KaleidoMania! applies transformational geometry to create visually attractive and interactive screenviews.

Exploring The Software:

Begin by going to the Investigations menu and opening "Instructions". You will want to practice manipulating the fundamental region in the source box.
• Change symmetries using the Symmetries button and matrix.
• Under the Pattern menu, explore "Show Grid" and "Show Fund Rgn".
• Under the Investigations menu, check out "The Tours", the "Symmetry Explorations", the Cyclic and Dihedral "Experiments", the "Isometry Puzzle" and the "Wallpaper Game".
• Explore the results of using the "Generate", "Slide Show" and "Full Screen" buttons.
• Use File > "New Source" and the palette of drawing tools to create some dramatic interaction with pattern generation.

Thinking About the Mathematics:

Understanding the notion of fundamental region is critical to the generation of the patterns.

1. Determine which patterns have fundamental regions that are:
   • triangular (regular or right)
   • rectangular
   • square, or
   • rhombic.

2. How is the shape of a fundamental region related to the symmetry type?
3. What is the meaning of the symbols on the edges and corners of fundamental regions as they appear in the source box?
4. Based on your explorations, how would you express in words exactly what a fundamental region is or does?