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M. C. Escher, was a Dutch graphicartist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture,and tessellations.

TRANSCRIPT

  • Contents

    1 M. C. Escher 11.1 Early life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Later life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Selected works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2 Another World (M. C. Escher) 72.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    3 Ascending and Descending 83.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    4 Atrani, Coast of Amal 94.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    5 Belvedere (M. C. Escher) 105.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    6 The Bridge (M. C. Escher) 116.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    7 Castrovalva (M. C. Escher) 127.1 In popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    8 Circle Limit III 138.1 Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    i

  • ii CONTENTS

    8.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.4 Printing details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.5 Exhibits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    9 Convex and Concave 169.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    10 Cube with Magic Ribbons 1710.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    11 Curl-up 1811.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1811.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    12 Dolphins (M. C. Escher) 2012.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    13 Drawing Hands 2113.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    14 Gravitation (M. C. Escher) 2214.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    15 Hand with Reecting Sphere 2315.1 Popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    16 House of Stairs 2416.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    17 Magic Mirror (M.C. Escher) 2517.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    18 Metamorphosis I 2618.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2618.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    19 Metamorphosis II 27

  • CONTENTS iii

    19.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    20 Metamorphosis III 2820.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2820.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2820.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    21 Print Gallery (M. C. Escher) 2921.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2921.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    22 Puddle (M. C. Escher) 3022.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3022.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    23 Regular Division of the Plane 3123.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    24 Relativity (M. C. Escher) 32

    25 Reptiles (M. C. Escher) 3325.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3325.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3325.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    26 Sky and Water I 3426.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3426.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    27 Sky and Water II 3527.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3527.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    28 Snakes (M. C. Escher) 3628.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    29 Stars (M. C. Escher) 3729.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3729.2 Inuences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3729.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3829.4 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

  • iv CONTENTS

    29.5 Collections and publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3829.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    30 Still Life and Street 3930.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3930.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3930.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    31 Still Life with Mirror 4031.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    32 Still Life with Spherical Mirror 4132.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4132.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    33 Three Spheres II 4233.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4233.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    34 Three Worlds (M. C. Escher) 4334.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4334.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4334.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    35 Tower of Babel (M. C. Escher) 4435.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4435.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4435.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    36 Waterfall (M. C. Escher) 4536.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4536.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4536.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4536.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    36.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4636.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4936.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

  • Chapter 1

    M. C. Escher

    Escher (1971)

    Maurits Cornelis Escher (/r/, Dutch: [murtskrnels r] ( );[1] 17 June 1898 27 March 1972),usually referred to asM. C. Escher, was a Dutch graphicartist. He is known for his often mathematically inspiredwoodcuts, lithographs, and mezzotints. These featureimpossible constructions, explorations of innity, archi-tecture, and tessellations.

    1.1 Early lifeMaurits Cornelis[2] was born in Leeuwarden, Friesland,in a house that forms part of the Princessehof CeramicsMuseum today. He was the youngest son of civil engineerGeorge Arnold Escher and his second wife, Sara Gleich-

    man. In 1903, the family moved to Arnhem, where heattended primary school and secondary school until 1918.Hewas a sickly child, and was placed in a special school atthe age of seven and failed the second grade.[3] Althoughhe excelled at drawing, his grades were generally poor.He also took carpentry and piano lessons until he wasthirteen years old. In 1919, Escher attended the HaarlemSchool of Architecture and Decorative Arts in Haarlem.He briey studied architecture, but he failed a numberof subjects (partly due to a persistent skin infection) andswitched to decorative arts.[3] He studied under SamuelJessurun de Mesquita, with whom he remained friendsfor years. In 1922, Escher left the school after havinggained experience in drawing and making woodcuts.

    1.2 Later life

    In 1922, an important year of his life, Escher trav-eled through Italy (Florence, San Gimignano, Volterra,Siena, Ravello) and Spain (Madrid, Toledo, Granada).He was impressed by the Italian countryside and bythe Alhambra, a fourteenth-century Moorish castle inGranada. The intricate decorative designs at Alhambra,which were based on geometrical symmetries featuringinterlocking repetitive patterns sculpted into the stonewalls and ceilings, were a powerful inuence on Eschersworks.[4] He returned to Italy regularly in the followingyears.In Italy, Escher met Jetta Umiker, whom he married in1924. The couple settled in Rome where their rst son,Giorgio (George) Arnaldo Escher, named after his grand-father, was born. Escher and Jetta later had two moresons: Arthur and Jan.[5]

    In 1935, the political climate in Italy (under Mussolini)became unacceptable to Escher. He had no interest inpolitics, nding it impossible to involve himself with anyideals other than the expressions of his own conceptsthrough his own particular medium, but he was averse tofanaticism and hypocrisy. When his eldest son, George,was forced at the age of nine to wear a Ballila uniform inschool, the family left Italy and moved to Chteau-d'x,Switzerland, where they remained for two years.[6]

