M. C. Escher

August 24, 2017 | Author: lev2468 | Category: Classical Geometry, Geometric Shapes, Space, Geometry, Mathematics
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Short Description

M. C. Escher, was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and ...

Description

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M. C. Escher

1

1.1

Early life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Later life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.3

Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.4

Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.5

Selected works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.6

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.7

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.8

Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.9

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Another World (M. C. Escher)

7

2.1

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Ascending and Descending

8

3.1

8

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Atrani, Coast of Amalfi

9

4.1

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Belvedere (M. C. Escher)

10

5.1

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

5.2

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

The Bridge (M. C. Escher)

11

6.1

11

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Castrovalva (M. C. Escher)

12

7.1

In popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

7.2

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

Circle Limit III

13

8.1

13

Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

ii

9

CONTENTS 8.2

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

8.3

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

8.4

Printing details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

8.5

Exhibits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

8.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

8.7

External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Convex and Concave

16

9.1

See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

9.2

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

10 Cube with Magic Ribbons 10.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Curl-up

17 17 18

11.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

11.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

12 Dolphins (M. C. Escher) 12.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Drawing Hands 13.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Gravitation (M. C. Escher)

20 20 21 21 22

14.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

14.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

15 Hand with Reflecting Sphere

23

15.1 Popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

15.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

15.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

16 House of Stairs 16.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Magic Mirror (M.C. Escher)

24 24 25

17.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

17.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

18 Metamorphosis I

26

18.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

18.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

19 Metamorphosis II

27

CONTENTS

iii

19.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

19.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

20 Metamorphosis III

28

20.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

20.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

20.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

21 Print Gallery (M. C. Escher)

29

21.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

21.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

22 Puddle (M. C. Escher)

30

22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

22.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

23 Regular Division of the Plane

31

23.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

23.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

24 Relativity (M. C. Escher)

32

25 Reptiles (M. C. Escher)

33

25.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

25.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

25.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

26 Sky and Water I

34

26.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

26.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

27 Sky and Water II

35

27.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

27.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

28 Snakes (M. C. Escher)

36

28.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

28.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

28.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

29 Stars (M. C. Escher)

37

29.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

29.2 Influences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

29.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

29.4 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

iv

CONTENTS 29.5 Collections and publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

29.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

30 Still Life and Street

39

30.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

30.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

30.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

31 Still Life with Mirror 31.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Still Life with Spherical Mirror

40 40 41

32.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

32.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

33 Three Spheres II

42

33.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

33.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

34 Three Worlds (M. C. Escher)

43

34.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

34.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

34.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

35 Tower of Babel (M. C. Escher)

44

35.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

35.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

35.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

36 Waterfall (M. C. Escher)

45

36.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

36.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

36.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

36.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . .

46

36.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

36.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

36.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

Chapter 1

M. C. Escher man. In 1903, the family moved to Arnhem, where he attended primary school and secondary school until 1918. He was a sickly child, and was placed in a special school at the age of seven and failed the second grade.[3] Although he excelled at drawing, his grades were generally poor. He also took carpentry and piano lessons until he was thirteen years old. In 1919, Escher attended the Haarlem School of Architecture and Decorative Arts in Haarlem. He briefly studied architecture, but he failed a number of subjects (partly due to a persistent skin infection) and switched to decorative arts.[3] He studied under Samuel Jessurun de Mesquita, with whom he remained friends for years. In 1922, Escher left the school after having gained experience in drawing and making woodcuts.

1.2 Later life In 1922, an important year of his life, Escher traveled through Italy (Florence, San Gimignano, Volterra, Siena, Ravello) and Spain (Madrid, Toledo, Granada). He was impressed by the Italian countryside and by the Alhambra, a fourteenth-century Moorish castle in Granada. The intricate decorative designs at Alhambra, which were based on geometrical symmetries featuring interlocking repetitive patterns sculpted into the stone Escher (1971) walls and ceilings, were a powerful influence on Escher’s works.[4] He returned to Italy regularly in the following Maurits Cornelis Escher (/ˈɛʃər/, Dutch: [ˈmʌurɪts years. kɔrˈneːlɪs ˈɛʃər] ( );[1] 17 June 1898 – 27 March 1972), In Italy, Escher met Jetta Umiker, whom he married in usually referred to as M. C. Escher, was a Dutch graphic 1924. The couple settled in Rome where their first son, artist. He is known for his often mathematically inspired Giorgio (George) Arnaldo Escher, named after his grandwoodcuts, lithographs, and mezzotints. These feature father, was born. Escher and Jetta later had two more impossible constructions, explorations of infinity, archisons: Arthur and Jan.[5] tecture, and tessellations. In 1935, the political climate in Italy (under Mussolini) became unacceptable to Escher. He had no interest in politics, finding it impossible to involve himself with any 1.1 Early life ideals other than the expressions of his own concepts through his own particular medium, but he was averse to Maurits Cornelis[2] was born in Leeuwarden, Friesland, fanaticism and hypocrisy. When his eldest son, George, in a house that forms part of the Princessehof Ceramics was forced at the age of nine to wear a Ballila uniform in Museum today. He was the youngest son of civil engineer school, the family left Italy and moved to Château-d'Œx, George Arnold Escher and his second wife, Sara Gleich- Switzerland, where they remained for two years.[6] 1

2 Escher, who had been very fond of and inspired by the landscapes in Italy, was decidedly unhappy in Switzerland. In 1937, the family moved again, to Uccle, a suburb of Brussels, Belgium. World War II forced them to move in January 1941, this time to Baarn, Netherlands, where Escher lived until 1970. Most of Escher’s better-known works date from this period. The sometimes cloudy, cold and wet weather of the Netherlands allowed him to focus intently on his work. For a time after undergoing surgery, 1962 was the only period in which Escher did not work on new pieces.

CHAPTER 1. M. C. ESCHER He worked primarily in the media of lithographs and woodcuts, though the few mezzotints he made are considered to be masterpieces of the technique. In his graphic art, he portrayed mathematical relationships among shapes, figures and space. Additionally, he explored interlocking figures using black and white to enhance different dimensions. Integrated into his prints were mirror images of cones, spheres, cubes, rings and spirals. Escher was left-handed.[7]

Escher moved to the Rosa Spier Huis in Laren in 1970, an artists’ retirement home in which he had his own studio. He died at the home on 27 March 1972, aged 73.

1.3 Works

Relativity, 1953

Although Escher did not have mathematical training— his understanding of mathematics was largely visual and intuitive—Escher’s work had a strong mathematical component, and more than a few of the worlds which he drew were built around impossible objects such as the Necker cube and the Penrose triangle. Many of Escher’s works employed repeated tilings called tessellations. Escher’s Drawing Hands, 1948 artwork is especially well liked by mathematicians and scientists, who enjoy his use of polyhedra and geometric In his early years, Escher sketched landscapes and nature. distortions. For example, in Gravity, multicolored turtles He also sketched insects, which appeared frequently in poke their heads out of a stellated dodecahedron. his later work. His first artistic work, completed in 1922, The mathematical influence in his work emerged around featured eight human heads divided in different planes. Later around 1924, he lost interest in “regular division” 1936, when he journeyed to the Mediterranean with the Adria Shipping Company. He became interested in order of planes, and turned to sketching landscapes in Italy with irregular perspectives that are impossible in natural form. and symmetry. Escher described his journey through the Mediterranean as “the richest source of inspiration I have Escher’s first print of an impossible reality was Still Life ever tapped.” and Street, 1937. His artistic expression was created from images in his mind, rather than directly from observations After his journey to the Alhambra, Escher tried to imand travels to other countries. Well known examples of prove upon the art works of the Moors using geometric his work include Drawing Hands, a work in which two grids as the basis for his sketches, which he then overlaid hands are shown, each drawing the other; Sky and Water, with additional designs, mainly animals such as birds and in which light plays on shadow to morph the water back- lions. ground behind fish figures into bird figures on a sky back- His first study of mathematics, which later led to its inground; and Ascending and Descending, in which lines of corporation into his art works, began with George Pólya's people ascend and descend stairs in an infinite loop, on academic paper on plane symmetry groups sent to him a construction which is impossible to build and possible by his brother Berend. This paper inspired him to learn to draw only by taking advantage of quirks of perception the concept of the 17 wallpaper groups (plane symmetry and perspective. groups). Using this mathematical concept, Escher cre-

1.3. WORKS ated periodic tilings with 43 colored drawings of different types of symmetry. From this point on he developed a mathematical approach to expressions of symmetry in his art works. Starting in 1937, he created woodcuts using the concept of the 17 plane symmetry groups.

3 division of a plane, which he applied in over 150 colored works. Other mathematical principles evidenced in his works include the superposition of a hyperbolic plane on a fixed 2-dimensional plane, and the incorporation of threedimensional objects such as spheres, columns and cubes into his works. For example, in a print called "Reptiles", he combined two and three-dimensional images. In one of his papers, Escher emphasized the importance of dimensionality and described himself as “irritated” by flat shapes: “I make them come out of the plane.”

Circle Limit III, 1959

In 1941, Escher summarized his findings in a sketchbook, which he labeled Regelmatige vlakverdeling in asymmetrische congruente veelhoeken (“Regular division of the plane with asymmetric congruent polygons”).[8] His intention in writing this was to aid himself in integrating mathematics into art. Escher is considered a research mathematician of his time because of his documentation with this paper, in which he studied color based division, Waterfall, 1961 and developed a system of categorizing combinations of shape, color and symmetrical properties. Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher’s interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher’s wood engravings Circle Limit I–IV demonstrate this concept. In 1959, Coxeter published his finding that these works were extraordinarily accurate: “Escher got it absolutely right to the millimeter.” Escher was awarded the Knighthood of the Order of Orange Nassau in 1955. Subsequently he regularly designed art for dignitaries around the world. In 1958, he published a book entitled Regular Division of the Plane, with reproductions of a series of woodcuts based on tessellations of the plane, in which he described the systematic buildup of mathematical designs in his artworks. He emphasized, "Mathematicians have opened the gate leading to an extensive domain.”

Sculpture of the small stellated dodecahedron that appears in Escher’s Gravitation. It can be found in front of the “Mesa+" building on the Campus of the University of Twente.

Escher also studied topology. He learned additional concepts in mathematics from the British mathematician Overall, his early love of Roman and Italian landscapes Roger Penrose. From this knowledge he created Waterand of nature led to his interest in the concept of regular fall and Up and Down, featuring irregular perspectives

4

CHAPTER 1. M. C. ESCHER

similar to the concept of the Möbius strip.

works. In 1980, this holding was sold to an American art Escher printed Metamorphosis I in 1937, which was a be- dealer and the Hague Museum. The Museum obtained ginning part of a series of designs that told a story through all of the documentation and the smaller portion of the the use of pictures. These works demonstrated a culmi- art works. nation of Escher’s skills to incorporate mathematics into The copyrights remained the possession of the three sons art. In Metamorphosis I, he transformed convex polygons – who later sold them to Cordon Art, a Dutch company. into regular patterns in a plane to form a human motif. Control of the copyrights was subsequently transferred to This effect symbolizes his change of interest from land- The M.C. Escher Company B.V. of Baarn, Netherlands, scape and nature to regular division of a plane. which licenses use of the copyrights on all of Escher’s art His piece Metamorphosis III is wide enough to cover all and on his spoken and written text. the walls in a room, and then loop back onto itself.

A related entity, the M.C. Escher Foundation of Baarn, After 1953, Escher became a lecturer at many organiza- promotes Escher’s work by organizing exhibitions, pubtions. A planned series of lectures in North America in lishing books and producing films about his life and work. 1962 was cancelled due to an illness, but the illustrations The primary institutional collections of original works and text for the lectures, written out in full by Escher, by M.C. Escher are the Escher Museum, a subsidiary of were later published as part of the book Escher on Es- the Haags Gemeentemuseum in The Hague; the National cher. In July 1969 he finished his last work, a woodcut Gallery of Art (Washington, DC); the National Gallery of called Snakes, in which snakes wind through a pattern of Canada (Ottawa); the Israel Museum (Jerusalem); Huis linked rings which fade to infinity toward both the center ten Bosch (Nagasaki, Japan); and the Boston Public Liand the edge of a circle. brary.

1.4 Legacy

Gödel, Escher, Bach by Douglas Hofstadter,[9] published in 1979, discusses the ideas of self-reference and strange loops, drawing on a wide range of artistic and scientific work, including the art of M. C. Escher and the music of J. S. Bach, to illustrate ideas behind Gödel’s incompleteness theorems.

