Syllabus
Lesson 1
 Math part 
 Art part 
Lesson 2
 Math part 
 Art part 
Lesson 3
 Math part 
 Art part 
Lesson 4
 Math part 
 Art part 
Lesson 5
 Math part 
 Art part 
Lesson 6
 Math part 
 Art part 
Lesson 7
 Math part 
 Art part 
Lesson 8
 Math part 
 Art part 
Final Project
Student's Work


Goals: To investigate issues of complexity, 3d illusion,
symmetries of scale.
 1.
Moorish slide. Escher on Moors.
"What a pity that the religion of the Moors forbade them to make images! it
seems to me that they sometimes came very close to the development of their
elements into more significant figures than the abstract geometric shapes that
they created. No Moorish artist has, as far as I know, ever dared (or didn't he
hit on the idea?) to use as building components concrete, recognizable figures
borrowed from nature, such as fished, birds, reptiles or human beings. This is
hardly believeable, for recognizability is so important to me
that I never could do without it."
 2.
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite"
Published in 1989 by HARRY N. ABRAMS, INC., New York
Symmetry Work 45, P. 42
Escher slide. Planar pattern.
"...in my opinion, it is not right that you make representations without
backgrounds. These are compositions in which background and figure take turns
changing functions. A continuing competition exists between the two, and it
isn't even possible to continue seeing one element as figure. Irresistibly
the elements functioning originally as background present themselves cyclically
as figures."
 J. W. Wagenaar, (private letter to M. C. Escher)
Leonardo, Pergamon Press, 1979, Vol. 12
From H.S.M. Coxeter: "The NonEuclidean Symmetry of Escher's Picture 'Circle
Limit III'
pp.1925 Fig. 1. Pattern whose symmetry group is (6,4,2)
"I first met Escher in September 1954, when an exhibition of his work was
sponsored by the International Congress of Mathematicians, meeting that year in
Amsterdam. Throughout the previous 17 years he had been making designs in which
a drawing of some animal is repeated as on wallpaper, with two remarkable
innovations: the basic unit (usually a single animal ...) is repeated not only
by translations but also by other isometries ..., and the replicas ingeniously
fit together so that there are no interstices. In the language of mathematics,
the basic unit is a fundamental region for a symmetry group.
In a letter of December 1958 he wrote: Did I ever thank you for sending me ...
`A symposium on Symmetry'? I was so pleased with this booklet and proud of the
two reproductions of my plane patterns! Though the text of your article on
`Crystal Symmetry and its Generalizations' is much too learned for a simple,
selfmade pattern man like me, some of the textillustrations and especially
Figure 7, page 11, gave me quite a shock...
If you could give me a simple explanation how to construct the following circles,
whose centres approach gradually from the outside till they reach the limit, i
shoudl be immensely pleased and very thankful to you! Are there other systems
besides this one to reach a circle limit?
Nevertheless I used your model for a large woodcut (of which I executed only
a sector of 120 degrees in wood, which I printed 3 times). I am sending you a
copy of it.
This picture was 'Circle limit I'."
M. C. Escher: "ESCHER on ESCHER Exploring the Infinite", P. 126
Published in 1989 by HARRY N. ABRAMS, INC., New York
Circle limit I, woodcut, 1958
 3. Escher is talking about ways of filling space. How would you describe the
difference between the approaches of Escher vs Penrose tiles? In math there is an
idea of homogenious space, space where every piece "looks" like every other piece.
The Islamic patterns with a lot of symmetry seem to have this property, whatever
it might be.
M. C. Escher: "ESCHER on ESCHER Exploring the Infinite", p. 116
Published in 1989 by HARRY N. ABRAMS, INC., New York
Figure 3
M.C.Escher: "ESCHER on ESCHER Exploring the Infinite", p. 117
Published in 1989 by HARRY N. ABRAMS, INC., New York
Regular Division of the Plane VI
PICTURE Slide of geometric fractal construction, Slide of analogous Escher print.
Is the Escher circle limit print more like a wallpaper pattern or is it more like
a Penrose tile? Why? I want to hear mathematical and artistic reasons. (We
look for ideas like regularity, intent of the artist to express infinity, etc.)
Ideas we can think about generalizing in math: tiling versus symmetry issues,
scaling. What are the symmetries for Escher? (We are looking for symmetries of
scale here.)
M.C.Escher:"ESCHER on ESCHER Exploring the Infinite", p. 42
Published in 1989 by HARRY N. ABRAMS, INC., New York
Circle limit IV,
New Escher circle limit print. What do you see?
