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February 2000
SIMMER Presentation

“Escher and You
presented by
Dr. Bruce A. Cload (University of Toronto)

“I never got a pass mark in math. The funny thing is I seem to latch on to mathematical theories without realizing what is happening.”                                                                                                          M.C. Escher

Escher (1898-1972), the Dutch Graphic Artist, is well known for his captivating prints of impossible staircases and colourful mosaics. Certain of his diagrams, his tessellations of the plane, are rich in rotational, reflectional, and translation symmetries and provide many attractive examples of the mathematical objects known as the plane periodic groups. The three dimensional analogs of these groups are important in x-ray crystallography, which played a pivotal role in the discovery of DNA.

            In this workshop, we will examine the concept of symmetry and compare, contrast, and classify the various symmetric structures in some of Escher’s diagrams. We will also discuss various computer and web teaching resources with which students can see, explore, and create their own Escher-like pictures. Part of the allure of this material is students can actually observe ‘mathematical representations’ in many everyday places—the stones on the library buildings, the tiles on the floor, the knitted pattern on a sweater. Through this, they can perceive mathematics as part of the world rather than as an abstract science.

Here are Questions for Exploration, References, and a short Handout:


Questions for Exploration

Escher Talk, Fields Institute

February, 2000

1. Geometry  

(a) What symmetries can you see in the different letters of the alphabet?

(b) What is a reflection followed by a rotation if the centre of the rotation is on the reflection line? If it is not on the reflection line? What happens if the rotation is performed first and then the reflection?

(c) Explore the composition of reflections, rotations, glide-reflections, and translations?

2. Linear Algebra (except for (b) which is Rea l Analysis)

(a) Symbolically, what is an isometry of the plane?

(b) If f(x,y) is an isometry of the plane, what is g(x,y) = f(x,y) – f(0,0)?  Why?

(c) Do reflections, rotations, translations, and glide-reflections act linearly? (i.e. Are they affine transformations?)

(d) If an isometry is a linear transformation, what type of matrix (or linear operator) represents the isometry?

(e) Compare and contrast the eigenvalues and eigenvectors of reflections and rotations. Can you distinguish between the two types of isometries based on this information?

(f) From the definition of an isometry of the plane (an isometry is a mapping of the plane onto the plane that preserves distance), delineate an argument that an isometry is affine.

(g) Synthesize this material to convince yourself that there are only four types of isometries.

3. Symmetry Group of a Figure

(a) What is the group operation? Why is it associative? What is the inverse of an isometry in this operation? What is the identity?

(b) Are symmetry groups usually abelian?

4. Some Irrelevant Complex Analysis

(a) If an analytic function is an isometry of the complex plane, what must it be?

(b) Find an isometry of the unit disc, which is not an isometry of the complex plane?

5. Putting it all together.

Outline in writing the mathematical complexity of Escher’s diagrams.


General References:

[1] Escher M.C., Exploring the Infinite, Abrams, NY, 1986.

[2] Escher M.C., The Graphic Work, Hawthorn Books, NY, 1960.

[3] Locher, J.L., The World of M.C. Escher, Abrams, NY, 1971.

[4] Schattschneider, D., Visions of Symmetry, Freeman, NY, 1990.

Mathematical References:

[5] Crowe, D., Symmetry, Rigid Motions, and Patterns, Arlington: COMPAP, 1986.

[6] Gallian, J., Contemporary Abstract Algebra, 4th ed., Houghton Mifflin Company, 1998.

General Escher:




Symmetry Groups:




Mathematical Symmetry:


Frieze Patterns:


X-ray Diffraction:


Computer Programs:

[16] Tesslemania Deluxe, MECC

[17] Tesslemania, MECC

Bruce Cload, PhD
Department of Mathematics
University of Toronto
Toronto, Ontario,
M6H 3G3
Tel.: 416-824-1575

You will need to have Acrobat Reader on your machine to enable you to read/print PDF files.
The Acrobat Reader is free to download from Adobe Systems.
Handout (in PDF format)
Questions for Exploration (in PDF Format)

References (in PDF format)

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This page was last updated: May 07, 2000