1.5 Selected works • Trees, ink (1920) • St. Bavo’s, Haarlem, ink (1920) • Flor de Pascua (The Easter Flower), woodcut/book illustrations (1921) • Eight Heads, woodcut (1922) The Escher Museum in The Hague

• Dolphins also known as Dolphins in Phosphorescent Sea, woodcut (1923)

See also: M. C. Escher in popular culture

• Tower of Babel, woodcut (1928)

The special way of thinking and the rich graphic work of M.C. Escher has had a continuous influence in science and art, as well as being referenced in popular culture. Ownership of the Escher intellectual property and of his unique art works have been separated from each other. In 1969, Escher’s business advisor, Jan W. Vermeulen, author of a biography in Dutch on the artist, established the M.C. Escher Stichting (M.C. Escher Foundation), and transferred into this entity virtually all of Escher’s unique work as well as hundreds of his original prints. These works were lent by the Foundation to the Hague Museum. Upon Escher’s death, his three sons dissolved the Foundation, and they became partners in the ownership of the art

• Street in Scanno, Abruzzi, lithograph (1930) • Castrovalva, lithograph (1930) • The Bridge, lithograph (1930) • Palizzi, Calabria, woodcut (1930) • Pentedattilo, Calabria, lithograph (1930) • Atrani, Coast of Amalfi, lithograph (1931) • Ravello and the Coast of Amalfi, lithograph (1931) • Covered Alley in Atrani, Coast of Amalfi, wood engraving (1931)

1.6. SEE ALSO

5

• Phosphorescent Sea, lithograph (1933)

• Order and Chaos (Contrast), lithograph (1950)

• Still Life with Spherical Mirror, lithograph (1934)

• Rippled Surface, woodcut and linoleum cut (1950)

• Hand with Reflecting Sphere also known as SelfPortrait in Spherical Mirror, lithograph (1935)

• Curl-up, lithograph (1951)

• Inside St. Peter’s, wood engraving (1935)

• House of Stairs, lithograph (1951)

• Portrait of G.A. Escher, lithograph (1935)

• House of Stairs II, lithograph (1951)

• “Hell”, lithograph, (copied from a painting by Hieronymus Bosch) (1935)

• Puddle, woodcut (1952)

• Regular Division of the Plane, series of drawings that continued until the 1960s (1936)

• Gravitation, (1952) • Dragon, woodcut lithograph and watercolor (1952)

• Still Life and Street (his first impossible reality), woodcut (1937)

• Cubic Space Division, lithograph (1952)

• Metamorphosis I, woodcut (1937)

• Relativity, lithograph (1953)

• Day and Night, woodcut (1938)

• Tetrahedral Planetoid, woodcut (1954)

• Cycle, lithograph (1938)

• Compass Rose (Order and Chaos II), lithograph (1955)

• Sky and Water I, woodcut (1938) • Sky and Water II, lithograph (1938)

• Convex and Concave, lithograph (1955)

• Metamorphosis II, woodcut (1939–1940)

• Three Worlds, lithograph (1955)

• Verbum (Earth, Sky and Water), lithograph (1942)

• Print Gallery, lithograph (1956)

• Reptiles, lithograph (1943)

• Mosaic II, lithograph (1957)

• Ant, lithograph (1943) • Encounter, lithograph (1944) • Doric Columns, wood engraving (1945) • Three Spheres I, wood engraving (1945)

• Cube with Magic Ribbons, lithograph (1957) • Belvedere, lithograph (1958) • Sphere Spirals, woodcut (1958)

• Magic Mirror, lithograph (1946)

• Circle Limit III, woodcut (1959)

• Three Spheres II, lithograph (1946)

• Ascending and Descending, lithograph (1960)

• Another World Mezzotint also known as Other World Gallery, mezzotint (1946)

• Waterfall, lithograph (1961)

• Eye, mezzotint (1946) • Another World also known as Other World, wood engraving and woodcut (1947) • Crystal, mezzotint (1947) • Up and Down also known as High and Low, lithograph (1947) • Drawing Hands, lithograph (1948) • Dewdrop, mezzotint (1948)

• Möbius Strip II (Red Ants) woodcut (1963) • Knot, pencil and crayon (1966) • Metamorphosis III, woodcut (1967–1968) • Snakes, woodcut (1969)

1.6 See also

• Stars, wood engraving (1948)

• Asteroid 4444 Escher was named in Escher’s honor in 1985.

• Double Planetoid, wood engraving (1949)

• Mathematics and art#M.C. Escher

6

CHAPTER 1. M. C. ESCHER

1.7 References [1] Duden Aussprachewörterbuch (6 ed.). Mannheim: Bibliographisches Institut & F.A. Brockhaus AG. 2005. ISBN 3-411-04066-1. [2] “We named him Maurits Cornelis after S.'s [Sara’s] beloved uncle Van Hall, and called him 'Mauk' for short ....”, Diary of Escher’s father, quoted in M. C. Escher: His Life and Complete Graphic Work, Abradale Press, 1981, p. 9. [3] Barbara E, PhD. Bryden. Sundial: Theoretical Relationships Between Psychological Type, Talent, And Disease. Gainesville, Fla: Center for Applications of Psychological Type. ISBN 0-935652-46-9. [4] Roza, Greg (2005). An Optical Artist: Exploring Patterns and Symmetry. Rosen Classroom. p. 20. ISBN 978-14042-5117-5. [5] “ESCHER”. Geom.uiuc.edu. Retrieved 7 December 2013. [6] Ernst, Bruno, The Magic Mirror of M.C. Escher, Taschen, 1978; p. 15

• Schattschneider, Doris & Walker, Wallace. (1987) M. C. Escher Kaleidocycles, Petaluma, California, Pomegranate Communications ISBN 0-906212-286. • Schattschneider, Doris (2004). M. C. Escher : Visions of Symmetry, New York, N.Y. : Harry N. Abrams, 2004. ISBN 0-8109-4308-5. • Schattschneider, Doris & Emmer, Michele, eds (2003). M. C. Escher’s Legacy: a Centennial Celebration; collection of articles coming from the M. C. Escher Centennial Conference, Rome, 1998 / Berlin; London: Springer-Verlag. ISBN 3-54042458-X (hbk). • “Escher, M. C.” in: The World Book Encyclopedia; 10th ed. 2001. Media • M. C. Escher, The Fantastic World of M. C. Escher, Video collection of examples of the development of his art, and interviews, Director, Michele Emmer.

[7] “The Official M.C. Escher Website – Biography”. Mcescher.com. Retrieved 7 December 2013. [8] Barry Cipra (1998). Paul Zorn, ed. What’s Happening in the Mathematical Sciences, Volume 4. American Mathematical Society. p. 103. ISBN 0-8218-0766-8. [9] Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, ISBN 0465-02656-7

1.9 External links • “M.C. Escher official website”. • “Math and the Art of M.C. Escher”. USA: SLU. • Artful Mathematics: The Heritage of M. C. Escher. USA: AMS.

1.8 Further reading

• Escherization problem and its solution. CA: University of Waterloo.

Books

• “Escher for Real”. IL: Technion. — physical replicas of some of Escher’s “impossible” designs

• Abrams (1995). The M. C. Escher Sticker Book. Harry N. Abrams. ISBN 0-8109-2638-5 . • Ernst, Bruno; Escher, M. C. (1995). The Magic Mirror of M. C. Escher (Taschen Series). Taschen America LLC. ISBN 1-886155-00-3 Escher’s art with commentary by Ernst on Escher’s life and art, including several pages on his use of polyhedra. • Escher, M. C. (1971) The Graphic Work of M. C. Escher, Ballantine. Includes Escher’s own commentary. • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0. • Locher, J. L., ed. (1981) M. C. Escher: His Life and Complete Graphic Work, Amsterdam • O'Connor, J. J. (17 June 2005) Escher. University of St Andrews, Scotland.

• “M.C. Escher: Life and Work”. USA: NGA. • “US Copyright Protection for UK Artists”. UK. Copyright issue regarding Escher from the Artquest Artlaw archive. • Schattschneider, Doris (June–July 2010). “The Mathematical Side of M. C. Escher” (PDF). Notices of the American Mathematical Society (USA) 57 (6): 706–18. Retrieved 9 July 2010. • Gallery of tessellations by M.C. Escher

Chapter 2

Another World (M. C. Escher) Another World, also known as Other World, is a woodcut print by the Dutch artist M. C. Escher first printed in January 1947. It depicts a cubic architectural structure made from brick. The structure is a paradox with an open archway on each of the five visible sides of the cube. The structure wraps around the vertical axis to enclose the viewer’s perspective. At the bottom of the image is an archway which we seem to be looking up from the base, and through it we can see space. At the top of that arch is another arch which is level with our perspective, and through it we are looking out over a lunar horizon. At the top of that arch is another arch which covers the top of the image. We are looking down at this arch from above and through it onto the lunar surface. Standing in each archway along the vertical axis is a metal sculpture of a bird with a humanoid face. In each side archway is a horn or cornucopia hanging on chains. It is interesting to note that the views from above and below are consistent, placing the statue so that it faces the horn, however the horizontal view reverses the relative positions of the statue and the horn, and rotates the horn 180 degrees. The previous month (December 1946), Escher created a mezzotint called Another World (Other World Gallery). The image in that print is the same as this one except that the arches continue on as an infinite corridor. The bird/human sculpture is a real sculpture which was given to Escher by his father-in-law. This sculpture first appears in Escher’s 1934 lithograph Still Life with Spherical Mirror.

2.1 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

2.2 External links • Other versions of Another World

7

Chapter 3

Ascending and Descending Ascending and Descending is a lithograph print by the Two earlier Escher pictures that feature stairs are House Dutch artist M. C. Escher first printed in March 1960. of Stairs and Relativity. The original print measures 14 in × 11 1 ⁄4 in (35.6 cm × 28.6 cm). The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identically dressed men appear on the staircase, one line ascending while the other descends. Two figures sit apart from the people on the endless staircase: one in a secluded courtyard, the other on a lower set of stairs. While most twodimensional artists use relative proportions to create an illusion of depth, Escher here and elsewhere uses conflicting proportions to create the visual paradox.

3.1 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

Ascending and Descending was influenced by, and is an artistic implementation of, the Penrose stairs, an impossible object; Lionel Penrose had first published his concept in the February 1958 issue of the British Journal of Psychology. Escher developed the theme further in his print Waterfall, which appeared in 1961. The two concentric processions on the stairs use enough people to emphasise the lack of vertical rise and fall. In addition, the shortness of the tunics worn by the people makes it clear that some are stepping up and some are stepping down. The structure is embedded in human activity. By showing an unaccountable ritual of what Escher calls an 'unknown' sect, Escher has added an air of mystery to the people who ascend and descend the stairs. Therefore, the stairs themselves tend to become incorporated into that mysterious appearance. There are 'free' people and Escher said of these: 'recalcitrant individuals refuse, for the time being, to take part in the exercise of treading the stairs. They have no use for it at all, but no doubt, sooner or later they will be brought to see the error of their non-conformity.' Escher suggests that not only the labours, but the very lives of these monk-like people are carried out in an inescapable, coercive and bizarre environment. Another possible source for the people’s looks is the Dutch idiom “a monk’s job”, which refers to a long and repetitive working activity with absolutely no practical purposes or results, and, by extension, to something completely useless.

8

Chapter 4

Atrani, Coast of Amalfi

Atrani in 2003.

Atrani, Coast of Amalfi is a lithograph print by the Dutch artist M. C. Escher, first printed in August 1931. Atrani is a small town and commune on the Amalfi Coast in the province of Salerno in the Campania region of southwestern Italy. Atrani is the second smallest town in Italy and was built right at the edge of the sea. This image of Atrani recurs several times in Escher’s work, most notably in his series of Metamorphosis prints: Metamorphosis I, II and III.

4.1 See also • Printmaking

4.2 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

9

Chapter 5

Belvedere (M. C. Escher) • Necker cube

Belvedere is a lithograph print by the Dutch artist M. C. Escher, first printed in May 1958. It shows a plausiblelooking building that is actually an impossible object. In this print, Escher uses two-dimensional images to depict objects free of the confines of the three-dimensional world. The image is of a rectangular three-story building. The upper two floors are open at the sides with the top floor and roof supported by pillars. From the viewer’s perspective, all the pillars on the middle floor are the same size at both the front and back, but the pillars at the back are set higher. The viewer also sees by the corners of the top floor that it is at a different angle than the rest of the structure. All these elements make it possible for all the pillars on the middle floor to stand at right angles, yet the pillars at the front support the back side of the top floor while the pillars at the back support the front side. This paradox also allows a ladder to extend from the inside of the middle floor to the outside of the top floor.

• M. C. Escher’s Waterfall

5.2 Sources

There is a man seated at the foot of the building holding an impossible cube. He appears to be constructing it from a diagram of a Necker cube at his feet with the intersecting lines circled. The window next to him is closed with an iron grille that is geometrically valid but practically impossible to assemble. The woman who is about to climb the steps of the building is modeled after a figure from the right panel of Hieronymus Bosch's 1500 triptych The Garden of Earthly Delights. This panel is individually titled Hell. A portion of Hell had earlier been recreated by Escher as a lithograph in 1935. The ridge in the background is part of Morrone Mountains in Abruzzo, that Escher had visited several times when living in Italy during the 1920s and 30s.

5.1 See also • Belvedere (structure) • Lithography • Paradox • Printmaking 10

• Escher’s Belvedere • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

Chapter 6

The Bridge (M. C. Escher) The Bridge is a lithograph print by the Dutch artist M. C. Escher, first printed in March 1930. It depicts a bridge connecting two sheer cliffs. On the top of the left hand cliff is a city. The chasm between the two cliffs is narrow but plummets out of view. In the distance is another outcrop with a city built on top. Both the rock and the architecture on this third outcrop are darker in colouration than in the foreground. The buildings appear to be modelled partly after southern Italian architecture. The rock is in blocky formations that appeared often during Escher’s Italian period and it is possible that the village seen is Assisi.

6.1 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

11

Chapter 7

Castrovalva (M. C. Escher) Castrovalva is a lithograph print by the Dutch artist M. C. Escher, first printed in February 1930. Like many of Escher’s early works, it depicts a place that he visited on a tour of Italy. It depicts the Abruzzo village of Castrovalva, which lies at the top of a sheer slope. The perspective is toward the northwest, from the narrow trail on the left which, at the point from which this view is seen, makes a hairpin turn to the right, descending to the valley. In the foreground at the side of the trail, there are several flowering plants, grasses, ferns, a beetle and a snail. In the expansive valley below there are cultivated fields and two more towns, the nearest of which is Anversa degli Abruzzi, with Casale in the distance.

7.1 In popular culture • In 1982 the name “Castrovalva” was used in a story in the BBC television series Doctor Who. The storyline also relied heavily on recursion, a favorite theme in Escher’s later and more famous works, and used ideas taken from Belvedere, Ascending and Descending, and Relativity to trap the protagonists in the city of Castrovalva. • The comic Kingdom of the Wicked is set in an imaginary world named Castrovalva.