    1

  • 2 CHAPTER 1. M. C. ESCHER

    Escher, who had been very fond of and inspired by thelandscapes in Italy, was decidedly unhappy in Switzer-land. In 1937, the family moved again, to Uccle, a suburbof Brussels, Belgium. World War II forced them to movein January 1941, this time to Baarn, Netherlands, whereEscher lived until 1970. Most of Eschers better-knownworks date from this period. The sometimes cloudy, coldand wet weather of the Netherlands allowed him to focusintently on his work. For a time after undergoing surgery,1962 was the only period in which Escher did not workon new pieces.Escher moved to the Rosa Spier Huis in Laren in 1970, anartists retirement home in which he had his own studio.He died at the home on 27 March 1972, aged 73.

    1.3 Works

    Drawing Hands, 1948

    In his early years, Escher sketched landscapes and nature.He also sketched insects, which appeared frequently inhis later work. His rst artistic work, completed in 1922,featured eight human heads divided in dierent planes.Later around 1924, he lost interest in regular divisionof planes, and turned to sketching landscapes in Italy withirregular perspectives that are impossible in natural form.Eschers rst print of an impossible reality was Still Lifeand Street, 1937. His artistic expression was created fromimages in his mind, rather than directly from observationsand travels to other countries. Well known examples ofhis work include Drawing Hands, a work in which twohands are shown, each drawing the other; Sky and Water,in which light plays on shadow to morph the water back-ground behind sh gures into bird gures on a sky back-ground; and Ascending and Descending, in which lines ofpeople ascend and descend stairs in an innite loop, ona construction which is impossible to build and possibleto draw only by taking advantage of quirks of perceptionand perspective.

    He worked primarily in the media of lithographs andwoodcuts, though the few mezzotints he made are con-sidered to be masterpieces of the technique. In hisgraphic art, he portrayed mathematical relationshipsamong shapes, gures and space. Additionally, he ex-plored interlocking gures using black and white to en-hance dierent dimensions. Integrated into his printswere mirror images of cones, spheres, cubes, rings andspirals. Escher was left-handed.[7]

    Relativity, 1953

    Although Escher did not have mathematical traininghis understanding of mathematics was largely visual andintuitiveEschers work had a strong mathematical com-ponent, and more than a few of the worlds which he drewwere built around impossible objects such as the Neckercube and the Penrose triangle. Many of Eschers worksemployed repeated tilings called tessellations. Eschersartwork is especially well liked by mathematicians andscientists, who enjoy his use of polyhedra and geometricdistortions. For example, in Gravity, multicolored turtlespoke their heads out of a stellated dodecahedron.The mathematical inuence in his work emerged around1936, when he journeyed to the Mediterranean with theAdria Shipping Company. He became interested in orderand symmetry. Escher described his journey through theMediterranean as the richest source of inspiration I haveever tapped.After his journey to the Alhambra, Escher tried to im-prove upon the art works of the Moors using geometricgrids as the basis for his sketches, which he then overlaidwith additional designs, mainly animals such as birds andlions.His rst study of mathematics, which later led to its in-corporation into his art works, began with George Plya'sacademic paper on plane symmetry groups sent to himby his brother Berend. This paper inspired him to learnthe concept of the 17 wallpaper groups (plane symmetrygroups). Using this mathematical concept, Escher cre-

  • 1.3. WORKS 3

    ated periodic tilings with 43 colored drawings of dier-ent types of symmetry. From this point on he developeda mathematical approach to expressions of symmetry inhis art works. Starting in 1937, he created woodcuts us-ing the concept of the 17 plane symmetry groups.

    Circle Limit III, 1959

    In 1941, Escher summarized his ndings in a sketchbook,which he labeled Regelmatige vlakverdeling in asym-metrische congruente veelhoeken (Regular division of theplane with asymmetric congruent polygons).[8] His in-tention in writing this was to aid himself in integratingmathematics into art. Escher is considered a researchmathematician of his time because of his documentationwith this paper, in which he studied color based division,and developed a system of categorizing combinations ofshape, color and symmetrical properties.Around 1956, Escher explored the concept of represent-ing innity on a two-dimensional plane. Discussions withCanadian mathematician H.S.M. Coxeter inspired Es-chers interest in hyperbolic tessellations, which are regu-lar tilings of the hyperbolic plane. Eschers wood engrav-ings Circle Limit IIV demonstrate this concept. In 1959,Coxeter published his nding that these works were ex-traordinarily accurate: Escher got it absolutely right tothe millimeter.Escher was awarded the Knighthood of the Order of Or-ange Nassau in 1955. Subsequently he regularly designedart for dignitaries around the world.In 1958, he published a book entitled Regular Divisionof the Plane, with reproductions of a series of woodcutsbased on tessellations of the plane, in which he describedthe systematic buildup of mathematical designs in his art-works. He emphasized, "Mathematicians have openedthe gate leading to an extensive domain.Overall, his early love of Roman and Italian landscapesand of nature led to his interest in the concept of regular

    division of a plane, which he applied in over 150 coloredworks. Other mathematical principles evidenced in hisworks include the superposition of a hyperbolic plane on axed 2-dimensional plane, and the incorporation of three-dimensional objects such as spheres, columns and cubesinto his works. For example, in a print called "Reptiles",he combined two and three-dimensional images. In oneof his papers, Escher emphasized the importance of di-mensionality and described himself as irritated by atshapes: I make them come out of the plane.

