Rotational and Reflectional Symmetry in Escher’s Sketches

Objective:

Look at symmetry in larger decorations. Learn to recognize rotations and reflections in wallpaper patterns.

Materials

 Printed version of the Symmetry in Escher's Sketches, dimmed: File:Symmetryineschersketches.pdf
• Printed copy of the Rotational and Reflectional Symmetry in Escher’s Sketches.

Exploration

Note: This exploration is based on the Regular Division of the Plane Drawings shown in Visions of Symmetry.

The plane filling patterns (also called wallpaper patterns) shown on pages 116 - 233 of Visions of Symmetry exhibit rotational and/or reflectional symmetries. We have not really defined what a wallpaper pattern is at this point, but we do not need to know much about them in order to investigate the symmetries present in these patterns. The patterns shown are only part of the infinite pattern. In other words, we need to think of these patterns as extending infinitely far in all directions. We will return to the concept of wallpaper patterns later in the course.

Example 1

Look at Sketch #3 (Weightlifter) on page 117. There is 2-fold rotational symmetry where the heads meet. There is 4-fold rotational symmetry where the elbows meet. There are lines of rotational reflection running through the centres of the bodies.

Example 2

Look at Sketch #5 (Strongmen) on page 119. There are no lines of reflection for this pattern. We do find rotational symmetry. Note that there is 2-fold rotational symmetry. Also note that there are two different rotational symmetries at work here. There is a 2-fold rotation around a point on the hand of the figure, and there is a 2-fold rotation about a point on the upper arm of the figure. These are considered different centers of rotation!

Checklist

Sketch Reflectional Symmetry (Yes or No) Rotational Symmetry
(2-, 3-, 4- or 6-fold; Where?)
Sketch #3 (Weightlifter) (pg 117)
Yes 2-fold at the head; 4-fold at the elbows
Sketch #4 (Chinese boy) (pg 118)
Sketch #5 (Strongman) (pg 119)
No 2-fold at hands and left arms
Sketch #12 (Butterfly) (pg 125)
Sketch #13 (Dragonfly) (pg 126)
Sketch #25 (Lizard) (pg 135)
Sketch #35 (Lizard) (pg 141)
Sketch #45 (Angel-Devil) (pg 150)
Sketch #85 (Lizard/Fish/Bat) (pg 184)
Sketch #91 (Beetles) (pg 190)