SIMMER Presentation

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. Esch**er*

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:

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**Escher
Talk, Fields Institute
**

**February,
2000
**

**(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?

**(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.

**(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?

**(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?

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

*********************************************************

**[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*, 4^{th} ed., Houghton
Mifflin Company, 1998.

**General Escher:
**

**[7] **http://www.worldofescher.com/gallery/

**[8] **http://www.etropolis.com/escher/

**[9] **http://www-sphys.unil.ch/escher/

**Symmetry Groups:
**

**[10] **http://aleph0.clarku.edu/~djoyce/wallpaper/

**[11] **http://www.geom.umn.edu/java/Kali/program.html

**[12]** http://www.photon.at/~werner/light/escher/

**Mathematical Symmetry:
**

**[13]** http://www.best.com/~xah/Wallpaper_dir/c0_WallPaper.html

**Frieze Patterns:
**

**[14] **http://www.geom.umn.edu/~lori/kali/friezepat.html

**X-ray Diffraction:
**

**[15] **http://www-wilson.ucsd.edu/education/xraydiff/xraydiff.html

**Computer Programs:
**

**[16] **Tesslemania
Deluxe, MECC

**[17] **Tesslemania,
MECC

Bruce Cload, PhD

Department of Mathematics

University of Toronto

Toronto, Ontario,

M6H 3G3

cload@math.utoronto.ca

Tel.: 416-824-1575

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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*