    Waterfall, 1961

    Sculpture of the small stellated dodecahedron that appears in Es-chersGravitation. It can be found in front of the Mesa+" build-ing on the Campus of the University of Twente.

    Escher also studied topology. He learned additional con-cepts in mathematics from the British mathematicianRoger Penrose. From this knowledge he created Water-fall and Up and Down, featuring irregular perspectives

  • 4 CHAPTER 1. M. C. ESCHER

    similar to the concept of the Mbius strip.Escher printedMetamorphosis I in 1937, which was a be-ginning part of a series of designs that told a story throughthe use of pictures. These works demonstrated a culmi-nation of Eschers skills to incorporate mathematics intoart. InMetamorphosis I, he transformed convex polygonsinto regular patterns in a plane to form a human motif.This eect symbolizes his change of interest from land-scape and nature to regular division of a plane.His piece Metamorphosis III is wide enough to cover allthe walls in a room, and then loop back onto itself.After 1953, Escher became a lecturer at many organiza-tions. A planned series of lectures in North America in1962 was cancelled due to an illness, but the illustrationsand text for the lectures, written out in full by Escher,were later published as part of the book Escher on Es-cher. In July 1969 he nished his last work, a woodcutcalled Snakes, in which snakes wind through a pattern oflinked rings which fade to innity toward both the centerand the edge of a circle.

    1.4 Legacy

    The Escher Museum in The Hague

    See also: M. C. Escher in popular culture

    The special way of thinking and the rich graphic workof M.C. Escher has had a continuous inuence in scienceand art, as well as being referenced in popular culture.Ownership of the Escher intellectual property and of hisunique art works have been separated from each other.In 1969, Eschers business advisor, Jan W. Vermeulen,author of a biography in Dutch on the artist, establishedtheM.C. Escher Stichting (M.C. Escher Foundation), andtransferred into this entity virtually all of Eschers uniquework as well as hundreds of his original prints. Theseworks were lent by the Foundation to the HagueMuseum.Upon Eschers death, his three sons dissolved the Founda-tion, and they became partners in the ownership of the art

    works. In 1980, this holding was sold to an American artdealer and the Hague Museum. The Museum obtainedall of the documentation and the smaller portion of theart works.The copyrights remained the possession of the three sons who later sold them to Cordon Art, a Dutch company.Control of the copyrights was subsequently transferred toThe M.C. Escher Company B.V. of Baarn, Netherlands,which licenses use of the copyrights on all of Eschers artand on his spoken and written text.A related entity, the M.C. Escher Foundation of Baarn,promotes Eschers work by organizing exhibitions, pub-lishing books and producing lms about his life and work.The primary institutional collections of original worksby M.C. Escher are the Escher Museum, a subsidiary ofthe Haags Gemeentemuseum in The Hague; the NationalGallery of Art (Washington, DC); the National Gallery ofCanada (Ottawa); the Israel Museum (Jerusalem); Huisten Bosch (Nagasaki, Japan); and the Boston Public Li-brary.Gdel, Escher, Bach by Douglas Hofstadter,[9] publishedin 1979, discusses the ideas of self-reference and strangeloops, drawing on a wide range of artistic and scienticwork, including the art of M. C. Escher and the music ofJ. S. Bach, to illustrate ideas behind Gdels incomplete-ness theorems.

    1.5 Selected works Trees, ink (1920) St. Bavos, Haarlem, ink (1920) Flor de Pascua (The Easter Flower), woodcut/bookillustrations (1921)

    Eight Heads, woodcut (1922) Dolphins also known as Dolphins in Phosphorescent

    Sea, woodcut (1923)

    Tower of Babel, woodcut (1928) Street in Scanno, Abruzzi, lithograph (1930) Castrovalva, lithograph (1930) The Bridge, lithograph (1930) Palizzi, Calabria, woodcut (1930) Pentedattilo, Calabria, lithograph (1930) Atrani, Coast of Amal, lithograph (1931) Ravello and the Coast of Amal, lithograph (1931) Covered Alley in Atrani, Coast of Amal, wood en-graving (1931)

  • 1.6. SEE ALSO 5

    Phosphorescent Sea, lithograph (1933) Still Life with Spherical Mirror, lithograph (1934) Hand with Reecting Sphere also known as Self-