7.2 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

12

Chapter 8

Circle Limit III

The (6,4,2) triangular hyperbolic tiling that inspired Escher

Circle Limit III, 1959

Circle Limit III is a woodcut made in 1959 by Dutch and fundamental doartist M. C. Escher, in which “strings of fish shoot up like as depicting the lines of reflection [5] mains of the (6,4,2) triangle group. rockets from infinitely far away” and then “fall back again whence they came”.[1] It is one of a series of four woodcuts by Escher depicting ideas from hyperbolic geometry. Dutch physicist 8.2 Geometry and mathematician Bruno Ernst called it “the best of the four”.[2] Escher seems to have believed that the white curves of his woodcut, which bisect the fish, represent hyperbolic lines in the Poincaré disk model of the hyperbolic plane, in which the whole hyperbolic plane is modeled as a disk in 8.1 Inspiration the Euclidean plane, and hyperbolic lines are modeled as Escher became interested in tesselations of the plane af- circular arcs perpendicular to the disk boundary. Indeed, that the fish move “perpendicularly to the ter a 1936 visit to the Alhambra in Granada, Spain,[3][4] Escher wrote [1] However, as Coxeter demonstrated, there is boundary”. and from the time of his 1937 artwork Metamorphosis I no hyperbolic arrangement of lines whose faces are alterhe had begun incorporating tessellated human and animal nately squares and equilateral triangles, as the figure de[4] figures into his artworks. In a 1958 letter from Escher picts. Rather, the white curves are hypercycles that meet to H. S. M. Coxeter, Escher wrote that he was inspired the boundary circle at angles of cos−1 ((21/4 − 2−1/4 )/2), to make his Circle Limit series by a figure in Coxeter’s [2] article “Crystal Symmetry and its Generalizations”.[2][3] approximately 80°. Coxeter’s figure depicts a tessellation of the hyperbolic plane by right triangles with angles of 30°, 45°, and 90° (a shape that is possible in hyperbolic geometry but not in Euclidean geometry); this tessellation may be interpreted

The symmetry axes of the triangles and squares that lie between the white lines are true hyperbolic lines. The squares and triangles of the woodcut have the same incidence pattern as the faces of the tritetragonal tiling of

13

14

CHAPTER 8. CIRCLE LIMIT III

8.4 Printing details The fish in Circle Limit III are depicted in four colors, allowing each string of fish to have a single color and each two adjacent fish to have different colors. Together with the black ink used to outline the fish, the overall woodcut has five colors. It is printed from five wood blocks, each of which provides one of the colors within a quarter of the disk, for a total of 20 impressions. The diameter of the outer circle, as printed, is 41.5cm.[10]

8.5 Exhibits As well as being included in the collection of the Escher Museum in The Hague, there is a copy of Circle Limit III in the collection of the National Gallery of Canada.[11] The tritetragonal tiling, a hyperbolic tiling of squares and equilateral triangles, overlaid on Escher’s image

the hyperbolic plane, but their geometry is not the same: in the tritetragonal tiling, the sides of the squares and triangles are hyperbolically straight line segments, while in Escher’s woodcut they are arcs of hypercycles, so that the smooth curves of Escher correspond to polygonal chains with corners in the tritetragonal tiling. The points at the centers of the quadrilaterals, where four fish meet at their fins, form the vertices of an order-8 triangular tiling, while the points where three fish fins meet and the points where three white lines cross together form the vertices of its dual, the octagonal tiling.[2] Similar tessellations by lines of fish may be constructed for other hyperbolic tilings formed by polygons other than triangles and squares, or with more than three white curves at each crossing.[6] Euclidean coordinates of circles containing the three most prominent white curves in the woodcut may be obtained by calculations in the field of rational numbers extended by the square roots of two and three.[7]

8.3 Symmetry Viewed as a pattern, ignoring the colors of the fish, in the hyperbolic plane, the woodcut has three-fold and fourfold rotational symmetry at the centers of its triangles and squares, respectively, and order-three dihedral symmetry (the symmetry of an equilateral triangle) at the points where the white curves cross. In John Conway's orbifold notation, this set of symmetries is denoted 433. Each fish provides a fundamental region for this symmetry group. Contrary to appearances, the fish do not have bilateral symmetry: the white curves of the drawing are not axes of reflection symmetry.[8][9]

8.6 References [1] Escher, as quoted by Coxeter (1979). [2] Coxeter, H. S. M. (1979), “The non-Euclidean symmetry of Escher’s picture 'Circle Limit III'", Leonardo 12: 19– 25, JSTOR 1574078. [3] Emmer, Michele (2006), “Escher, Coxeter and symmetry”, International Journal of Geometric Methods in Modern Physics 3 (5-6): 869–879, doi:10.1142/S0219887806001594, MR 2264394. [4] Schattschneider, Doris (2010), “The mathematical side of M. C. Escher”, Notices of the AMS 57 (6): 706–718. [5] An elementary analysis of Coxeter’s figure, as Escher might have understood it, is given by Casselman, Bill (June 2010), How did Escher do it?, AMS Feature Column. Coxeter expanded on the mathematics of triangle group tessellations, including this one in Coxeter, H. S. M. (1997), “The trigonometry of hyperbolic tessellations”, Canadian Mathematical Bulletin 40 (2): 158–168, doi:10.4153/CMB-1997-019-0, MR 1451269. [6] Dunham, Douglas, “More “Circle Limit III” patterns”, The Bridges Conference: Mathematical Connections in Art, Music, and Science, London, 2006. [7] Coxeter, H. S. M. (2003), “The trigonometry of Escher’s woodcut Circle Limit III", M.C.Escher’s Legacy: A Centennial Celebration, Springer, pp. 297–304, doi:10.1007/3540-28849-X_29. [8] Conway, J. H. (1992), “The orbifold notation for surface groups”, Groups, Combinatorics & Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser. 165, Cambridge: Cambridge Univ. Press, pp. 438–447, doi:10.1017/CBO9780511629259.038, MR 1200280. Conway wrote that “The work Circle Limit III is equally intriguing” (in comparison to Circle Limit IV, which has a different symmetry group), and uses is it as an example of this symmetry group.

8.7. EXTERNAL LINKS

[9] Herford, Peter (1999), “The geometry of M. C. Escher’s circle-Limit-Woodcuts”, Zentralblatt fü Didaktik der Mathematik 31 (5): 144–148, doi:10.1007/BF02659805. Paper presented to the 8th International Conference on Geometry, Nahsholim (Israel), March 7–14, 1999. [10] Escher, M. C. (2001), M. C. Escher: The Graphic Work, Taschen, p. 10. [11] Circle Limit III, National Gallery of Canada, retrieved 2013-07-09.

8.7 External links • Douglas Dunham Department of Computer Science University of Minnesota, Duluth • Examples Based on Circle Limits III and IV, 2006:More “Circle Limit III” Patterns, 2007:A “Circle Limit III” Calculation

15

Chapter 9

Convex and Concave Convex and Concave is a lithograph print by the Dutch artist M. C. Escher, first printed in March 1955. It depicts an ornate architectural structure with many stairs, pillars and other shapes. The relative aspects of the objects in the image are distorted in such a way that many of the structure’s features can be seen as both convex shapes and concave impressions. This is a very good example of Escher’s mastery in creating illusion of “Impossible Architectures”. The windows, roads, stairs and other shapes can be perceived as opening out in seemingly impossible ways and positions. Even the image on the flag is of reversible cubes. One can view these features as concave by viewing the image upside-down. Note that all additional elements and decoration on the left are consistent with a viewpoint from above, while those on the right with a viewpoint from below: hiding half the image makes it very easy to switch between convex and concave.

9.1 See also • Printmaking

9.2 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

16

Chapter 10

Cube with Magic Ribbons Cube with Magic Ribbons is a lithograph print by the Dutch artist M. C. Escher first printed in 1957. It depicts two interlocking bands wrapped around the frame of a cube. The cube framework by itself is perfectly possible but the interlocking of the “magical” bands within it is impossible. This print is significant for being the first Escher drawing to use a true impossible object.

10.1 References • Ernst, Bruno (2006), “Optical Illusions”, Impossible Worlds: 2 in 1 Adventures with Impossible Objects, Cologne: Taschen, ISBN 3-8228-5410-7

17

Chapter 11

Curl-up This article is about the lithograph print. exercise, see Crunch (exercise).

For the

(spontaneous generation), because of the absence, in nature, of wheel shaped, living creatures with the ability to roll themselves forward. The accompanying 'beastie' depiction, referred to as 'revolving bitch' or 'roll paunch' in laymen’s terms, subsequently anticipates the need with sensitivity. Biological details are still few: is it a mammal, a reptile, or an insect? It has a long, drawn-out, horned, sectioned body and three sets of legs; the ends of which look like the human foot. In the middle of the fat, round head, that is provided with a strong, bent parrots beak; they have bulb-shaped eyes, which, placed on posts, protrude far out from both sides of the head. In the stretched out position, the animal can, slow and cautiously, with the use of his six legs, move forward over a variety of terrains (it can potentially climb or descend steep stairs, plow through bushes, or scramble over boulders). However, when it must cover a great distance, and has a relatively flat path to his disposal, he pushes his head to the ground and rolls himself up with lightning speed, at which time he pushes himself off with his legs- for as much as they can still touch the ground. In the rolled up state it exhibits the form of a discus, of which the eye posts are the central axle. By pushing off alternately with one of his three pairs of legs, he can achieve great speeds. It is also sometimes desirable during the rolling (i.e. The descent of an incline, or coasting to a finish) to hold up the legs and 'freewheel' forward. Whenever it wants, it can return again to the walking position in two ways: first abruptly, by suddenly extending his body, but then it’s lying on his back with his legs in the air, and second through gradual deceleration (braking with his feet) and slowly unrolling backwards in standing position.

Curl-up or Wentelteefje (original Dutch title) is a lithograph print by M. C. Escher, first printed in November 1951. This is the only work by Escher consisting largely of text. The text, which is written in Dutch, describes an imaginary species called Pedalternorotandomovens centroculatus articulosus, also known as “wentelteefje” or “rolpens”. He says this creature came into existence because of the absence in nature of wheel shaped, living creatures with the ability to roll themselves forward. The creature is elongated and armored with several keratinized joints. It has six legs, each with what appears to be a human foot. It has a disc-shaped head with a parrot-like beak and eyes on stalks on either side. It can either crawl over a variety of terrain with its six legs or press its beak to the ground and roll into a wheel shape. It can then roll, gaining acceleration by pushing with its legs. On slopes it can tuck its legs in and roll freely. This rolling can end in one of two ways; by abruptly unrolling in motion, which leaves the creature belly-up, or by braking to a stop with its legs and slowly unrolling backwards. The word wentelteefje is Dutch for French toast, “wentel” meaning “to turn over”. Rolpens is a dish made with chopped meat wrapped in a roll and then fried or baked. “Een pens” means “belly”, often used in the phrase beerbelly. There is a diagonal gap through the text containing an illustration showing the step by step process of the creature rolling into a wheel. This creature appears in two more prints completed later the same month, House of Stairs and House of Stairs II.

11.1 Translation The translation of the surrounding text is as follows:

11.2 See also • Printmaking

The Pedalternorotandomovens Centroculatus Articulosus (curl-up) came into existence

• Rotating locomotion in living systems 18

11.3. SOURCES

11.3 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

19

Chapter 12

Dolphins (M. C. Escher) Dolphins also known as a Dolphins in Phosphorescent Sea is a woodcut print by the Dutch artist M. C. Escher. This work was first printed in February, 1923. Escher had been fascinated by the glowing outlines of ocean waves breaking at night and this image depicts the outlines made by a school of dolphins swimming and breaching ahead of the bow of a ship. The glow was created by bioluminescent dinoflagellates.

12.1 Sources • Lewis, J.L. (2002). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

20

Chapter 13

Drawing Hands Drawing Hands is a lithograph by the Dutch artist M. C. Escher first printed in January 1948. It depicts a sheet of paper out of which, from wrists that remain flat on the page, two hands rise, facing each other and in the paradoxical act of drawing one another into existence. Although Escher used paradoxes in his works often, this is one of the most obvious examples. It is referenced in the book Gödel, Escher, Bach, by Douglas Hofstadter, who calls it an example of a strange loop. It is also used in Structure and Interpretation of Computer Programs by Harold Abelson and Gerald Jay Sussman as an allegory for the eval and apply functions of programming language interpreters in computer science, which feed each other.

13.1 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

21

Chapter 14

Gravitation (M. C. Escher) Gravitation (also known as Gravity) is a mixed media work by the Dutch artist M. C. Escher completed in June 1952. It was first printed as a black-and-white lithograph and then coloured by hand in watercolour. It depicts a nonconvex regular polyhedron known as the small stellated dodecahedron. Each facet of the figure has a trapezoidal doorway. Out of these doorways protrude the heads and legs of twelve turtles without shells, who are using the object as a common shell. The turtles are in six coloured pairs (red, orange, yellow, magenta, green and indigo) with each turtle directly opposite its counterpart.

14.1 See also • Printmaking

14.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

22

Chapter 15

Hand with Reflecting Sphere Hand with Reflecting Sphere also known as Self-Portrait in Spherical Mirror is a lithograph print by Dutch artist M. C. Escher, first printed in January 1935. The piece depicts a hand holding a reflective sphere. In the reflection, most of the room around Escher can be seen, and the hand holding the sphere is revealed to be Escher’s. Self-portraits in reflective, spherical surfaces are common in Escher’s work, and this image is the most prominent and famous example. In much of his self-portraiture of this type, Escher is in the act of drawing the sphere, whereas in this image he is seated and gazing into it. On the walls there are several framed pictures, one of which appears to be of an Indonesian shadow puppet.

