
Lesson 7 Art part: Escher 

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. Dutch artist, MC Escher,18931972 What observations can you make about Escher from this work of art?
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
Symmetry work 22
Images
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.
Published in 1989 by HARRY N. ABRAMS, INC., New York Regular Decision of the Plane II
Symmetry, Printing, mirror image
Published in 1989 by HARRY N. ABRAMS, INC., New York Symmetry Work 85
Contrasts, opposites, use of color
Published in 1989 by HARRY N. ABRAMS, INC., New York Symmetry Work 63
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 42 Symmetry Work 45
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 36 Sky and Water
Escher added a new color to a design only when he included a new image in the tesselation.
Published in 1989 by HARRY N. ABRAMS, INC., New York Symmetry Work 70
Themes of Boundlessness
Published in 1989 by HARRY N. ABRAMS, INC., New York carved Beechwood Ball with Fish
Circle limits.
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.
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, nonEuclidean 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
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 4850) "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"
Published in 1989 by HARRY N. ABRAMS, INC., New York Metamorphose II
Fantasy
Mirror image  reflection
Published in 1989 by HARRY N. ABRAMS, INC., New York Drawing Hands
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 59 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..."
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 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.
Published in 1989 by HARRY N. ABRAMS, INC., New York Print Gallery
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 68 Convex Concave
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 73 Another World
M. C. Escher:"ESCHER on ESCHER Exploring the Infinite", p. 74 High and Low
Why do mathematicians enjoy his work?
Making Planar Tessellations:Regular Polygons  triangles, grids, hexagons all tesselate the plane by themselves
Tessellation slide A. In this case we will deal exclusively with the 2dimensional plane. "All 2dimensional 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 Slide 1 page 135. In slide 587 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 slide 6, 7 page 138,139 "figure 591 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 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 midpoint 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.

© Copyright 1996, Pippa Drew and Dorothy Wallace, Dartmouth College