    Portrait in Spherical Mirror, lithograph (1935)

    Inside St. Peters, wood engraving (1935) Portrait of G.A. Escher, lithograph (1935) Hell, lithograph, (copied from a painting byHieronymus Bosch) (1935)

    Regular Division of the Plane, series of drawings thatcontinued until the 1960s (1936)

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

    Metamorphosis I, woodcut (1937) Day and Night, woodcut (1938) Cycle, lithograph (1938) Sky and Water I, woodcut (1938) Sky and Water II, lithograph (1938) Metamorphosis II, woodcut (19391940) Verbum (Earth, Sky and Water), lithograph (1942) Reptiles, lithograph (1943) Ant, lithograph (1943) Encounter, lithograph (1944) Doric Columns, wood engraving (1945) Three Spheres I, wood engraving (1945) Magic Mirror, lithograph (1946) Three Spheres II, lithograph (1946) Another World Mezzotint also known as Other World

    Gallery, mezzotint (1946)

    Eye, mezzotint (1946) AnotherWorld also known asOtherWorld, wood en-graving and woodcut (1947)

    Crystal, mezzotint (1947) Up and Down also known as High and Low, litho-graph (1947)

    Drawing Hands, lithograph (1948) Dewdrop, mezzotint (1948) Stars, wood engraving (1948) Double Planetoid, wood engraving (1949)

    Order and Chaos (Contrast), lithograph (1950)

    Rippled Surface, woodcut and linoleum cut (1950)

    Curl-up, lithograph (1951)

    House of Stairs, lithograph (1951)

    House of Stairs II, lithograph (1951)

    Puddle, woodcut (1952)

    Gravitation, (1952)

    Dragon, woodcut lithograph and watercolor (1952)

    Cubic Space Division, lithograph (1952)

    Relativity, lithograph (1953)

    Tetrahedral Planetoid, woodcut (1954)

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

    Convex and Concave, lithograph (1955)

    Three Worlds, lithograph (1955)

    Print Gallery, lithograph (1956)

    Mosaic II, lithograph (1957)

    Cube with Magic Ribbons, lithograph (1957)

    Belvedere, lithograph (1958)

    Sphere Spirals, woodcut (1958)

    Circle Limit III, woodcut (1959)

    Ascending and Descending, lithograph (1960)

    Waterfall, lithograph (1961)

    Mbius Strip II (Red Ants) woodcut (1963)

    Knot, pencil and crayon (1966)

    Metamorphosis III, woodcut (19671968)

    Snakes, woodcut (1969)

    1.6 See also Asteroid 4444 Escher was named in Eschers honorin 1985.

    Mathematics and art#M.C. Escher

  • 6 CHAPTER 1. M. C. ESCHER

    1.7 References[1] Duden Aussprachewrterbuch (6 ed.). Mannheim: Bibli-

    ographisches Institut & F.A. Brockhaus AG. 2005. ISBN3-411-04066-1.

    [2] We named him Maurits Cornelis after S.'s [Saras]beloved uncle Van Hall, and called him 'Mauk' for short...., Diary of Eschers father, quoted inM. C. Escher: HisLife and Complete Graphic Work, Abradale Press, 1981,p. 9.

    [3] Barbara E, PhD. Bryden. Sundial: Theoretical Relation-ships Between Psychological Type, Talent, And Disease.Gainesville, Fla: Center for Applications of Psycholog-ical Type. ISBN 0-935652-46-9.

    [4] Roza, Greg (2005). An Optical Artist: Exploring Patternsand Symmetry. Rosen Classroom. p. 20. ISBN 978-1-4042-5117-5.

    [5] ESCHER. Geom.uiuc.edu. Retrieved 7 December2013.

    [6] Ernst, Bruno, TheMagic Mirror of M.C. Escher, Taschen,1978; p. 15

    [7] The Ocial M.C. Escher Website Biography.Mcescher.com. Retrieved 7 December 2013.

    [8] Barry Cipra (1998). Paul Zorn, ed. Whats Happening inthe Mathematical Sciences, Volume 4. American Mathe-matical Society. p. 103. ISBN 0-8218-0766-8.

    [9] Hofstadter, Douglas R. (1999) [1979], Gdel, Escher,Bach: An Eternal Golden Braid, Basic Books, ISBN 0-465-02656-7

    1.8 Further readingBooks

    Abrams (1995). The M. C. Escher Sticker Book.Harry N. Abrams. ISBN 0-8109-2638-5 .

    Ernst, Bruno; Escher, M. C. (1995). The MagicMirror of M. C. Escher (Taschen Series). TaschenAmerica LLC. ISBN 1-886155-00-3 Eschers artwith commentary by Ernst on Eschers life and art,including several pages on his use of polyhedra.

    Escher, M. C. (1971) The Graphic Work of M. C.Escher, Ballantine. Includes Eschers own commen-tary.

