Symmetries of Patterns

Why there is no five-fold symmetry

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Pattern symmetries: translation, rotation, reflection and glide reflection.
• There are exactly 17 wallpaper symmetry groups.
• Periodic patterns can only have rotational symmetry of order 2, 3, 4 or 6.

Steps

1. Four ways to move a pattern
2. Moving a pattern onto itself
3. Footprints
4. Seven friezes
5. Name that frieze
6. Wallpaper groups
7. The crystallographic restriction