Up: The Shape of Space Curriculum Materials: Symmetry and Tiling

Symmetry and 2-Dimensional Space

The Shape of Space Curriculum Materials

Objectives

Students will be able to relate tiling to fundamental domains.

Materials

Symmetry and 2-Dimensional Space Activity Sheet

Introduction

Show the students a repeating pattern and ask what could be a representation of the pattern. Where do students find a repeating pattern in this design? What is the smallest stamp one could make which could generate this pattern? Is there more than one option for the making of this stamp?

Notice how a pattern is repeated. This is a tiling, a pattern that covers an area with no gaps and no overlaps. If we transfer this tiling to a transparency, it is possible to slide the transparency a certain distance until the transparency exactly matches the tiling everywhere.

In a periodic tiling of the plane you can identify a fundamental domain. A fundamental domain is the smallest tile with which you could cover the plane by using some combination of glides, rotations, reflections, and glide reflections. Different combinations of these operations produce different tilings.

Activity

Hand out the activity sheet and have students work on finding fundamental domains on the patterns. Wrap up the activity by asking students the questions below.

Questions

Are there any other options for the choice of fundamental domain? [Yes for all but #4]
Do all patterns have a fundamental domain? Why or why not? [All on the activity sheet do, but not all patterns are periodic. Show an example.]
Do all patterns have the same shape fundamental domain? Why or why not? [No, parallelograms, hexagons and triangles are just a few of the possible shaped fundamental domains you could tile the plan with.]
Do all fundamental domains of a tiling have the same area? [yes]

Extensions

Have students look for fundamental domains in wallpaper samples.

Up: The Shape of Space Curriculum Materials: Symmetry and Tiling

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