    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 andComplete Graphic Work, Amsterdam

    O'Connor, J. J. (17 June 2005) Escher. Universityof St Andrews, Scotland.

    Schattschneider, Doris & Walker, Wallace. (1987)M. C. Escher Kaleidocycles, Petaluma, California,Pomegranate Communications ISBN 0-906212-28-6.

    Schattschneider, Doris (2004). M. C. Escher : Vi-sions of Symmetry, New York, N.Y. : Harry N.Abrams, 2004. ISBN 0-8109-4308-5.

    Schattschneider, Doris & Emmer, Michele, eds(2003). M. C. Eschers Legacy: a Centennial Cele-bration; collection of articles coming from the M.C. Escher Centennial Conference, Rome, 1998 /Berlin; London: Springer-Verlag. ISBN 3-540-42458-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 ofhis art, and interviews, Director, Michele Emmer.

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

    Escherization problem and its solution. CA: Univer-sity of Waterloo.

    Escher for Real. IL: Technion. physical repli-cas of some of Eschers impossible designs

    M.C. Escher: Life and Work. USA: NGA. US Copyright Protection for UK Artists. UK.Copyright issue regarding Escher from the ArtquestArtlaw archive.

    Schattschneider, Doris (JuneJuly 2010). TheMathematical Side of M. C. Escher (PDF). Noticesof the American Mathematical Society (USA) 57 (6):70618. Retrieved 9 July 2010.

    Gallery of tessellations by M.C. Escher

  • Chapter 2

    Another World (M. C. Escher)

    AnotherWorld, also known asOtherWorld, is a woodcutprint by the Dutch artist M. C. Escher rst printed in Jan-uary 1947.It depicts a cubic architectural structure made from brick.The structure is a paradox with an open archway on eachof the ve visible sides of the cube. The structure wrapsaround the vertical axis to enclose the viewers perspec-tive. At the bottom of the image is an archway whichwe seem to be looking up from the base, and through itwe can see space. At the top of that arch is another archwhich is level with our perspective, and through it we arelooking out over a lunar horizon. At the top of that archis another arch which covers the top of the image. Weare looking down at this arch from above and through itonto the lunar surface.Standing in each archway along the vertical axis is a metalsculpture of a bird with a humanoid face. In each sidearchway is a horn or cornucopia hanging on chains. Itis interesting to note that the views from above and be-low are consistent, placing the statue so that it faces thehorn, however the horizontal view reverses the relativepositions of the statue and the horn, and rotates the horn180 degrees.The previous month (December 1946), Escher created amezzotint called Another World (Other World Gallery).The image in that print is the same as this one except thatthe arches continue on as an innite corridor.The bird/human sculpture is a real sculpture which wasgiven to Escher by his father-in-law. This sculpture rstappears in Eschers 1934 lithograph Still Life with Spher-ical 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 theDutch artist M. C. Escher rst printed in March 1960.The original print measures 14 in 11 14 in (35.6 cm 28.6 cm). The lithograph depicts a large building roofedby a never-ending staircase. Two lines of identicallydressed men appear on the staircase, one line ascendingwhile the other descends. Two gures sit apart from thepeople on the endless staircase: one in a secluded court-yard, the other on a lower set of stairs. While most two-dimensional artists use relative proportions to create anillusion of depth, Escher here and elsewhere uses con-icting proportions to create the visual paradox.Ascending and Descending was inuenced by, and isan artistic implementation of, the Penrose stairs, animpossible object; Lionel Penrose had rst published hisconcept in the February 1958 issue of the British Journalof Psychology. Escher developed the theme further in hisprintWaterfall, which appeared in 1961.The two concentric processions on the stairs use enoughpeople to emphasise the lack of vertical rise and fall. Inaddition, the shortness of the tunics worn by the peoplemakes it clear that some are stepping up and some arestepping down.The structure is embedded in human activity. By show-ing an unaccountable ritual of what Escher calls an 'un-known' sect, Escher has added an air of mystery to thepeople who ascend and descend the stairs. Therefore, thestairs themselves tend to become incorporated into thatmysterious appearance.There are 'free' people and Escher said of these: 'recalci-trant individuals refuse, for the time being, to take part inthe exercise of treading the stairs. They have no use forit at all, but no doubt, sooner or later they will be broughtto see the error of their non-conformity.'Escher suggests that not only the labours, but the verylives of these monk-like people are carried out in an in-escapable, coercive and bizarre environment. Anotherpossible source for the peoples looks is the Dutch id-iom a monks job, which refers to a long and repetitiveworking activity with absolutely no practical purposes orresults, and, by extension, to something completely use-less.

    Two earlier Escher pictures that feature stairs are Houseof Stairs and Relativity.

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

    8

  • Chapter 4

    Atrani, Coast of Amal

    Atrani in 2003.

