Patterns on the Plane

Patterns on the Plane (9-12th grade)

Presenter: Ruth Vanderpool, Ph.D

Activity: This presentation follows Chapter 2 & 3 of John Conway’s The Symmetries of Things closely. Ideally this activity would take 90 minutes.

Ten minutes is spent introducing the idea of planar patterns and in particular mirror symmetry. After seeing a few examples students are given a few minutes to make their own planar patterns. Notation (called a signature) for the mirror symmetry is introduced is introduced to the class and students are given time to identify their own creations. After this, rotational symmetry will be introduced along with its signature. After practicing a few examples a sheet of more examples will be handed out to let the students practice further. Higher level students may be told of the two remaining patterns whose signatures have yet to be introduced.

The natural question, “Do I have all the symmetries for this pattern?” will then be addressed. Students will be shown the ‘cost’ of each digit in a pattern’s signature and shown that the total cost is always $2. The remaining time will be given to identifying the signatures of planar patterns around the room and discussing how many different kinds of planar patterns there are.

Materials I would need from the classroom:

Overhead projector to project examples


1. (Assigned before visit) Find as many repeating patterns on a plane as you can.

2. (Assigned after visit) Can you find a plane pattern that isn’t of the form ‘blue’*’red’?


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