# Escher's Wallpaper Groups Exploration

Objective: Practice recognizing the symmetries in wallpaper patterns. Learn to use the flow chart to identify wallpaper symmetry groups.

## Materials

• Printed copy of the table to fill out.

## Exploration

Throughout his career, Escher sketched repeating patterns that he used as the basis for much of his artwork. These patterns are numbered from 1-137 and are displayed in the book Visions of Symmetry, as well as at Regular Division of the Plane Drawings.

In this exploration you will look at the patterns from Escher's sketchbook and use the flowchart to determine their wallpaper symmetry groups.

First, identify the largest order rotation for the tessellation. Next, check if there are reflections and/or glide-reflections. Finally, use the flowchart to determine the symmetry group.

The first few examples are done for you:

• Sketch #1 (Lions): The largest order rotation is 2. There are no reflections or glide-reflections. Following the flow chart we end up at p2.
• Sketch #2 (Lions): The largest order rotation is 2. There are no reflections, but there are glide-reflections. Following the flow chart we end up at pgg.
• Sketch #3 (Weightlifters): 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.)
• Sketch #4 (Men): 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.)

Now continue and fill in the rest of the table.

Escher Drawing Largest Rotation Reflection Glide-Reflection Symmetry Group
Sketch #1 (Lions) 2-fold No No p2
Sketch #2 (Lions) 2-fold No Yes ppg
Sketch #3 (Weightlifters) 4-fold Yes Yes p4g
Sketch #4 (Men) 3-fold Yes No p31m
Sketch #6 (Camels)
Sketch #9 (Birds)
Sketch #13 (Dragonflies)
Sketch #16 (Dogs)
Sketch #20 (Fish)
Sketch #28 (Birds)
Sketch #37 (Bugs)
Sketch #42 (Shells/Starfish)
Sketch #43 (Flowers/Leaves)
Sketch #44 (Birds)
Sketch #45 (Angels/Devils)
Sketch #70 (Moths)
Sketch #81 (Bat/Bird/Moth)
Sketch #91 (Bugs)

Handin: The completed table.

Escher's Wallpaper Groups Solutions (Instructors only)