    Atrani, Coast of Amal is a lithograph print by the Dutchartist M. C. Escher, rst printed in August 1931. Atraniis a small town and commune on the Amal Coast inthe province of Salerno in the Campania region of south-western Italy. Atrani is the second smallest town in Italyand was built right at the edge of the sea. This image ofAtrani recurs several times in Eschers work, most notablyin his series ofMetamorphosis prints: Metamorphosis I, IIand 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)

    Belvedere is a lithograph print by the Dutch artist M. C.Escher, rst printed in May 1958. It shows a plausible-looking building that is actually an impossible object.In this print, Escher uses two-dimensional images to de-pict objects free of the connes of the three-dimensionalworld. The image is of a rectangular three-story build-ing. The upper two oors are open at the sides with thetop oor and roof supported by pillars. From the viewersperspective, all the pillars on themiddle oor are the samesize at both the front and back, but the pillars at the backare set higher. The viewer also sees by the corners of thetop oor that it is at a dierent angle than the rest of thestructure. All these elements make it possible for all thepillars on the middle oor to stand at right angles, yet thepillars at the front support the back side of the top oorwhile the pillars at the back support the front side. Thisparadox also allows a ladder to extend from the inside ofthe middle oor to the outside of the top oor.There is a man seated at the foot of the building holdingan impossible cube. He appears to be constructing it froma diagram of a Necker cube at his feet with the intersect-ing lines circled. The window next to him is closed withan iron grille that is geometrically valid but practicallyimpossible to assemble.The woman who is about to climb the steps of the build-ing is modeled after a gure from the right panel ofHieronymus Bosch's 1500 triptychTheGarden of EarthlyDelights. This panel is individually titled Hell. A portionof Hell had earlier been recreated by Escher as a litho-graph in 1935.The ridge in the background is part of Morrone Moun-tains in Abruzzo, that Escher had visited several timeswhen living in Italy during the 1920s and 30s.

    5.1 See also Belvedere (structure) Lithography Paradox Printmaking

    Necker cube M. C. EschersWaterfall

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

    10

  • Chapter 6

    The Bridge (M. C. Escher)

    The Bridge is a lithograph print by the Dutch artist M. C.Escher, rst printed in March 1930.It depicts a bridge connecting two sheer clis. On the topof the left hand cli is a city. The chasm between the twoclis is narrow but plummets out of view. In the distanceis another outcrop with a city built on top. Both the rockand the architecture on this third outcrop are darker incolouration than in the foreground. The buildings appearto be modelled partly after southern Italian architecture.The rock is in blocky formations that appeared often dur-ing Eschers Italian period and it is possible that the vil-lage 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, rst printed in February 1930. Like many ofEschers early works, it depicts a place that he visited ona tour of Italy.It depicts the Abruzzo village of Castrovalva, which liesat the top of a sheer slope. The perspective is toward thenorthwest, from the narrow trail on the left which, at thepoint from which this view is seen, makes a hairpin turnto the right, descending to the valley. In the foregroundat the side of the trail, there are several owering plants,grasses, ferns, a beetle and a snail. In the expansive valleybelow there are cultivated elds and two more towns, thenearest of which is Anversa degli Abruzzi, with Casale inthe distance.

    7.1 In popular culture In 1982 the name Castrovalva was used in a storyin the BBC television seriesDoctor Who. The story-line also relied heavily on recursion, a favorite themein Eschers later and more famous works, and usedideas taken from Belvedere, Ascending and Descend-ing, and Relativity to trap the protagonists in the cityof Castrovalva.

    The comic Kingdom of the Wicked is set in an imag-inary 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

    Circle Limit III, 1959

    Circle Limit III is a woodcut made in 1959 by Dutchartist M. C. Escher, in which strings of sh shoot up likerockets from innitely far away and then fall back againwhence they came.[1]

    It is one of a series of four woodcuts by Escher depict-ing ideas from hyperbolic geometry. Dutch physicistand mathematician Bruno Ernst called it the best of thefour.[2]

    8.1 InspirationEscher became interested in tesselations of the plane af-ter a 1936 visit to the Alhambra in Granada, Spain,[3][4]and from the time of his 1937 artwork Metamorphosis Ihe had begun incorporating tessellated human and animalgures into his artworks.[4] In a 1958 letter from Escherto H. S. M. Coxeter, Escher wrote that he was inspiredto make his Circle Limit series by a gure in Coxetersarticle Crystal Symmetry and its Generalizations.[2][3]Coxeters gure depicts a tessellation of the hyperbolicplane by right triangles with angles of 30, 45, and 90(a shape that is possible in hyperbolic geometry but not inEuclidean geometry); this tessellation may be interpreted

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

    as depicting the lines of reection and fundamental do-mains of the (6,4,2) triangle group.[5]

