Lesson 7 Art part:   Escher


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:

Explore formal and psychological themes in MC Eschers work.
Experiment with designing a planar tesselation using a recognizable image.

Dutch artist, MC Escher,1893-1972

What observations can you make about Escher from this work of art?

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 70
Published in 1989 by HARRY N. ABRAMS, INC., New York

Order and Chaos

Major themes in Escher's work are contrast, duality, transformation, infinity and spatial paradoxes. He uses symmetry to order this world of duality and paradox. In the slide above Escher explores the duality of order vs. chaos. We shall see how this idea influences his work, both formally and psychologically.

Tiling
Escher was a great master of tessellation (the regular division of the plane, or tiling). He created symmetrical designs and planar tesselations, which he described as congruent, convex polygons joined together."

M.C.Escher:"ESCHER on ESCHER Exploring the Infinite", p. 45

Symmetry work 22

Images
Escher always preferred to use only animate images in his tiled patterns.

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 47
Published in 1989 by HARRY N. ABRAMS, INC., New York

Symmetry work 25

In 1922 Escher visited the Alhambra palace and saw the wall tilings of the Moors. He was excited to find other artists who had been captivated by tilings, but also made this revealing comment: "What a pity their religion forbade them to make graven images." Escher's notebooks soon became full of repeating patterns inspired by the Moors. Imagery gave his patterns a different psychological character from the serene designs of Islam.

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 105
Published in 1989 by HARRY N. ABRAMS, INC., New York

Regular Decision of the Plane II

Symmetry, Printing, mirror image
While, Escher's work includes representation, it is still involved with the language of visual symmetry and order.
Symmetry is integral to the medium of printmaking and graphic arts. The impression of a woodblock is a reflection or mirror image of the design carved into the block. Multiplicity and repetition are functions of printing as well. Thus, Escher chose a medium that naturally expressed two motions of symmetry: reflection and translation. These elements of symmetry also showed Escher's strong love of order. The technical difficulty of woodcutting suited Escher's fastidious nature too.

M.C.Escher:"ESCHER on ESCHER Exploring the Infinite", p. 38
Published in 1989 by HARRY N. ABRAMS, INC., New York

Symmetry Work 85

Contrasts, opposites, use of color
Contrast is necessary in tessellation. Contrasts in color, value and tone make contour lines between shapes, so we can distinguish between forms. Contrast epitomized the dualism of negative and positive space. This visual duality informs Escher's work psychologically in pieces such as: "Pessimist and Optimist," "Angels and Devils" and "Sky and Water."

M.C.Escher:"ESCHER on ESCHER Exploring the Infinite", p. 46
Published in 1989 by HARRY N. ABRAMS, INC., New York

Symmetry Work 63

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 42
Published in 1989 by HARRY N. ABRAMS, INC., New York

Symmetry Work 45

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 36
Published in 1989 by HARRY N. ABRAMS, INC., New York

Sky and Water

Escher added a new color to a design only when he included a new image in the tesselation.

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 28
Published in 1989 by HARRY N. ABRAMS, INC., New York

Symmetry Work 70

Themes of Boundlessness
Escher was also fascinated by the concept of infinity, which led him into explorations of space beyond the two dimensional plane. He carved the surface of this six inch ball with twelve identical fishes to show that a "fragmentary" plane could be filled endlessly. "When you turn this ball in your hands, fish after fish appears in endless succession. Though their number is restricted, they symbolize the idea of boundlessness in a manner that is not obtainable."

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 37
Published in 1989 by HARRY N. ABRAMS, INC., New York

carved Beechwood Ball with Fish

Circle limits.
Escher began to experiment with varying the scale of his tiled images. By gradually reducing the scale of each figure toward the center or outer edges, he showed that an "infinitely small limit" could be reached, "symbolizing an infinity of number." "The slide below shows a metamorphosis of hexagons into reptiles, from the center (where the elements are infinitely small) toward the borders."

M.C.Escher:"ESCHER on ESCHER Exploring the Infinite", p. 39
Published in 1989 by HARRY N. ABRAMS, INC., New York

Development II

Escher continued to work with themes of infinity and created a series of circle limit" prints.

