# Escher's Wallpaper Groups Exploration

Objective: Learn to recognize the lattice in a wallpaper pattern. Learn to use the flow chart to identify wallpaper symmetry groups.

## Materials

 Printed version of the Escher's Wallpaper Groups Exploration: File:Eschertessellations.pdf
• Printed copy of the Escher's Wallpaper Groups Exploration.

## Exploration

In this exploration we will look at the wallpaper patterns in Visions of Symmetry and use the flowchart to determine their wallpaper group. First we will identify the largest order rotation for the tessellation, and check if there are reflections and glide-reflections. This is the data we need to use the flowchart.

Several examples from Visions of Symmetry:

• Drawing #1: The largest order rotation is 2. There are no reflections or glide-reflections. Following the flow chart we end up at p2.
• Drawing #2: The largest order rotation is 2. There are no reflections, but there are glide-reflections. Following the flow chart we end up at pgg.
• Drawing #3: The largest order rotation is 4. There are reflections, and there are glide-reflections. Following the flow chart we end up at p4g. (The mirror lines meet at 90 degree angles, not at 45 degree angles.)
• Drawing #4: The largest order rotation is 3. There are reflections, but there are no glide-reflections. Following the flow chart we end up at p31m. (The centers of rotation at the hands do NOT lie on the mirror lines.)
Escher Drawing Rotation Reflection Glide-Reflection Symmetry group
Visions of Symmetry #1, pg 116 2-fold No No p2
Visions of Symmetry #2, pg 116 2-fold No Yes ppg
Visions of Symmetry #3, pg 117 4-fold Yes Yes p4g
Visions of Symmetry #4, pg 118 3-fold Yes No p31m
Visions of Symmetry #6, pg 120
Visions of Symmetry #9, pg 123
Visions of Symmetry #13, pg 126
Visions of Symmetry #16, pg 128
Visions of Symmetry #20, pg 131
Visions of Symmetry #28, pg 137
Visions of Symmetry #37, pg 145
Visions of Symmetry #42, pg 148
Visions of Symmetry #43, pg 149
Visions of Symmetry #44, pg 151
Visions of Symmetry #45, pg 150
Visions of Symmetry #70, pg 142
Visions of Symmetry #81, pg 181
Visions of Symmetry #91, pg 190

Handin: A sheet with answers to all questions.

Escher's Wallpaper Groups Solutions (Instructors only)