DISCOVERING SYMMETRY WITH KALEIDOTILE

SYMMETRIES IN THE PLANE

Students will find reflective symmetries of planar patterns.
Students will understand the meaning of the symmetry group names in the KaleidoTile program.
Students will find the angle sum of triangles in the plane.

KaleidoTile software
protractors, if you are using the Challenge Topic
Discovering Symmetry with KaleidoTile: Symmetries in the Plane Activity Sheet

Review the definition of line of reflection: When you place a mirror on a line of reflection, the image in the mirror is precisely the portion of the pattern lying behind the mirror.

Review the construction of regular polygons with two mirrors and tilings of the plane using three mirrors, as described in Reflecting on Reflecting 3. Make sure the following ideas are discussed:

• Review the symmetries of the regular polygons, including dihedral symmetry and lines of reflection.
• The base triangle of a kaleidoscope is the triangle cut off by the three mirrors.

If you wish to include the Challenge Topic, discuss the angle sum of a triangle in the plane and review angle measure in degrees and radians, as appropriate.

• Have the students load the KaleidoTile program.
• Have students work through the exercises on the Discovering Symmetry with KaleidoTile: Symmetries in the Plane Activity Sheet in groups of 2-4.
• The Challenge Topic emphasizes measurement of angles in the plane, and should be assigned only if students have covered this material.
• Allow the students to determine their own methods for measuring angles. Protractors may be useful, but not necessary.
• Angles can be measured in either degrees or radians.

Discuss the following points with the class:
• KaleidoTile's dual edges represent the lines of reflection of planar patterns.
• There is an exact correspondence between the construction of planar patterns using mirrors and the corresponding patterns on KaleidoTile. KaleidoTile's base triangle corresponds to the base triangle of the mirrors and the dual edges correspond to the mirrors.

Instruct the students to write answers to the following questions:

• What do the symmetry group names in KaleidoTile mean?
• Predict the characteristics of a pattern in the symmetry group (2, 3, 5).

Use the program Kali to examine dihedral symmetry and the symmetries of regular polygons.