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 42
Published in 1989 by HARRY N. ABRAMS, INC., New York

Circle Limit IV

In regard to this theme he says "...this is no simple question, but a complicated, non-Euclidean problem, much too difficult for a layman, as I am." He sought the help of English mathematician, H.S.M. Coxeter." page 42

Image stories
Escher pursued themes of transformation in works he called "Image Stories" which involved images transforming from one state into another. In another version of "Pessimist and Optimist", he explains, "... on a gray wall, these human figures increase their mutual contrast toward the center ... each kind detaches themselves from the wall surface and walks into space ... Thus going round they can't help meeting in the foreground ... the black pessimist keeps his finger raised in a gesture of warning, but the white optimist cheerfully comes to his encounter, and so they finally shake hands."

M.C.Escher:"ESCHER on ESCHER Exploring the Infinite", p. 46
Published in 1989 by HARRY N. ABRAMS, INC., New York

Encounter

This concept eventually led Escher to create a thirteen foot "woodcut strip" called "Metamorphosis" which shows a gradual progression of transformation. The word "metamorphose" becomes a springboard for the images. Escher describes the following: "Placed horizontally and vertically in the plane, with the letters o and m as points of intersection the words are gradually transformed into a mosaic of black and white squares which in turn develop into reptiles. Now the rhythm changes...by and by each each figure simplifies into a regular hexagon. At this point an association of ideas occurs: hexagons are reminiscent of the cells of a honeycomb, and no sooner has this thought occurred than a bee larva begins to stir in every cell... and so forth, continuing with images of birds, fish and even a city, and then back to abstract shapes and letters."(pages 48-50) "The structural world of Escher's art is fundamental to his language of symmetry ... his work ultimately refers to space and time ... [and] more than just a light hearted game ...[it] approach[es] something that is primeval and eternal, a groping for the secret of existence, much more than just filling the plane"

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p.48-53
Published in 1989 by HARRY N. ABRAMS, INC., New York

Metamorphose II

Fantasy
The theme of fantasy vs. reality appears throughout Escher's work and is a natural offshoot of his preoccupation with time and space. We can see the beginnings of his fantasy world in prints that play with reflection not only as a symmetry operation, but also as a pyschological or metaphysical experience. "All the prints in the last ten years of his life are symbolic in nature. [They express] a longing to approach infinity as purely and as closely as possible, the idea of endlessness represented on a piece of paper.

Mirror image - reflection
Escher employed mirror images to express metaphysical ideas as well as to create a sense of order. Reflection could demonstrate dualities of the conscious and unconscious world, reality and fantasy, truth and falsehood, and, ultimately, conflicting spatial points of view.

M. C. Escher: "ESCHER on ESCHER Exploring the Infinite", p. 66
Published in 1989 by HARRY N. ABRAMS, INC., New York

Drawing Hands

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 59
Published in 1989 by HARRY N. ABRAMS, INC., New York

Still Life with reflecting Shere

In "Hand with Reflecting Sphere", a spherical mirror is resting on a left hand. But as a print is the reverse of the original stone drawing, it is his right hand you see depicted ... the spherical reflection compresses the whole environment in one circular image. "You are immovably the focus of your world..."

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 60
Published in 1989 by HARRY N. ABRAMS, INC., New York

Hand with reflecting Shere

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 61
Published in 1989 by HARRY N. ABRAMS, INC., New York

Eye

In "The Eye" he chose the image of the skull, "with whom we are confronted whether we like it or not."

The psychological irony of these images carries over into a spatial and temporal one with prints like the "Print Gallery." Here Escher uses 3 dimensional illusion on the 2 dimensional plane to create double exposures of time and space. In this work the boy in the picture actually sees himself as a "detail of the images within the composition" -- the present coexisting with the future and past in the same space, and reality consolidated with depiction.

Escher also deals with impossible combinations of time and space in "Convex and Concave", "Another World", and "High and Low," where "Notions of up and down are interchangable."

In "High and Low" and "Convex and Concave", surprising juxtapositions of time and space are populated with people in "different worlds [who] are unaware of each other." We feel a sense of loneliness in much of Escher's art because he expresses his own alienation.