15.1 Popular culture Frank O'Connor, the manager of the Halo video game series, made an illustration that references this work. It appears in the Halo Graphic Novel. In Disney’s TRON: Legacy, Jeff Bridges’ Character, CLU, is seen holding a reflective apple in which he sees his own reflection. This may be in homage to Escher, as there are two octahedra on a nearby shelf, and much of the digital world is made up of tessellations, a subject largely focused on by Escher.

15.2 See also • Still Life with Spherical Mirror • Three Spheres II • Lithography

15.3 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

23

Chapter 16

House of Stairs For other works titled “House of Stairs”, see House of Stairs (disambiguation). House of Stairs is a lithograph print by the Dutch artist M. C. Escher first printed in November 1951. This print measures 18⅝" × 9⅜". It depicts the interior of a tall structure crisscrossed with stairs and doorways. A total of 46 "wentelteefje" (imaginary creatures created by Escher) are crawling on the stairs. The wentelteefje has a long, armored body with six legs, humanoid feet, a parrot-like beak and eyes on stalks. Some are seen to roll in through doors, wound in a wheel shape and then unroll to crawl up the stairs, while others crawl down stairs and wind up to roll out. The wentelteefje first appeared earlier the same month in the lithograph Curl-up. Later that month, House of Stairs was extended to a vertical length of 55½" in a print titled House of Stairs II by repeating and mirroring some of the architecture and creatures.

16.1 References • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

24

Chapter 17

Magic Mirror (M.C. Escher) This article is about the lithograph by M. C. Escher. For other uses of Magic mirror, see Magic mirror. Magic Mirror is a lithograph print by the Dutch artist M. C. Escher first printed in January, 1946. It depicts a mirror standing vertically on wooden supports on a tiled surface. The perspective is looking down at an angle at the right hand side of the mirror. There is a sphere at each side of the mirror. The main focus of the image is a procession of small griffin (winged lion) sculptures that emerge from the surface of the mirror and trail away from it in single file. Both the angular reflection of the tiles and the offset between the reflection of the sphere in front of the mirror and the sphere behind it prove it is a mirror. Yet the reflection of the griffin procession continues to emerge from behind the mirror. The griffin processions of both sides loop around to the front and enter a tessellated pattern on the tile surface.

17.1 See also • Reptiles • Regular Division of the Plane • Printmaking • Paradox

17.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

25

Chapter 18

Metamorphosis I Metamorphosis I is a woodcut print by the Dutch artist M. C. Escher which was first printed in May, 1937. This piece measures 19.5 by 90.8 centimetres (7.7 in × 35.7 in) and is printed on two sheets. The concept of this work is to morph one image into a tessellated pattern, then gradually to alter the outlines of that pattern to become an altogether different image. From left to right, the image begins with a depiction of the coastal Italian town of Atrani (see Atrani, Coast of Amalfi). The outlines of the architecture then morph to a pattern of three-dimensional blocks. These blocks then slowly become a tessellated pattern of cartoon-like figures in oriental attire.

18.1 See also • Metamorphosis II • Metamorphosis III • Regular Division of the Plane • Printmaking

18.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

26

Chapter 19

Metamorphosis II Metamorphosis II is a woodcut print by the Dutch artist M. C. Escher. It was created between November, 1939 and March, 1940. This print measures 19.2 by 389.5 centimetres (7.6 in × 153.3 in) and was printed from 20 blocks on 3 combined sheets. Like Metamorphosis I, the concept of this piece is to morph one image into a tessellated pattern and then slowly alter that pattern eventually to become a new image. The process begins left to right with the word metamorphose (the Dutch form of the word metamorphosis) in a black rectangle, followed by several smaller metamorphose rectangles forming a grid pattern. This grid then becomes a black and white checkered pattern, which then becomes tessellations of reptiles, a honeycomb, insects, fish, birds and a pattern of three-dimensional blocks with red tops. These blocks then become the architecture of the Italian coastal town of Atrani (see Atrani, Coast of Amalfi). In this image Atrani is linked by a bridge to a tower in the water, which is actually a rook piece from a chess set. There are other chess pieces in the water and the water becomes a chess board. The chess board leads to a checkered wall, which then returns to the word metamorphose.

19.1 See also • Metamorphosis I • Metamorphosis III • Regular Division of the Plane • Tessellation • Printmaking

19.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

27

Chapter 20

Metamorphosis III Metamorphosis III is a woodcut print by the Dutch artist M. C. Escher created during 1967 and 1968. Measuring 19 cm × 680 cm (7½ × 268 inches - 22'4”), this is Escher’s largest print. It was printed on thirty-three blocks on six combined sheets and mounted on canvas. This print was partly coloured by hand.

20.3 Sources

It begins identically to Metamorphosis II, with the word metamorphose (the Dutch form of the word metamorphosis) forming a grid pattern and then becoming a black-and-white checkered pattern. Then the first set of new imagery begins. The angles of the checkered pattern change to elongated diamond shapes. These then become an image of flowers with bees. This image then returns to the diamond pattern and back into the checkered pattern. It then resumes with the Metamorphosis II imagery until the bird pattern. The birds then become sailing boats. From the sailing boats the image changes to a second fish pattern. Then from the fish to horses. The horses then become a second bird pattern. The second bird pattern then becomes black-and-white triangles, which then become envelopes with wings. These winged envelopes then return to the black-and-white triangles and then to the original bird pattern. It then resumes with the Metamorphosis II print until its conclusion.

20.1 See also • Metamorphosis I • Metamorphosis II • Atrani, Coast of Amalfi • Regular Division of the Plane • Tessellation • Printmaking

20.2 External links • Images of Metamorphosis III and other well known works at mcescher.com 28

• Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

Chapter 21

Print Gallery (M. C. Escher) Print Gallery (Dutch: Prentententoonstelling) is a lithograph printed in 1956 by the Dutch artist M. C. Escher. It depicts a man in a gallery viewing a print of a seaport, and among the buildings in the seaport is the very gallery in which he is standing. In the book Gödel, Escher, Bach, Douglas Hofstadter explains it as a strange loop showing three kinds of “in-ness": the gallery is physically in the town (“inclusion”); the town is artistically in the picture (“depiction”); the picture is mentally in the person (“representation”).

21.2 External links

Escher’s signature is on a circular void in the center of the work. In 2003, two Dutch mathematicians, Bart de Smit and Hendrik Lenstra, reported a way of filling in the void by treating the work as drawn on an elliptic curve over the field of complex numbers. They deem an idealized version of Print Gallery to contain a copy of itself, rotated clockwise by about 157.63 degrees and shrunk by a factor of about 22.58.[1] Print Gallery has been discussed in relation to postmodernism by a number of writers, including Silvio Gaggi,[2] Barbara Freedman,[3] Stephen Bretzius,[4] and Marie-Laure Ryan.[5]

21.1 References [1] de Smit, B. (2003). “The Mathematical Structure of Escher’s Print Gallery”. Notices of the American Mathematical Society 50 (4): 446–451. [2] Gaggi, Silvio (1989). Modern/Postmodern: A Study in Twentieth-Century Arts and Ideas. University of Pennsulvania Press. pp. 44–45. ISBN 0-8122-8154-3. [3] Freedman, Barbara (1991). Staging the gaze: postmodernism, psychoanalysis, and Shakespearean comedy. Cornell University Press. pp. 124–126. ISBN 0-8014-9737X. [4] Bretzius, Stephen (1997). Shakespeare in theory: the postmodern academy and the early modern theater. University of Michigan Press. p. 57. ISBN 0-472-10853-0. [5] Ryan, Marie-Laure (2000). Narrative as virtual reality: immersion and interactivity in literature and electronic media. Johns Hopkins University Press. p. 165. ISBN 08018-6487-9.

29

• HarryCarry5 (Jul 26, 2009). Escher’s Print Gallery Explained. YouTube. • Artful Mathematics: The Heritage of M. C. Escher, by Bart de Smit and Hendrik Lenstra

Chapter 22

Puddle (M. C. Escher) Puddle is a woodcut print by the Dutch artist M. C. Escher, first printed in February 1952. Since 1936, Escher’s work had become primarily focused on paradoxes, tessellation and other abstract visual concepts. This print, however, is a realistic depiction of a simple image that portrays two perspectives at once. It depicts an unpaved road with a large pool of water in the middle of it at twilight. Turning the print upside-down and focusing strictly on the reflection in the water, it becomes a depiction of a forest with a full moon overhead. The road is soft and muddy and in it there are two distinctly different sets of tire tracks, two sets of footprints going in opposite directions and two bicycle tracks. Escher has thus captured three elements: the water, sky and earth.

22.1 See also • Three Worlds • Printmaking

22.2 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

30

Chapter 23

Regular Division of the Plane 23.1 Sources 23.2 Further reading • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0. • Schattsneider, Doris (2004) M.C. Escher: Visions of Symmetry Harry N. Abrams, Inc. ISBN 0-81094308-5.

Regular Division of the Plane III, woodcut, 1957 - 1958.

Regular Division of the Plane is a series of drawings by the Dutch artist M. C. Escher which began in 1936. These images are based on the principle of tessellation, irregular shapes or combinations of shapes that interlock completely to cover a surface or plane. The inspiration for these works began in 1936 with a visit to the Alhambra, a fourteenth-century Moorish castle near Granada, Spain. Escher had visited the Alhambra once before in 1922 but in this visit he had spent several days studying and sketching the ornate tile designs there. In 1958 Escher published his book The Regular Division of the Plane. This book included several woodcut prints to demonstrate the concept, but the series of drawings continued until the late 1960s, ending at drawing #137. While not Escher’s most artistically important works, some of these patterns are among Escher’s most famous, having been used for a number of commercial products, including neckties. 31

Chapter 24

Relativity (M. C. Escher) Relativity is a lithograph print by the Dutch artist M. C. Escher, first printed in December 1953. It depicts a world in which the normal laws of gravity do not apply. The architectural structure seems to be the centre of an idyllic community, with most of its inhabitants casually going about their ordinary business, such as dining. There are windows and doorways leading to park-like outdoor settings. All of the figures are dressed in identical attire and have featureless bulb-shaped heads. Identical characters such as these can be found in many other Escher works. In the world of Relativity, there are three sources of gravity, each being orthogonal to the two others. Each inhabitant lives in one of the gravity wells, where normal physical laws apply. There are sixteen characters, spread between each gravity source, six in one and five each in the other two. The apparent confusion of the lithograph print comes from the fact that the three gravity sources are depicted in the same space. The structure has seven stairways, and each stairway can be used by people who belong to two different gravity sources. This creates interesting phenomena, such as in the top stairway, where two inhabitants use the same stairway in the same direction and on the same side, but each using a different face of each step; thus, one descends the stairway as the other climbs it, even while moving in the same direction nearly side-by-side. In the other stairways, inhabitants are depicted as climbing the stairways upside-down, but based on their own gravity source, they are climbing normally. Each of the three parks belongs to one of the gravity wells. All but one of the doors seem to lead to basements below the parks. Though physically possible, such basements are certainly unusual and add to the surreal effect of the picture. This is one of Escher’s most popular works and has been used in a variety of ways, as it can be appreciated both artistically and scientifically. Interrogations about perspective and the representation of three-dimensional images in a two-dimensional picture are at the core of Escher’s work, and Relativity represents one of his greatest achievements in this domain.

32

Chapter 25

Reptiles (M. C. Escher) Reptiles is a lithograph print by the Dutch artist M. C. Escher first printed in March 1943. It depicts a desk on which is a drawing of a tessellated pattern of reptiles. The reptiles at one edge of the drawing come to life and crawl around the desk and over the objects on it to eventually re-enter the drawing at its opposite edge. The desk is littered with ordinary objects, as well as a metal dodecahedron that the reptiles climb over. Although only the size of small lizards, these reptiles appear to have tusks and the one standing on the dodecahedron blows smoke from its nostrils. Like many of Escher’s works, this image was intended to depict a paradoxical and slightly humorous concept with no real philosophical meaning. There were, however, many popular misconceptions about the image’s meaning. Once a woman telephoned Escher and told him that she thought the image was a “striking illustration of reincarnation". The most common myth revolves around a small book on the desk with the letters JOB printed on it. Many people believed it to be the biblical Book of Job, when in fact it was a book of JOB brand cigarette papers. A colorized version of the lithograph was used by rock band Mott the Hoople as the sleeve artwork for its eponymous first album, released in 1969.

25.1 See also • Regular Division of the Plane

25.2 References • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

25.3 External links • Decoration with Escher Lizard by William Chow.

33

Chapter 26

Sky and Water I Sky and Water I is a woodcut print by the Dutch artist M. C. Escher first printed in June 1938.

• M. C. Escher—29 Master Prints; Harry N. Abrams, Inc., Publishers.

The basis of this print is a regular division of the plane consisting of birds and fish. Both prints have the horizontal series of these elements—fitting into each other like the pieces of a jigsaw puzzle—in the middle, transitional portion of the prints. In this central layer the pictorial elements are equal: birds and fish are alternately foreground or background, depending on whether the eye concentrates on light or dark elements. The birds take on an increasing three-dimensionality in the upward direction, and the fish, in the downward direction. But as the fish progress upward and the birds downward they gradually lose their shapes to become a uniform background of sky and water, respectively.

• Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

According to Escher: “In the horizontal center strip there are birds and fish equivalent to each other. We associate flying with sky, and so for each of the black birds the sky in which it is flying is formed by the four white fish which encircle it. Similarly swimming makes us think of water, and therefore the four black birds that surround a fish become the water in which it swims.” This print has been used in physics, geology, chemistry, and in psychology for the study of visual perception. In the pictures a number of visual elements unite into a simple visual representation, but separately each forms a point of departure for the elucidation of a theory in one of these disciplines.

26.1 See also • Printmaking • Sky and Water II • Tessellation

26.2 Sources • M. C. Escher—The Graphic Work; BenediktTaschen Publishers. 34

Chapter 27

Sky and Water II Sky and Water II is a lithograph print by the Dutch artist M. C. Escher first printed in 1938. It is similar to the woodcut Sky and Water I, which was first printed only months earlier.

27.1 See also • Tessellation • Printmaking

27.2 Sources • M. C. Escher—The Graphic Work; BenediktTaschen Publishers. • M. C. Escher—29 Master Prints; Harry N. Abrams, Inc., Publishers.

35

Chapter 28

Snakes (M. C. Escher) Snakes is a woodcut print by the Dutch artist M. C. Escher first printed in July 1969. It depicts a disc made up of interlocking circles that grow progressively smaller towards the center and towards the edge. There are three snakes laced through the edge of the disc. Snakes has rotational symmetry of order 3, comprising a single wedge-shaped image repeated three times in a circle. This means that it was printed from three blocks that were rotated on a pin to make three impressions each. Close inspection reveals the central mark left by the pin. The image is printed in three colours: green, brown and black. In several earlier works Escher explored the limits of infinitesimal size and infinite number, for example the Circle Limit series, by actually carrying through the rendering of smaller and smaller figures to the smallest possible sizes. By contrast, in Snakes, the infinite diminution of size – and infinite increase in number – is only suggested in the finished work. Nevertheless, the print shows very clearly how this rendering would have been carried out to the limits of human visibility. This was Escher’s last print.

28.1 See also • Printmaking

28.2 References • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

28.3 External links • A 3-dimensional animation based on Escher’s print • A video of the artist making the print.

36

Chapter 29

Stars (M. C. Escher) Stars is a wood engraving print by the Dutch artist M. C. Escher first printed in October 1948, depicting two chameleons in a polyhedral cage floating through space.[1][2]

29.1 Description The print depicts a hollowed-out compound of three octahedra, a polyhedral compound composed of three regular octahedra, floating in space. Numerous other polyhedra and polyhedral compounds float in the background; the four largest are, on the upper left, the compound of cube and octahedron; on the upper right, the stella octangula; on the lower left, a compound of two cubes; and on the lower right, a solid version of the same octahedron 3compound. The smaller polyhedra visible within the print also include all of the five Platonic solids and the rhombic dodecahedron.[3][4] Two chameleons are contained within the cage-like shape of the central compound; Escher writes that they were chosen as its inhabitants “because they are able to cling by their legs and tails to the beams of their cage as it swirls through space”.[5] The chameleon on the left sticks out his tongue, perhaps in commentary; Coxeter observes that the tongue has an unusual spiral-shaped tip.[4]

A rhombicuboctahedron drawn by Leonardo da Vinci in a similar style to Escher

Escher’s interest in geometry is well known, but he was also an avid amateur astronomer, and in the early 1940s became a member of the Dutch Association for Meteorology and Astronomy. He owned a 6 cm refracting Although most published copies of Stars are telescope, and recorded several observations of binary monochromatic, with white stars and chameleons stars.[2] on a black background, the copy in the National Gallery of Canada is tinted in different shades of turquoise, The use of polyhedra to model heavenly bodies can be traced back to Plato, who wrote in Timaeus that the yellow, green, and pale pink.[6] constellations were arranged in the form of a regular dodecahedron. Later, Johannes Kepler theorized that the distribution of distances of the planets from the sun could 29.2 Influences be explained by the shapes of the five Platonic solids. Escher, also, regularly depicted polyhedra in his artworks The design for Stars was likely influenced by Escher’s own relating to astronomy and other worlds.[2] interest in both geometry and astronomy, by a long history of using geometric forms to model the heavens, and by Escher drew the octahedral compound of Stars in a a drawing style used by Leonardo da Vinci. However, beveled wire-frame style that was previously used by for Luca Pacioli's although the polyhedral shape depicted in Stars had been Leonardo da Vinci in his illustrations [4][3][7] book, De divina proportione. studied before in mathematics, it was most likely invented independently for this image by Escher without reference H. S. M. Coxeter reports that the shape of the central chameleon cage in Stars had previously been described, to those studies. 37

38 with a photograph of a model of the same shape, in 1900 by Max Brückner. However, Escher would not have been aware of this reference and Coxeter writes that “It is remarkable that Escher, without any knowledge of algebra or analytic geometry, was able to rediscover this highly symmetrical figure.”[4]

CHAPTER 29. STARS (M. C. ESCHER) As well as being exhibited in the Escher Museum, copies of Stars are in the permanent collections of the Mildred Lane Kemper Art Museum[13] and the National Gallery of Canada.[6]

29.6 References 29.3 Analysis

[1] Locher, J. L. (2000), The Magic of M. C. Escher, Harry N. Abrams, Inc., p. 100, ISBN 0-8109-6720-0.

Martin Beech interprets the many polyhedral compounds within Stars as corresponding to double stars and triple star systems in astronomy.[2] Beech writes that, for Escher, the mathematical orderliness of polyhedra depicts the “stability and timeless quality” of the heavens, and similarly Marianne L. Teuber writes that Stars “celebrates Escher’s identification with Johannes Kepler’s neoPlatonic belief in an underlying mathematical order in the universe”.[8]

[2] Beech, Martin, “Escher’s Stars", Journal of the Royal Astronomical Society of Canada 86: 169–177, Bibcode:1992JRASC..86..169B.

Alternatively, Howard W. Jaffe interprets the polyhedral forms in Stars crystallographically, as “brilliantly faceted jewels” floating through space, with its compound polyhedra representing crystal twinning.[9] However, R. A. Dunlap points out the contrast between the order of the polyhedral forms and the more chaotic biological nature of the chameleons inhabiting them.[10] In the same vein, Beech observes that the stars themselves convey tension between order and chaos: despite their symmetric shapes, the stars are scattered apparently at random, and vary haphazardly from each other.[2] As Escher himself wrote about the central chameleon cage, “I shouldn't be surprised if it wobbles a bit.”[2]

29.4 Related works

[3] Hart, George W. (1996), “The Polyhedra of M.C. Escher”, Virtual Polyhedra. [4] Coxeter, H. S. M. (1985), “A special book review: M. C. Escher: His life and complete graphic work”, The Mathematical Intelligencer 7 (1): 59–69, doi:10.1007/BF03023010. Coxeter’s analysis of Stars is on pp. 61–62. [5] Escher, M. C. (2001), M.C. Escher, the graphic work, Taschen, p. v, ISBN 978-3-8228-5864-6. [6] Stars, National Gallery of Canada, retrieved 2011-11-19. [7] Calter, Paul (1998), “The Platonic Solids”, Lecture Notes: Geometry in Art and Architecture, Dartmouth College. [8] Teuber, M. L. (July 1974), “Sources of ambiguity in the prints of Maurits C. Escher”, Scientific American 231: 90– 104, doi:10.1038/scientificamerican0774-90. [9] Jaffe, Howard W. (1996), “About the frontispiece”, Crystal Chemistry and Refractivity, Dover, p. vi, ISBN 978-0-486-69173-2. [10] Dunlap, R. A. (1992), “Fivefold symmetry in the graphic art of M. C. Escher”, in Hargittai, István, Fivefold Symmetry (2nd ed.), World Scientific, pp. 489–504, ISBN 978-981-02-0600-0.

A closely related woodcut, Study for Stars, completed in August 1948,[2][11] depicts wireframe versions of several [11] Locher (2000), p. 99. of the same polyhedra and polyhedral compounds, floating in black within a square composition, but without [12] Clute, John; Grant, John (1999), The encyclopedia of fantasy (2nd ed.), Macmillan, p. 322, ISBN 978-0-312the chameleons. The largest polyhedron shown in Study 19869-5. for Stars, a stellated rhombic dodecahedron, is also one of two polyhedra depicted prominently in Escher’s 1961 [13] Artwork detail, Kemper Museum, retrieved 2011-11-19. print Waterfall.[3] Escher’s later work Four Regular Solids (Stereometric Figure) returned to the theme of polyhedral compounds, depicting a more explicitly Keplerian form in which the compound of the cube and octahedron is nested within the compound of the dodecahedron and icosahedron.[10]

29.5 Collections and publications Stars was used as cover art for the 1962 anthology Best Fantasy Stories edited by Brian Aldiss.[12]

Chapter 30

Still Life and Street Still Life and Street is a woodcut print by the Dutch artist M. C. Escher which was first printed in March, 1937. It was his first print of an impossible reality. In this artwork we have two quite distinctly recognizable realities bound together in a natural, and yet at the same time a completely impossible, way. Looked at from the window, the houses make book-rests between which tiny dolls are set up. Looked at from the street, the books stand yards high and a gigantic tobacco jar stands at the crossroads. A small street in Savona, Italy, was the inspiration for this work.[1] Escher said it was one of his favorite drawings but thought he could have drawn it better. This image is a classic example of Escher’s plays on perspective. In it, the horizontal plane of the table continues into the distance to become the street, and the rows of books on the table are seen to lean against the tall buildings that line the street.

30.1 See also • Printmaking

30.2 References [1] “World of Escher Gallery”. Retrieved February 23, 2010.

30.3 Sources • Locher, J.L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

39

Chapter 31

Still Life with Mirror Still Life with Mirror is a lithograph by the Dutch artist M. C. Escher which was created in 1934.[1] The reflection of the mirror mingles together two completely unrelated spaces and introduces the outside world of the small town narrow street in Abruzzi into internal world of the bedroom.[2] This work of Escher is closely related to his later application of mirror effect in 1937 Still Life and Street.[3] Escher manipulates the scale in different parts of the print to achieve the effect of smooth connection between worlds.[4]

31.1 References [1] Doris Schattschneider; Michele Emmer (19 September 2005). M.C. Escher’s Legacy: A Centennial Celebration : Collection of Articles Coming from the M.C. Escher Centennial Conference, Rome, 1998. Springer. p. 219. ISBN 978-3-540-20100-7. Retrieved 17 June 2013. [2] Bruno Ernst (1994). The Magic Mirror of M.C. Escher. Barnes & Noble. pp. 22, 74. ISBN 978-1-56619-770-0. Retrieved 14 July 2013. [3] Norman Rockwell; M. C. Escher; J. C. Locher (1 June 1984). The World of M. C. Escher. Penguin USA. p. 7. ISBN 978-0-451-79959-3. Retrieved 18 July 2013. [4] Castner, Henry (2013). “The Robinson XI Projection”. Cartographic Perspectives. pp. 63–65. Retrieved 19 July 2013.

40

Chapter 32

Still Life with Spherical Mirror Still Life with Spherical Mirror is a lithography print by the Dutch artist M. C. Escher first printed in November 1934. It depicts a setting with rounded bottle and a metal sculpture of a bird with a human face seated atop a newspaper and a book. The background is dark but in the bottle can be seen the reflection of Escher’s studio and Escher himself sketching the scene. Self-portraits in reflective spherical surfaces can be found in Escher’s early ink drawings and in his prints as late as the 1950s. The metal bird/human sculpture is real and was given to Escher by his father-in-law. This sculpture appears again in Escher’s later prints Another World Mezzotint (Other World Gallery) (1946) and Another World (1947).

32.1 See also • Printmaking

32.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

41

Chapter 33

Three Spheres II Three Spheres II is a lithograph print by the Dutch artist M. C. Escher first printed in April 1946. As the title implies, it depicts three spheres resting on a flat surface. The sphere on the left is transparent with a photorealistic depiction of the refracted light cast through it towards the viewer and onto the flat surface. The sphere in the center is reflective. Its reflection is a self-replicating image of Escher in his studio drawing the three spheres. In the reflection one can clearly see the image of the three spheres on the paper Escher is drawing on: in the center sphere of that image, one can vaguely make out the reflection of Escher’s studio, which is depicted in the main image. This process is implied to be infinite, recursive. The sphere on the right is opaque and diffuse, i.e. neither specularly reflective nor transparent.

33.1 See also • Still Life with Spherical Mirror • Hand with Reflecting Sphere • Printmaking

33.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

42

Chapter 34

Three Worlds (M. C. Escher) Three Worlds is a lithograph print by the Dutch artist M. C. Escher first printed in December 1955. Three Worlds depicts a large pool or lake during the autumn or winter months, the title referring to the three visible perspectives in the picture: the surface of the water on which leaves float, the world above the surface, observable by the water’s reflection of a forest, and the world below the surface, observable in the large fish swimming just below the water’s surface. Escher also created a picture named “Two Worlds”.

34.1 See also • Puddle • Printmaking The picture is based on true optical effects, reflection and refraction. The angle of incidence is the line between the reflection of the trees and refraction allowing the view of the fish.

34.2 Sources • Locher, J. L. (2000). The Magic of M. C. Escher. Harry N. Abrams, Inc. ISBN 0-8109-6720-0.

34.3 External links • Gallery of Eschers images

43

Chapter 35

Tower of Babel (M. C. Escher) Tower of Babel is a 1928 woodcut by M. C. Escher. It depicts the Babylonians attempting to build a tower to reach God, a story that is recounted in Genesis 11:9. God frustrated their attempts by creating a confusion of languages so the builders could no longer understand each other and the work halted. Although Escher dismissed his works before 1935 as of little or no value as they were “for the most part merely practice exercises”, some of them, including the Tower of Babel, chart the development of his interest in perspective and unusual viewpoints that would become the hallmarks of his later, more famous, work.

