
3. Describe the four kinds of symmetry (isometries).
4. Define the symmetry group of a tiling.
5. How many different wallpaper tilings are there?
6. What are the basic construction elements of a Penrose tiling?
 
Part 2:
On the Geometry Center page, click on the link to “Kali,” an interactive tiling software.  When you arrive there, select a wallpaper group and then click and drag the mouse inside the large white drawing window.  Double click to stop drawing.  Then you may click and drag the mouse inside the window again to add more detail in another color; double click to stop drawing, and so on… 

Experiment with different wallpaper groups, frieze groups and rosette groups to get a feel for the effects produced by the various prototiles.