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 67
Published in 1989 by HARRY N. ABRAMS, INC., New York

Print Gallery

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 68
Published in 1989 by HARRY N. ABRAMS, INC., New York

Convex Concave

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 73
Published in 1989 by HARRY N. ABRAMS, INC., New York

Another World

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 74
Published in 1989 by HARRY N. ABRAMS, INC., New York

High and Low

Why do mathematicians enjoy his work?
Why do you think Escher experienced little recognition until the late 1960s?
When you look closely at a work by Escher, how does it make you feel?
How do you see Escher's approach in the architectural pieces? Does he provide an ordered or disordered world?
What is positive about it?

Making Planar Tessellations:

Regular Polygons - triangles, grids, hexagons all tesselate the plane by themselves

Tessellation
What is a tessellation? We can define it as a closed shape or polygon that repeats on all sides without leaving any gaps.

slide A.

In this case we will deal exclusively with the 2-dimensional plane. "All 2-dimensional tesselations have one important property: they can be extended in the plane infinitely in every direction." Basic to all 2 dimensional tesselations are the three motions of symmetry: translation, rotation, and reflection.

Translation is the most basic motion of tessellation. Certain shapes naturally create translating tesselations. For instance, squares, triangles, parallelograms and hexagrams all tessellate.

Slide b, c and d.

Escher made slight changes to these tessellating shapes until they became recognizable forms. Let's first begin by modifying a simple parallelogram in translation.

Modifying by translation
In figures 5-86 and 5-87,s lines have been translated from one side of the parallelogram to the other. "The process is broken down into four steps in figure 5-86. We start with a parallelogram, modify one side, translate the modified line to the opposite, parallel, congruent side; and finally, delete the orignal sides. The resulting shape tessellates the plane."

Slide 1 page 135.

In slide 5-87 the top and bottom sides were also changed.

In order to translate a modified line from one side to the other in a polygon, the 2 sides must be parallel and congruent

slide 2, page 136.

In this example the modification moves within and outside of the basic line.

Escher created the following tessellation "Pegasus" with this method.

p186, and 187 Slides 3, 4

Hexagons are another polygon that will regularly tessellate. Since the irregular hexagon below has three pairs of opposite parallel and congruent sides, we can modify it by the same principles and create an image that tesselates.

Slide 5 page 137.

Tessellation by Rotation at midpoints of Sides
Triangles are another shape that tessellate. However, since they have no parallel and congruent sides, another modification technique is used.
"You can modify [a scalene triangle] by rotation of one or more modified half-sides 180 degrees around the midpoint of a side.

slide 6, 7 page 138,139

"figure 5-91 shows an example of this type of modification. This technique can also be used with four sided polygons.

Slide 8. page 140.

Escher demonstrates this type of tessellation with a quadralateral in "Fish"

Slide 9, 10. page 192, 193. fish,

A more complex example of this is Escher's Lizard, based on a grid of paralellograms.

Slide 11, 12 Page 198, 199.

Modifying Polygons by rotation about Vertices
Lizard

slide 13,14 page 202.

If we separate one frog from the grid, it's clear that it is part of a modified square. Modifying by rotation about vertices created the frog shape. With this shape turned in four directions we can see a 4 fold rotation around a mid-point place at the snout of each frog.

Slide 14 page 203.

In the crab design below, we have a design which is "modified by translation, rotation about the midpoint of a side, and reflection to create the crab shape." page 221.

slide 15 page 221. Escher colore this design.

slide 16, page 220.

Lets try to make an Escher tesselation, using the simplest technique: modification by tranlation.

Interlocking Patterns

It is relatively easy to obtain a pattern made from a single interlocking piece. It is much more difficult to develop the piece so that it looks like a natural form. The development these interlocking pieces can become a fascinating game!

Start with a parallelogram (square or rectangle) and modify opposite sides in exactly the same way to create an interlocking pattern. You add to the bottom, the area taken from the top, and to the left side you add the area taken from the right side. The resulting piece is a fundamental region which will fit with itself to fill the plane without gaps or overlaps. Use your imagination to develop both negative and positive spaces into recognizable forms. Carry out assignment in gouache, suitable for portfolio presentation.



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