35.3 Notes

In contrast to many other depictions of the biblical story, such as those by Pieter Brueghel the Elder (The Tower of Babel) and Gustave Doré (The Confusion of Tongues), Escher depicts the tower as a geometrical structure and places the viewpoint above the tower. This allows him to exercise his skill with perspective, but he also chose to centre the picture around the top of the tower as the focus for the climax of the action. He later commented:

Some of the builders are white and others black. The work is at a standstill because they are no longer able to understand one another. Seeing as the climax of the drama takes place at the summit of the tower which is under construction, the building has been shown from above as though from a bird’s eye view[1]

35.1 See also • Belvedere • Waterfall

35.2 References [1] Finkel, I. L.; Seymour, M. J., eds. (2009). Babylon. Oxford University Press. ISBN 978-0-19-538540-3.

44

• Miranda Fellows (1995). The Life and Works of Escher. Bristol: Paragon Book Service. ISBN 0-75251175-0.

Chapter 36

Waterfall (M. C. Escher) Waterfall (Waterval) is a lithograph print by the Dutch artist M. C. Escher first printed in October 1961. It shows an apparent paradox where water from the base of a waterfall appears to run downhill along the water path before reaching the top of the waterfall.

that Escher drew in ink as a study in 1942. The background seems to be a climbing expanse of terraced farmland.

This drawing seemingly depicts a violation of the principle of conservation of energy of physics, due to the fact While most two-dimensional artists use relative propor- that the water gains kinetic energy, but does not lose any tions to create an illusion of depth, Escher here and elsegravitational potential energy. where uses conflicting proportions to create a visual paradox. The waterfall’s leat has the structure of two Penrose triangles. A Penrose triangle is an impossible object designed by Oscar Reutersvärd in 1934, and independently 36.2 References by Roger Penrose in 1958.[1] [1] Penrose, L. S.; Penrose, R. (1958). “Impossible objects: A special type of visual illusion”. British Journal of Psychology 49 (1): 31–33. doi:10.1111/j.20448295.1958.tb00634.x. PMID 13536303.

36.1 Description The image depicts a village or small city with an elevated aqueduct and waterwheel as the main feature. The aqueduct begins at the waterwheel and flows behind it. The walls of the aqueduct step downward, suggesting that it slopes downhill. The aqueduct turns sharply three times, first to the left, then straight forward and finally to the left again. The viewer looks down at the scene diagonally, which means that from the viewer’s perspective the aqueduct appears to be slanted upward. The viewer is also looking across the scene diagonally from the lower right, which means that from the viewer’s perspective the two left-hand turns are directly in line with each other, while the waterwheel, the forward turn and the end of the aqueduct are all in line. The second left-hand turn is supported by pillars from the first, while the other two corners are supported by a tower of pillars that begins at the waterwheel. The water falls off the edge of the aqueduct and over the waterwheel in an infinite cycle; in his notes on the picture, Escher points out that some water must be periodically added to this apparent perpetual motion machine to compensate for evaporation. The two support towers continue above the aqueduct and are topped by two compound polyhedra. The one on the left is a compound of three cubes. The one on the right is a stellation of a rhombic dodecahedron (or a compound of three nonregular octahedra) and is known as Escher’s solid.

36.3 External links

Below the mill is a garden of bizarre, giant plants. This is actually a magnified view of a cluster of moss and lichen 45

• Escher’s Solid—from Wolfram MathWorld • Escher’s Solid Includes a great deal of metric data • The Polyhedra of M.C. Escher from George W. Hart

46

CHAPTER 36. WATERFALL (M. C. ESCHER)

36.4 Text and image sources, contributors, and licenses 36.4.1

Text

• M. C. Escher Source: http://en.wikipedia.org/wiki/M.%20C.%20Escher?oldid=643000547 Contributors: Tarquin, Koyaanis Qatsi, Jeronimo, Ap, Sjc, Andre Engels, Scipius, Danny, Tsja, Rootbeer, Cyrek, Tedernst, Leandrod, Patrick, JohnOwens, Michael Hardy, Gabbe, Wapcaplet, Ixfd64, GTBacchus, Paul A, Looxix, Ahoerstemeier, GGano, Suisui, Angela, Jebba, Andres, Raven in Orbit, Conti, Schneelocke, Gamma, Lenaic, Dcoetzee, WhisperToMe, Timc, Topbanana, MD87, PuzzletChung, Robbot, Murray Langton, 1984, Fredrik, Romanm, Gandalf61, Chris Roy, P0lyglut, Gidonb, Blainster, Diderot, LGagnon, Iaen, CdaMVvWgS, Intangir, Giftlite, Graeme Bartlett, Christopher Parham, Gtrmp, Aratuk, Folks at 137, Zigger, Bradeos Graphon, Snowdog, Curps, NeoJustin, Alison, FriedMilk, Vodka, Solipsist, Bobblewik, Mateuszica, Quadell, Noe, Antandrus, Piotrus, Jossi, MacGyverMagic, Tomruen, Urhixidur, Joyous!, Gerrit, Adashiel, Trevor MacInnis, Piotras, Mike Rosoft, D6, MichaelMcGuffin, Justin Foote, Slady, Vinoir, Discospinster, Helohe, Zaheen, Rich Farmbrough, Guanabot, Pmaccabe, Ardonik, Smyth, Ericamick, Bender235, ESkog, Mashford, JoeSmack, Theinfo, Kwamikagami, Shanes, RoyBoy, Adambro, Renice, Bobo192, Valve, BrokenSegue, Viriditas, Dee Earley, Slinky Puppet, Rje, Twexler, Sam Korn, Haham hanuka, Erri4a, Ogress, MCiura, Alansohn, Jeltz, Andrewpmk, Paleorthid, Ricky81682, Burn, WikiParker, Snowolf, Blobglob, Gbeeker, Randy Johnston, ShawnVW, Ianblair23, Redvers, Dismas, Alexander Maier, Simetrical, Jeffrey O. Gustafson, Bjones, David Haslam, Samsoncity, Mandavi, WadeSimMiser, Iisryan, CharlesC, Liface, Dysepsion, Mandarax, Matilda, Graham87, Sparkit, Kbdank71, ConradKilroy, Wikix, Nightscream, Koavf, Astropithicus, Isantala, Loudenvier, Staecker, TheRingess, MZMcBride, SeanMack, Afterwriting, Fred Bradstadt, FlaBot, M4tth, Naraht, Michaelbluejay, RobertG, DannyWilde, Nihiltres, Uglinessman, Klosterdev, Ayla, DevastatorIIC, Nicapicella, Alphachimp, Bmicomp, King of Hearts, Jeffr, Chobot, SirGrant, DVdm, Random user 39849958, Cactus.man, Hall Monitor, B., Sophitus, Ben Tibbetts, JPD, YurikBot, Wavelength, Sceptre, Jack Cain, Phantomsteve, Todasco, Kyorosuke, Neilbeach, Oni Lukos, Anomalocaris, K.C. 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HexaChord, Angryapathy, Felix Folio Secundus, Addbot, BONKEROO, Wingspeed, Fieldday-sunday, Mr. Wheely Guy, CanadianLinuxUser, Robbinsville, Jaeger123, Derekliu730, Eivindbot, DFS454, Chzz, Favonian, Vysotsky, Tassedethe, Tide rolls, Avono, Gail, Jarble, Trotter, Quantumobserver, Legobot, Luckas-bot, Yobot, General Doggy, Ptbotgourou, Fraggle81, GreatInDayton, Anypodetos, Dra0003, Pepinzoe, Chelley.t, Jim1138, Piano non troppo, AdjustShift, Kingpin13, Warriorow, Materialscientist, ImperatorExercitus, Eightheads, Dromioofephesus, OllieFury, ArthurBot, Nifky?, NW’s Public Sock, Xqbot, Capricorn42, Jesusjones22, Gilo1969, Jmundo, Abce2, Frankie0607, RibotBOT, Bellerophon, Garshgrang, Sensesused, Shadowjams, Dan6hell66, Captain-n00dle, FrescoBot, Dogposter, Pepper, Tony Wills, Xhaoz, DivineAlpha, Dseg33, WQUlrich, Wandererlad, Bone head89, Toaster oven09, Thaeralali, Loopygrumpkins, Tomcat7, RedBot, Nderadio96, Abc518, TobeBot, Professor Fiendish, Sheogorath, Benmurdoch, Lotje, Javierito92, Vrenator, Clarkcj12, Nemesis of Reason, Bluefist, Bramley007, Specs112, Tbhotch, DARTH SIDIOUS 2, Alph Bot, Samuelhong55789237, ArtquestLondon, EmausBot, Kchappell32, Fæ, Josve05a, Gz33, Wayne Slam, HandsomeFella, Spicemix, ClueBot NG, Jnorton7558, Frietjes, Concord113, Timmmm4, Mark K Adams, Helpful Pixie Bot, TetraEleven, Lawandeconomics1, T0134125H, Mark Arsten, OttawaAC, Dainomite, Snow Blizzard, FatWhite&Nerdi2000, Glacialfox, Fred C. Anderson, PaulLouisM, SergeantHippyZombie, Cyberbot II, Dexbot, Franck Holland, Mogism, Lugia2453, VIAFbot, Graphium, Hippocamp, Epicgenius, Red-eyed demon, Nonsenseferret, Alexandre Candalaft, EvergreenFir, DavidLeighEllis, ReconditeRodent, Luvartanddesign, Patbdwll, Guyonaudo, Stormmeteo, Monkbot, Xxylelxx and Anonymous: 920 • Another World (M. C. Escher) Source: http://en.wikipedia.org/wiki/Another%20World%20(M.%20C.%20Escher)?oldid=585947649 Contributors: Fuelbottle, Mdob, MakeRocketGoNow, Justin Foote, Deror avi, Platypus222, Nihiltres, Anomalocaris, Attilios, SmackBot, Gary2863, Unint, Michaelbusch, Bentendo24, Cydebot, Soetermans, WinBot, Goldenrowley, Evaunit666, Johnbod, GrahamHardy, GlassFET, Fantastic fred, Vanished user 82345ijgeke4tg, Anticipation of a New Lover’s Arrival, The, Addbot, Lightbot, Anypodetos, Tb94114, Citation bot, Petropoxy (Lithoderm Proxy), Lunaibis, Wikipelli, ClueBot NG and Anonymous: 10