Do you see a 3D thing? If you can, what shape is
it? Do you think you could make something that shape? (How fast would it have
to move away from you? How fast would it have to grow sideways?) We draw a
picture of an infinite cylindrical bullet on the board. Imagine the bottom up view:
what do you see? What would have to happen to get a circle limit picture this way?
In the circle limit print above, do you see motion? Escher calls this the "flow
of traffic"? What about the use of animal images here? Do they strengthen the
idea of motion or weaken it? How does this compare with Islamic designs?
Do you see regularity? How about the ways the animals meet at noses and tails?
How about the angles of meeting? This is what mathematicians call homogeneity.
 5. OPTIONAL DISCUSSION OF RELATIVITY Issues of space. Ask class to explain
gravity: What keeps the moon in orbit around the earth? How does it work?
Instructor indulges in 17th century skepticism here. Even Isaac Newton found the
idea of "force at a distance" disturbing. What is in between the moon and the
earth? What is space like? How does the moon know that the earth is there? Can
anyone think of other examples like force at a distance? What is the difference
between "force at a distance" and E.S.P.? There are lot's of interesting
historical questions here about what a new idea is and how ideas come to be
accepted. Now we think space is curved, just like a circle limit print, and we
think of the curvature as the source of gravity. Gravity is given by the shape
of space.
 6. Escher reading on "The impossible". Circle limit slide.
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite" p. 41
Published in 1989 by HARRY N. ABRAMS, INC., New York
Square Limit
 7. Escher circle limit. Shlain quote.
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 43
Published in 1989 by HARRY N. ABRAMS, INC., New York
Circle Limit III
"In the case of the visual arts, in addition to illuminating, imitating, and
interpreting reality, a few artists create a language of symbols for things for
which there are yet to be words. Just as Sigmund Freud in his Civilization
and Its Discontents compared the progress of a civilization's entire people
to the development of a single individual, I propose that the radical innovations
of art embody the preverbal stages of new concepts that will eventually change a
civilization. Whether for an infant or a society on the verge of change, a new
way to think about reality begins with the assimilation of unfamiliar images.
This collation leads to abstract ideas that only later give rise to a descriptive
language."
 Leonard Shlain, Art and Physics, Parallel Visions in Space, Time
and Light
" There is in art a clairvoyance for which we have not yet found a name, and
still less an explanation."
 John Russell, The Meanings of Modern Art
In this case it was backwards. First the math, then the physics, then the art.
 8. Actually, the spacetime thing we live in is shaped very much like one
of E's prints. And mathematicians have been messing with an object like the
circle limit picture for a while.
(Lobatchevsky, 1825 more or less, Poincare, late 1800's). Picture.
 9. OPTIONAL DISCUSSION OF MODULAR GROUP Want to hear about the group action
that gives these symmetries? We give a short history of noneuclidean geometry,
followed by a description of the Poincare upper half plane, as far as we dare to
go. Just flying, here. But we do point out that you can get five fold symmetry,
with repeats, this way. Escher knew this! I think. I can't find an example of a
woodblock like that, though. Major facts about this type of space: this 2d
surface can't fit in 3d space without self intersection, and light would travel
not along geodesics but along horocycles.
 10. The University of Minnesota Geometry
Center website, has software that lets you build your own circle limit groups.
 11. Escher on math:
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 57
Published in 1989 by HARRY N. ABRAMS, INC., New York
Cubic Space Filling
"Allow me to say that Father Bach has been a strong inspiration to me, and
that many a print reached definite form in my mind while I was listening to the
lucid, logical language he speaks, while I was drinking the clear wine he pours.
When one speaks about 'lucid' and 'logical', one thinks involuntarily of
mathematics. In high school in Arnhem I was a particularly poor student in
arithmentic and algebra because I had, and still have, great trouble with the
abstractions of numbers and letters. Things went a little better in geometry
when I was called upon to use my imagination, but i never excelled in this
subject either while in school.
But our path through life can take strange turns."
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 57
Published in 1989 by HARRY N. ABRAMS, INC., New York
Depth
"Although I am even now still a layman in the area of mathematics, and
although I lack theoretical knowledge, the mathematicians, and in particular the
crystallographers, have had considerable influence on my work of the last twenty
years. The laws of the phenomena around usorder, regularity, cyclical
repetition, and renewalshave assumed greater and greater importance for me.
The awareness of their presence gives me peace and provides me with support. I
try in my prints to testify that we live in a beautiful and orderly world, and
not in a formless chaos, as it sometimes seems."
 12. We usually conclude this lesson by watching Not Knot,
a movie from the Geometry Center, which gives you the sensation of
travelling through a three dimensional circle limit print.