    8.2 GeometryEscher seems to have believed that the white curves of hiswoodcut, which bisect the sh, represent hyperbolic linesin the Poincar disk model of the hyperbolic plane, inwhich the whole hyperbolic plane is modeled as a disk inthe Euclidean plane, and hyperbolic lines are modeled ascircular arcs perpendicular to the disk boundary. Indeed,Escher wrote that the sh move perpendicularly to theboundary.[1] However, as Coxeter demonstrated, there isno hyperbolic arrangement of lines whose faces are alter-nately squares and equilateral triangles, as the gure de-picts. Rather, the white curves are hypercycles that meetthe boundary circle at angles of cos1((21/4 21/4)/2),approximately 80.[2]

    The symmetry axes of the triangles and squares that liebetween the white lines are true hyperbolic lines. Thesquares and triangles of the woodcut have the same in-cidence pattern as the faces of the tritetragonal tiling of

    13

  • 14 CHAPTER 8. CIRCLE LIMIT III

    The tritetragonal tiling, a hyperbolic tiling of squares and equi-lateral triangles, overlaid on Eschers image

    the hyperbolic plane, but their geometry is not the same:in the tritetragonal tiling, the sides of the squares and tri-angles are hyperbolically straight line segments, while inEschers woodcut they are arcs of hypercycles, so that thesmooth curves of Escher correspond to polygonal chainswith corners in the tritetragonal tiling. The points atthe centers of the quadrilaterals, where four sh meetat their ns, form the vertices of an order-8 triangulartiling, while the points where three sh ns meet and thepoints where three white lines cross together form thevertices of its dual, the octagonal tiling.[2] Similar tessel-lations by lines of sh may be constructed for other hy-perbolic tilings formed by polygons other than trianglesand squares, or with more than three white curves at eachcrossing.[6]

    Euclidean coordinates of circles containing the threemostprominent white curves in the woodcut may be obtainedby calculations in the eld of rational numbers extendedby the square roots of two and three.[7]

    8.3 Symmetry

    Viewed as a pattern, ignoring the colors of the sh, in thehyperbolic plane, the woodcut has three-fold and four-fold rotational symmetry at the centers of its triangles andsquares, respectively, and order-three dihedral symme-try (the symmetry of an equilateral triangle) at the pointswhere the white curves cross. In John Conway's orbifoldnotation, this set of symmetries is denoted 433. Each shprovides a fundamental region for this symmetry group.Contrary to appearances, the sh do not have bilateralsymmetry: the white curves of the drawing are not axesof reection symmetry.[8][9]

    8.4 Printing detailsThe sh in Circle Limit III are depicted in four colors,allowing each string of sh to have a single color and eachtwo adjacent sh to have dierent colors. Together withthe black ink used to outline the sh, the overall woodcuthas ve colors. It is printed from ve wood blocks, eachof which provides one of the colors within a quarter ofthe disk, for a total of 20 impressions. The diameter ofthe outer circle, as printed, is 41.5cm.[10]

    8.5 ExhibitsAs well as being included in the collection of the EscherMuseum in The Hague, there is a copy of Circle Limit IIIin the collection of the National Gallery of Canada.[11]

    8.6 References[1] Escher, as quoted by Coxeter (1979).

    [2] Coxeter, H. S. M. (1979), The non-Euclidean symmetryof Eschers picture 'Circle Limit III'", Leonardo 12: 1925, JSTOR 1574078.

    [3] Emmer, Michele (2006), Escher, Coxeter andsymmetry, International Journal of GeometricMethods in Modern Physics 3 (5-6): 869879,doi:10.1142/S0219887806001594, MR 2264394.

    [4] Schattschneider, Doris (2010), The mathematical side ofM. C. Escher, Notices of the AMS 57 (6): 706718.

    [5] An elementary analysis of Coxeters gure, as Eschermight have understood it, is given by Casselman, Bill(June 2010), How did Escher do it?, AMS Feature Col-umn. Coxeter expanded on the mathematics of trian-gle group tessellations, including this one in Coxeter, H.S. M. (1997), The trigonometry of hyperbolic tessella-tions, Canadian Mathematical Bulletin 40 (2): 158168,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 Escherswoodcut Circle Limit III",M.C.Eschers Legacy: A Centen-nial Celebration, Springer, pp. 297304, doi:10.1007/3-540-28849-X_29.

    [8] Conway, J. H. (1992), The orbifold notation for surfacegroups, Groups, Combinatorics & Geometry (Durham,1990), London Math. Soc. Lecture Note Ser. 165,Cambridge: Cambridge Univ. Press, pp. 438447,doi:10.1017/CBO9780511629259.038, MR 1200280.Conway wrote that The work Circle Limit III is equallyintriguing (in comparison to Circle Limit IV, which has adierent symmetry group), and uses is it as an example ofthis symmetry group.

  • 8.7. EXTERNAL LINKS 15

    [9] Herford, Peter (1999), The geometry of M. C. Escherscircle-Limit-Woodcuts, Zentralblatt f Didaktik derMathematik 31 (5): 144148, doi:10.1007/BF02659805.Paper presented to the 8th International Conference onGeometry, Nahsholim (Israel), March 714, 1999.