36.4. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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• Ascending and Descending Source: http://en.wikipedia.org/wiki/Ascending%20and%20Descending?oldid=628867960 Contributors: Hyacinth, AnonMoos, Cluth, Chowbok, Ouro, Perey, Justin Foote, Kinitawowi, Rje, Deror avi, Stephen, Mandarax, Nightscream, Nihiltres, Anomalocaris, ErkDemon, Jpbowen, Eptin, SmackBot, InverseHypercube, Sirgregmac, Valenciano, Danny Beaudoin, Peter Horn, Ugo1970, Cydebot, Cruncher, Goldenrowley, Fetchcomms, Quanyails, Boffob, Pernambuco, Yonidebot, Johnbod, Dark Ermac, Barraki, GrahamHardy, VolkovBot, Tanveerbadar, Matty B 1000, BlueVelvet86, Abaroth, ClueBot, Chimino, Addbot, BONKEROO, Lightbot, Zorrobot, Yobot, Anypodetos, Calle, Redgirly123, Herr Satz, Shaydalton, Enki H., Jujutacular, Lunaibis, Jncobbs, My name is not dave and Anonymous: 30 • Atrani, Coast of Amalfi Source: http://en.wikipedia.org/wiki/Atrani%2C%20Coast%20of%20Amalfi?oldid=602810480 Contributors: Altenmann, Fuelbottle, MakeRocketGoNow, Mike Rosoft, Justin Foote, Nihiltres, Cactus.man, Cydebot, Oreo Priest, WinBot, Goldenrowley, Derlay, Johnbod, 5theye, GrahamHardy, FlagSteward, DerBorg, Addbot, Lightbot, Luckas-bot, Anypodetos, Lunaibis and Anonymous: 5 • Belvedere (M. C. Escher) Source: http://en.wikipedia.org/wiki/Belvedere%20(M.%20C.%20Escher)?oldid=581938560 Contributors: AnonMoos, Nyh, Jorge Stolfi, Sonett72, MakeRocketGoNow, Mike Rosoft, Justin Foote, Aris Katsaris, Tomheaton, Rje, Deror avi, Banpei, Sparkit, Vkrnt, Platypus222, Nihiltres, Tavilis, Anomalocaris, ErkDemon, ThreeBlindMice, Cydebot, Goldenrowley, David Eppstein, PCock, Johnbod, GrahamHardy, CryptWizard, Abaroth, Mack-the-random, Addbot, Lightbot, Zorrobot, Anypodetos, HRoestBot, Joesodacapn, EmausBot, Lunaibis, ClueBot NG and Anonymous: 11 • The Bridge (M. C. Escher) Source: http://en.wikipedia.org/wiki/The%20Bridge%20(M.%20C.%20Escher)?oldid=585948883 Contributors: Blainster, Fuelbottle, Jason Quinn, Blankfaze, MakeRocketGoNow, Justin Foote, MockTurtle, Deror avi, Tabletop, Sparkit, Nihiltres, JM.Beaubourg, Cactus.man, Cydebot, Goldenrowley, Johnbod, GrahamHardy, Ntropia, Addbot, LaaknorBot, Lightbot, Anypodetos, Doc Quintana, Lunaibis and Anonymous: 4 • Castrovalva (M. C. Escher) Source: http://en.wikipedia.org/wiki/Castrovalva%20(M.%20C.%20Escher)?oldid=637772444 Contributors: Dino, Wikiborg, Andrewman327, Fuelbottle, Khaosworks, Kuralyov, MakeRocketGoNow, Parmadil, Justin Foote, Deror avi, Splintax, Nihiltres, Cyclone49, Tamfang, DavidCooke, Cydebot, Nick Number, Goldenrowley, Michig, Fray Pentaro, Johnbod, 5theye, GrahamHardy, Pesak, Addbot, Lightbot, Luckas-bot, Anypodetos, FrescoBot, Lunaibis, The Herald and Anonymous: 9 • Circle Limit III Source: http://en.wikipedia.org/wiki/Circle%20Limit%20III?oldid=632226953 Contributors: Hyacinth, Tomruen, Ericoides, David Eppstein, Piledhigheranddeeper, Double sharp and Chris857 • Convex and Concave Source: http://en.wikipedia.org/wiki/Convex%20and%20Concave?oldid=640805094 Contributors: Hadal, Tobias Bergemann, MakeRocketGoNow, Justin Foote, Foobaz, Deror avi, Camw, Sparkit, Nihiltres, Mar-Vell, SmackBot, Commander Keane bot, Asydwaters, OrphanBot, BocoROTH, Tunmise, Kingfish, Cydebot, Goldenrowley, Quanyails, David Eppstein, STBot, Johnbod, GrahamHardy, Aspects, Martarius, ClueBot, Tameamseo, Addbot, Lightbot, Anypodetos, Cresix, Lunaibis, ClueBot NG, 1234Wearecool, Melcous and Anonymous: 14 • Cube with Magic Ribbons Source: http://en.wikipedia.org/wiki/Cube%20with%20Magic%20Ribbons?oldid=558088417 Contributors: Smjg, Justin Foote, Bender235, DreamGuy, GregorB, Nihiltres, Anomalocaris, ErkDemon, SmackBot, Durova, OrphanBot, Omega9380, DanielRigal, Cydebot, Johnbod, Fleebo, GrahamHardy, Phyte, Lightbot, Anypodetos, JazzieIce!, Lunaibis, Helpful Pixie Bot and Anonymous: 7 • Curl-up Source: http://en.wikipedia.org/wiki/Curl-up?oldid=565754208 Contributors: Lee M, Quadell, MakeRocketGoNow, Justin Foote, Rich Farmbrough, Chibimagic, Deror avi, Reinoutr, Sparkit, HopDavid, Nihiltres, SpectrumDT, Cactus.man, Anomalocaris, SmackBot, Keegan, [email protected], Tawkerbot2, Cydebot, Goldenrowley, EdwinGroothuis, Sluzzelin, Swpb, Mausy5043, Johnbod, GrahamHardy, Serprex, Ainlina, ClueBot, Addbot, Lightbot, Anypodetos, Ripohopeteg, Lunaibis, ClueBot NG, Sapphiredenise26 and Anonymous: 9 • Dolphins (M. C. Escher) Source: http://en.wikipedia.org/wiki/Dolphins%20(M.%20C.%20Escher)?oldid=624705226 Contributors: Fuelbottle, Quadell, MakeRocketGoNow, Justin Foote, Nihiltres, Cactus.man, SmackBot, GoodDay, Ser Amantio di Nicolao, Cydebot, MarshBot, Goldenrowley, Sluzzelin, Magioladitis, Johnbod, GrahamHardy, Flyer22, Life of Riley, Addbot, Imbrickle, Favonian, Lightbot, Anypodetos, Storm42, Petropoxy (Lithoderm Proxy), Lunaibis, ZéroBot, Orange Suede Sofa, ClueBot NG, Faizan and Anonymous: 15 • Drawing Hands Source: http://en.wikipedia.org/wiki/Drawing%20Hands?oldid=623449380 Contributors: Aratuk, MakeRocketGoNow, Trevor MacInnis, Justin Foote, Discospinster, BrokenSegue, Alansohn, Mandarax, Deltabeignet, Ianthegecko, Nihiltres, Anomalocaris, Grafen, Dyefade, Kubra, Verne Equinox, Doc Strange, Bluebot, Tunmise, Hvn0413, Twas Now, Outriggr, HonztheBusDriver, Cydebot, Thijs!bot, Mentifisto, Goldenrowley, Spencer, Bongwarrior, Jonathanzung, Johnbod, Fountains of Bryn Mawr, Fleebo, SixteenBitJorge, GrahamHardy, Tomer T, SieBot, Flyer22, ClueBot, Adgjladgjl, Addbot, Mentisock, Tide rolls, Lightbot, Maxis ftw, Recognizance, TheWatchDude, Meaghan, Gorko, TheMesquito, Lunaibis, ZéroBot, Coasterlover1994, Smartie2thaMaxXx, ClueBot NG, TCN7JM, VeryCrocker, Aryanbhargava and Anonymous: 51 • Gravitation (M. C. Escher) Source: http://en.wikipedia.org/wiki/Gravitation%20(M.%20C.%20Escher)?oldid=558086749 Contributors: Ajd, Tomruen, MakeRocketGoNow, Canterbury Tail, Justin Foote, MBisanz, Rje, DanielLC, Axl, Tchalvak, Ardfern, Sparkit, HopDavid, Nihiltres, King of Hearts, Anomalocaris, Tamfang, Tunmise, FlocciNonFacio, Cydebot, Oreo Priest, Goldenrowley, Steelpillow, Sluzzelin, Johnbod, 49erDuck, TnTGamer, Addbot, Lightbot, Anypodetos, Lunaibis, Grapple X and Anonymous: 9 • Hand with Reflecting Sphere Source: http://en.wikipedia.org/wiki/Hand%20with%20Reflecting%20Sphere?oldid=623449139 Contributors: Wwwwolf, Angela, Darkwind, Auric, Fuelbottle, MakeRocketGoNow, Justin Foote, Thu, SlimVirgin, Deror avi, AySz88, Nihiltres, Fragglet, Bloodofox, Jonathan.s.kt, Attilios, Ohbusiness, GrafZahl, JeffyP, John, Mgiganteus1, Outriggr, Cydebot, Christian75, DBaba, WinBot, Goldenrowley, Kainino, Phort99, Johnbod, 5theye, Skier Dude, GrahamHardy, Dendodge, Jake178656, Radical.bison, ClueBot, Sun Creator, Egmontaz, Rror, Addbot, Lithoderm, Anypodetos, Flanker0007, Lunaibis, Fæ, ClueBot NG, Skarmenadius, ChromeAce, Ele boz, Lugia2453, VeryCrocker and Anonymous: 52 • House of Stairs Source: http://en.wikipedia.org/wiki/House%20of%20Stairs?oldid=558086556 Contributors: Lee M, Samsara, Iaen, Alan Liefting, Chowbok, MakeRocketGoNow, Justin Foote, Axl, Deror avi, Sparkit, Nihiltres, Argyrios Saccopoulos, SpectrumDT, SpuriousQ, Gaius Cornelius, Anomalocaris, Ketsuekigata, SmackBot, Tunmise, Yadaman, JohnI, Cydebot, Xantharius, Goldenrowley, Albmont, Rettetast, GrahamHardy, Addbot, Lightbot, Anypodetos, DiverDave, Armbrust, Lunaibis, Benabbes ilyes and Anonymous: 12 • Magic Mirror (M.C. Escher) Source: http://en.wikipedia.org/wiki/Magic%20Mirror%20(M.C.%20Escher)?oldid=585982369 Contributors: Ixfd64, Fuelbottle, Jason Quinn, MakeRocketGoNow, Justin Foote, ShawnVW, Deror avi, Nihiltres, RussBot, Anomalocaris, CmdrObot, Jibi44, Cydebot, Goldenrowley, Mrob27, Johnbod, 5theye, GrahamHardy, Station1, Lightbot, Yobot, Anypodetos, Rameshngbot, EmausBot, Lunaibis, Silv the Something, Cold Season and Anonymous: 5

48

CHAPTER 36. WATERFALL (M. C. ESCHER)

• Metamorphosis I Source: http://en.wikipedia.org/wiki/Metamorphosis%20I?oldid=596993215 Contributors: Fuelbottle, Jason Quinn, MakeRocketGoNow, Justin Foote, Rje, Nihiltres, Cactus.man, Wavelength, Anomalocaris, Attilios, Cydebot, Oreo Priest, WinBot, Goldenrowley, Johnbod, GrahamHardy, Addbot, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), Lunaibis and Anonymous: 3 • Metamorphosis II Source: http://en.wikipedia.org/wiki/Metamorphosis%20II?oldid=596993563 Contributors: Wikiborg, Hadal, Fuelbottle, Jason Quinn, MakeRocketGoNow, Justin Foote, Rje, FlaBot, Nihiltres, Cactus.man, Wavelength, Anomalocaris, Ljump12, Attilios, SmackBot, Cydebot, Oreo Priest, WinBot, Goldenrowley, Sluzzelin, KConWiki, Johnbod, GlassCobra, Addbot, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), Lunaibis and Anonymous: 9 • Metamorphosis III Source: http://en.wikipedia.org/wiki/Metamorphosis%20III?oldid=558089664 Contributors: Merphant, Wikiborg, MakeRocketGoNow, Justin Foote, Vystrix Nexoth, KBi, Unixer, Sparkit, Nihiltres, Cactus.man, Anomalocaris, Attilios, SmackBot, Cydebot, Goldenrowley, Sluzzelin, Johnbod, Cometstyles, GrahamHardy, Falcon8765, Adamfinmo, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), Lunaibis and Anonymous: 14 • Print Gallery (M. C. Escher) Source: http://en.wikipedia.org/wiki/Print%20Gallery%20(M.%20C.%20Escher)?oldid=573733602 Contributors: Anomalocaris, Katharineamy, GrahamHardy, Regregex, Jsfouche, Topher385, Kilom691, Σ, RjwilmsiBot, AvicBot, Helpful Pixie Bot, BG19bot, EagerToddler39 and Anonymous: 5 • Puddle (M. C. Escher) Source: http://en.wikipedia.org/wiki/Puddle%20(M.%20C.%20Escher)?oldid=629578776 Contributors: Ahoerstemeier, MakeRocketGoNow, Mike Rosoft, Justin Foote, Roo72, JRM, Swamp Ig, Ardfern, Mandarax, Sparkit, Pispöki, Nihiltres, JM.Beaubourg, Hellbus, StuRat, Smurrayinchester, OrphanBot, Tunmise, Mphinney, Cydebot, Goldenrowley, Johnbod, Americanfreedom, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), MegaSloth, EmausBot, Lunaibis, ClueBot NG, Pratyya Ghosh and Anonymous: 16 • Regular Division of the Plane Source: http://en.wikipedia.org/wiki/Regular%20Division%20of%20the%20Plane?oldid=631124333 Contributors: Charles Matthews, Hyacinth, Altenmann, Fuelbottle, Smjg, MakeRocketGoNow, Justin Foote, Rich Farmbrough, Jonathunder, Ringbang, GregorB, Cactus.man, Nlu, SmackBot, CBM, Cydebot, WinBot, Goldenrowley, Sluzzelin, Sophie means wisdom, Johnbod, Dekaptein, Chillum, Lightbot, Anypodetos, Measles, Petropoxy (Lithoderm Proxy), Shadowjams, WindingPaths, KLBot2, Ele boz, Brad7777, Khazar2, Reatlas and Anonymous: 10 • Relativity (M. C. Escher) Source: http://en.wikipedia.org/wiki/Relativity%20(M.%20C.%20Escher)?oldid=638896886 Contributors: GTBacchus, AnonMoos, Andycjp, MakeRocketGoNow, Justin Foote, Vague Rant, Smyth, Root4(one), Bobo192, Cmdrjameson, Alansohn, Netkinetic, Deror avi, Kevin Lomax, Wikipedian231, Mandarax, Graham87, Nightscream, Nihiltres, Sceptre, Huw Powell, Jlittlet, Haoie, BazookaJoe, JQF, SmackBot, EvilCouch, Ze miguel, DWaterson, Verne Equinox, Amatulic, Raymondluxuryacht, Joerite, Valenciano, Dream out loud, Incarania, Denimcat, Marm, Rogerbrent, Andyroo316, Ryulong, Andreworkney, Outriggr, Senorelroboto, Matt. P, Cydebot, Anonymi, Jonathon Black, Thijs!bot, Sagaciousuk, Davidhorman, Luna Santin, Goldenrowley, JAnDbot, Gfwellman, Sho222, Partymetroid, Pomte, Yonidebot, Johnbod, JRNorbergé, Dark Ermac, Barraki, GrahamHardy, Deor, Philip Trueman, Johnred32, Enigmaman, Antixt, Shadow Falcon, StAnselm, Mikemoral, PurpleTigerFish, Thorrstein, Martarius, ClueBot, Binksternet, Graoully, Jusdafax, Haz7po5, NuclearWarfare, Thinboy00P, Addbot, Proxima Centauri, Lightbot, Anypodetos, Anonymous from the 21th century, Jim1138, Materialscientist, ArthurBot, The Slee, Pink cloudy sky, Regancy42, John Cline, Wackywace, Dffgd, ClueBot NG, Deebo993, Sanfazer, Rm1271 and Anonymous: 102 • Reptiles (M. C. Escher) Source: http://en.wikipedia.org/wiki/Reptiles%20(M.%20C.%20Escher)?oldid=637131761 Contributors: Fuelbottle, MakeRocketGoNow, Justin Foote, Discospinster, Axl, WBardwin, Nihiltres, Bombe, Anomalocaris, Tanet, SmackBot, Tamfang, Cydebot, Thijs!bot, WinBot, Goldenrowley, Sluzzelin, CapnPrep, Captain Infinity, Johnbod, GrahamHardy, IPSOS, Addbot, Lightbot, Anypodetos, Lunaibis, Sd31263, DASHBotAV and Anonymous: 13 • Sky and Water I Source: http://en.wikipedia.org/wiki/Sky%20and%20Water%20I?oldid=589454687 Contributors: Jason Quinn, Alexf, Justin Foote, Deror avi, Jeff3000, Nihiltres, Anomalocaris, HeartofaDog, Hmains, TenPoundHammer, LadyofShalott, Cydebot, N5iln, Sluzzelin, Maias, Ferritecore, Spider-X, Alex LaPointe, Johnbod, Nave.notnilc, ClueBot, Anypodetos, Ciphers, Petropoxy (Lithoderm Proxy), Lotje, DARTH SIDIOUS 2, Lunaibis, ClueBot NG, Adamrce and Anonymous: 14 • Sky and Water II Source: http://en.wikipedia.org/wiki/Sky%20and%20Water%20II?oldid=590865416 Contributors: Jason Quinn, Justin Foote, Deror avi, Nihiltres, Anomalocaris, Hmains, Betacommand, LadyofShalott, Cydebot, Sluzzelin, Maias, Johnbod, GrahamHardy, Solo1234, Lithoderm, Anypodetos, Lunaibis, ClueBot NG and Anonymous: 1 • Snakes (M. C. Escher) Source: http://en.wikipedia.org/wiki/Snakes%20(M.%20C.%20Escher)?oldid=571526080 Contributors: GTBacchus, MakeRocketGoNow, Justin Foote, Vinoir, ESkog, Sparkit, Nihiltres, Anomalocaris, Amatulic, OrphanBot, Tunmise, Cottingham, John, Heavy1974, Pithecanthropus, Cydebot, Moterola4, Goldenrowley, Egpetersen, Sluzzelin, Johnbod, Jdubsonetrillion, GrahamHardy, Kryptocow, Favonian, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), Lunaibis, ClueBot NG and Anonymous: 8 • Stars (M. C. Escher) Source: http://en.wikipedia.org/wiki/Stars%20(M.%20C.%20Escher)?oldid=621494526 Contributors: Hadal, MakeRocketGoNow, Justin Foote, Rich Farmbrough, Rjwilmsi, HopDavid, Nihiltres, Anomalocaris, HeartofaDog, SchfiftyThree, OrphanBot, Tunmise, Lmcelhiney, Cydebot, Lossenelin, Goldenrowley, David Eppstein, Mausy5043, Johnbod, GrahamHardy, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), Metricmike, Lunaibis, ClueBot NG, Helpful Pixie Bot and Anonymous: 7 • Still Life and Street Source: http://en.wikipedia.org/wiki/Still%20Life%20and%20Street?oldid=585949818 Contributors: Cluth, Academic Challenger, Fuelbottle, Jason Quinn, MakeRocketGoNow, Justin Foote, GregorB, Matt Deres, Nihiltres, Cactus.man, SmackBot, Krispos42, Cydebot, Oreo Priest, AntiVandalBot, WinBot, Goldenrowley, EagleFan, Johnbod, SixteenBitJorge, Philip Trueman, ClueBot, Rai27, Addbot, Lithoderm, Lightbot, Anypodetos, Petropoxy (Lithoderm Proxy), Markeilz, Ondokuzmart, Lunaibis, ClueBot NG, Ele boz, Bzweebl and Anonymous: 17 • Still Life with Mirror Source: http://en.wikipedia.org/wiki/Still%20Life%20with%20Mirror?oldid=587023752 Contributors: Nihiltres, Woodshed, David Eppstein and AgadaUrbanit • Still Life with Spherical Mirror Source: http://en.wikipedia.org/wiki/Still%20Life%20with%20Spherical%20Mirror?oldid=585949405 Contributors: Fuelbottle, Jason Quinn, MakeRocketGoNow, Justin Foote, Deror avi, FlaBot, Nihiltres, Anomalocaris, Attilios, Cydebot, Christian75, WinBot, Goldenrowley, Johnbod, GrahamHardy, Vanished user 82345ijgeke4tg, SilverVishnu, Addbot, Lightbot, Anypodetos, Sketchmoose, Lunaibis, ZéroBot and Anonymous: 3 • Three Spheres II Source: http://en.wikipedia.org/wiki/Three%20Spheres%20II?oldid=590949977 Contributors: Kku, Fuelbottle, MakeRocketGoNow, Justin Foote, Deror avi, Nihiltres, Neilbeach, Anomalocaris, GraemeL, Hmains, TimBentley, GoodDay, Cydebot, WinBot, Goldenrowley, Bakabaka, Johnbod, GrahamHardy, Ryan032, Addbot, Lightbot, Anypodetos, Lunaibis and Anonymous: 7

