Lesson 7 Math part:   Escher


Syllabus


Lesson 1
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Art part |

Lesson 2
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Lesson 3
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Art part |

Lesson 4
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Art part |

Lesson 5
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Lesson 6
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Lesson 7
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Lesson 8
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Final Project

Student's Work



 

Goals:

To investigate issues of complexity, 3-d 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

    Symmetry Work 45

    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 Non-Euclidean Symmetry of Escher's Picture 'Circle Limit III'
    pp.19-25 Fig. 1. Pattern whose symmetry group is (6,4,2)

    symmetry group (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, self-made pattern man like me, some of the text-illustrations 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

    Circle limit I

  • 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

    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

    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,

    Circle limit IV

    New Escher circle limit print. What do you see?

    Do you see a 3-D 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

    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

    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 space-time 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 2-d surface can't fit in 3-d 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

    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

    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 us--order, regularity, cyclical repetition, and renewals--have 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.



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