36.4. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

49

• Three Worlds (M. C. Escher) Source: http://en.wikipedia.org/wiki/Three%20Worlds%20(M.%20C.%20Escher)?oldid=587410640 Contributors: MakeRocketGoNow, Justin Foote, Vinoir, Sparkit, BD2412, Nihiltres, Ismail, Anomalocaris, Smurrayinchester, SmackBot, OrphanBot, Tunmise, Hetar, Cydebot, Lugnuts, Shirt58, Goldenrowley, TimVickers, Larry Rosenfeld, VoABot II, ExplicitImplicity, J.delanoy, Uncle Dick, Johnbod, GrahamHardy, BlackCab, Lightbot, Yobot, Anypodetos, Clarkcj12, Lunaibis, Delusion23 and Anonymous: 9 • Tower of Babel (M. C. Escher) Source: http://en.wikipedia.org/wiki/Tower%20of%20Babel%20(M.%20C.%20Escher)?oldid= 635381358 Contributors: Tom harrison, Justin Foote, Bender235, Deror avi, BD2412, Nihiltres, SmackBot, Hmains, Ceoil, HisSpaceResearch, Neelix, Cydebot, Yomangani, R'n'B, Johnbod, GrahamHardy, Yomangan, Good Olfactory, Addbot, Anypodetos, Citation bot, Petropoxy (Lithoderm Proxy), Super Goku V, Lunaibis, Everything Else Is Taken, ZéroBot, SporkBot, Petrb, Bzweebl and Anonymous: 3 • Waterfall (M. C. Escher) Source: http://en.wikipedia.org/wiki/Waterfall%20(M.%20C.%20Escher)?oldid=628867978 Contributors: Lee M, Hyacinth, AndrewKepert, Bearcat, Ajd, MakeRocketGoNow, Justin Foote, Tomheaton, Alphax, Gbeeker, ReyBrujo, Deror avi, GregorB, Sparkit, BD2412, Rjwilmsi, HopDavid, Vkrnt, Nihiltres, Nivix, JM.Beaubourg, RussBot, Gaius Cornelius, Anomalocaris, ErkDemon, Zagalejo, SmackBot, Bazonka, Ghiraddje, Tunmise, BlackTerror, Reade, Poa, Hvn0413, Cydebot, Stuston, Thijs!bot, Eleuther, Goldenrowley, CommonsDelinker, Pbroks13, Silas S. Brown, Fountains of Bryn Mawr, Cometstyles, Joanenglish, GrahamHardy, Robert Stanforth, Srushe, JohnTopShelf, Binksternet, Mattgirling, Trivialist, M4gnum0n, Marius Vordal, Addbot, Lithoderm, Adrian 1001, Tide rolls, Lightbot, Zorrobot, Anypodetos, AnomieBOT, Ciphers, Mauro Lanari, Truth or consequences-2, Erik9bot, FrescoBot, Dinamik-bot, RjwilmsiBot, EmausBot, Lunaibis, ZéroBot, Wmayner, Thine Antique Pen, ClueBot NG, Wolf of Thor, Nineran and Anonymous: 52

36.4.2

Images

• File:2C_3_1979.JPG Source: http://upload.wikimedia.org/wikipedia/commons/4/4c/2C_3_1979.JPG License: CC BY-SA 3.0 Contributors: Gerard Caris Original artist: Gerard Caris • File:Ascending_and_Descending.jpg Source: http://upload.wikimedia.org/wikipedia/en/6/66/Ascending_and_Descending.jpg License: Fair use Contributors: Official M. C. Escher website Original artist: ? • File:Atrani.JPG Source: http://upload.wikimedia.org/wikipedia/commons/f/f7/Atrani.JPG License: CC-BY-SA-3.0 Contributors: ? Original artist: ? • File:Ballerina-icon.jpg Source: http://upload.wikimedia.org/wikipedia/commons/3/3a/Ballerina-icon.jpg License: CC-BY-SA-3.0 Contributors: • Snowdance.jpg Original artist: Snowdance.jpg: Rick Dikeman • File:Belvedere.jpg Source: http://upload.wikimedia.org/wikipedia/en/5/52/Belvedere.jpg License: Fair use Contributors: Official M. C. Escher website Original artist: ? • File:Circle_limits_III_with_overlay.png Source: License: Fair use Contributors: Compare to File:Escher_Circle_Limit_III.jpg Original artist: Tomruen

http://upload.wikimedia.org/wikipedia/en/f/f4/Circle_limits_III_with_overlay.png

• File:Commons-logo.svg Source: http://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Original artist: ? • File:DrawingHands.jpg Source: http://upload.wikimedia.org/wikipedia/en/b/ba/DrawingHands.jpg License: Fair use Contributors: ? Original artist: ? • File:Escher’{}s_Relativity.jpg Source: http://upload.wikimedia.org/wikipedia/en/a/a3/Escher%27s_Relativity.jpg License: Fair use Contributors: ? Original artist: ? • File:Escher,_Regular_Division_of_the_Plane_III.jpg Source: http://upload.wikimedia.org/wikipedia/en/9/96/Escher%2C_Regular_ Division_of_the_Plane_III.jpg License: Fair use Contributors: Cybermuse Original artist: ? • File:Escher.jpg Source: http://upload.wikimedia.org/wikipedia/commons/6/6a/Escher.jpg License: CC BY-SA 3.0 Contributors: Ga het na (Nationaal Archief NL) Original artist: Photographer: Hans Peters (ANEFO) • File:Escher_Circle_Limit_III.jpg Source: http://upload.wikimedia.org/wikipedia/en/5/55/Escher_Circle_Limit_III.jpg License: Fair use Contributors: [1] Original artist: ? • File:Escher_Museum.jpg Source: http://upload.wikimedia.org/wikipedia/commons/b/b4/Escher_Museum.jpg License: CC BY 2.0 Contributors: Flickr: Escher Museum Original artist: Andrew Crump • File:Escher_Waterfall.jpg Source: http://upload.wikimedia.org/wikipedia/en/e/e8/Escher_Waterfall.jpg License: Fair use Contributors: ? Original artist: ? • File:Flag_of_the_Netherlands.svg Source: http://upload.wikimedia.org/wikipedia/commons/2/20/Flag_of_the_Netherlands.svg License: Public domain Contributors: Own work Original artist: Zscout370 • File:Hokusai-fuji7.png Source: http://upload.wikimedia.org/wikipedia/commons/7/7f/Hokusai-fuji7.png License: Public domain Contributors: Jim Breen’s Ukiyo-E Gallery - Hokusai Original artist: Katsushika Hokusai ( ) • File:Hyperbolic_domains_642.png Source: http://upload.wikimedia.org/wikipedia/commons/6/60/Hyperbolic_domains_642.png License: Public domain Contributors: KaleidoTile Original artist: Tom Ruen • File:Leonardo_polyhedra.png Source: http://upload.wikimedia.org/wikipedia/commons/1/18/Leonardo_polyhedra.png License: Public domain Contributors: ? Original artist: ?

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CHAPTER 36. WATERFALL (M. C. ESCHER)

• File:Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg Source: http://upload.wikimedia.org/wikipedia/commons/9/ 95/Marble_floor_mosaic_Basilica_of_St_Mark_Vencice.jpg License: Public domain Contributors: http://www.georgehart.com/ virtual-polyhedra/uccello.html Original artist: Paolo Uccello • File:Optical_Illustion-Ambiguous_Patterns.svg Source: http://upload.wikimedia.org/wikipedia/commons/5/51/Optical_ Illustion-Ambiguous_Patterns.svg License: CC-BY-SA-3.0 Contributors: Converted from orignal png Original artist: Alex Turner • File:Padlock-silver.svg Source: http://upload.wikimedia.org/wikipedia/commons/f/fc/Padlock-silver.svg License: CC0 Contributors: http://openclipart.org/people/Anonymous/padlock_aj_ashton_01.svg Original artist: This image file was created by AJ Ashton. Uploaded from English WP by User:Eleassar. Converted by User:AzaToth to a silver color. • File:Question_book-new.svg Source: http://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0 Contributors: Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist: Tkgd2007 • File:Speaker_Icon.svg Source: http://upload.wikimedia.org/wikipedia/commons/2/21/Speaker_Icon.svg License: Public domain Contributors: ? Original artist: ? • File:Text_document_with_red_question_mark.svg Source: http://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_ with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svg from the Tango project. Original artist: Benjamin D. Esham (bdesham) • File:Universiteit_Twente_Mesa_Plus_Escher_Object.jpg Universiteit_Twente_Mesa_Plus_Escher_Object.jpg License: Original artist: Berteun Damman (I self)

Source: http://upload.wikimedia.org/wikipedia/commons/3/33/ Public domain Contributors: Made with a Minolta DImage G600

• File:WPVA-khamsa.svg Source: http://upload.wikimedia.org/wikipedia/commons/c/cd/WPVA-khamsa.svg License: CC BY 3.0 Contributors: Vectorized version of Image:WPVA-khamsa.png by User:Sparkit Original artist: • first version Fluff • File:Wikiquote-logo.svg Source: http://upload.wikimedia.org/wikipedia/commons/f/fa/Wikiquote-logo.svg License: Public domain Contributors: ? Original artist: ?

36.4.3

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