    [10] Escher, M. C. (2001), M. C. Escher: The Graphic Work,Taschen, p. 10.

    [11] Circle Limit III, National Gallery of Canada, retrieved2013-07-09.

    8.7 External links Douglas Dunham Department of Computer ScienceUniversity of Minnesota, Duluth

    Examples Based on Circle Limits III andIV, 2006:More Circle Limit III Patterns,2007:A Circle Limit III Calculation

  • Chapter 9

    Convex and Concave

    Convex and Concave is a lithograph print by the Dutchartist M. C. Escher, rst printed in March 1955.It depicts an ornate architectural structure with manystairs, pillars and other shapes. The relative aspects ofthe objects in the image are distorted in such a way thatmany of the structures features can be seen as both con-vex shapes and concave impressions. This is a very goodexample of Eschers mastery in creating illusion of Im-possible Architectures. The windows, roads, stairs andother shapes can be perceived as opening out in seem-ingly impossible ways and positions. Even the image onthe ag is of reversible cubes. One can view these featuresas concave by viewing the image upside-down.Note that all additional elements and decoration on theleft are consistent with a viewpoint from above, whilethose on the right with a viewpoint from below: hidinghalf the image makes it very easy to switch between con-vex 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 theDutch artist M. C. Escher rst printed in 1957. It de-picts two interlocking bands wrapped around the frameof a cube. The cube framework by itself is perfectly pos-sible but the interlocking of the magical bands within itis impossible. This print is signicant for being the rstEscher 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. For theexercise, see Crunch (exercise).

    Curl-up or Wentelteefje (original Dutch title) is alithograph print by M. C. Escher, rst printed in Novem-ber 1951.This is the only work by Escher consisting largely of text.The text, which is written in Dutch, describes an imagi-nary species called Pedalternorotandomovens centrocula-tus articulosus, also known as wentelteefje or rolpens.He says this creature came into existence because of theabsence in nature of wheel shaped, living creatures withthe ability to roll themselves forward.The creature is elongated and armored with severalkeratinized joints. It has six legs, each with what ap-pears to be a human foot. It has a disc-shaped head witha parrot-like beak and eyes on stalks on either side.It can either crawl over a variety of terrain with its six legsor press its beak to the ground and roll into a wheel shape.It can then roll, gaining acceleration by pushing with itslegs. On slopes it can tuck its legs in and roll freely. Thisrolling can end in one of two ways; by abruptly unrollingin motion, which leaves the creature belly-up, or by brak-ing to a stop with its legs and slowly unrolling backwards.The word wentelteefje is Dutch for French toast, wen-tel meaning to turn over. Rolpens is a dish made withchopped meat wrapped in a roll and then fried or baked.Een pens means belly, often used in the phrase beer-belly.There is a diagonal gap through the text containing an il-lustration showing the step by step process of the creaturerolling into a wheel. This creature appears in two moreprints completed later the same month, House of Stairsand House of Stairs II.

    11.1 TranslationThe translation of the surrounding text is as follows:

    The Pedalternorotandomovens Centrocula-tus Articulosus (curl-up) came into existence

    (spontaneous generation), because of the ab-sence, in nature, of wheel shaped, living crea-tures with the ability to roll themselves forward.The accompanying 'beastie' depiction, referredto as 'revolving bitch' or 'roll paunch' in lay-mens terms, subsequently anticipates the needwith sensitivity. Biological details are still few:is it a mammal, a reptile, or an insect? It hasa long, drawn-out, horned, sectioned body andthree sets of legs; the ends of which look likethe human foot. In the middle of the fat, roundhead, that is provided with a strong, bent par-rots beak; they have bulb-shaped eyes, which,placed on posts, protrude far out from both sidesof the head. In the stretched out position, the an-imal can, slow and cautiously, with the use ofhis six legs, move forward over a variety of ter-rains (it can potentially climb or descend steepstairs, plow through bushes, or scramble overboulders). However, when it must cover a greatdistance, and has a relatively at path to his dis-posal, he pushes his head to the ground and rollshimself upwith lightning speed, at which time hepushes himself o with his legs- for as much asthey can still touch the ground. In the rolled upstate it exhibits the form of a discus, of which theeye posts are the central axle. By pushing o al-ternately with one of his three pairs of legs, hecan achieve great speeds. It is also sometimesdesirable during the rolling (i.e. The descent ofan incline, or coasting to a nish) to hold upthe legs and 'freewheel' forward. Whenever itwants, it can return again to the walking positionin two ways: rst abruptly, by suddenly extend-ing his body, but then its lying on his back withhis legs in the air, and second through gradualdeceleration (braking with his feet) and slowlyunrolling backwards in standing position.

    11.2 See also Printmaking Rotating locomotion in living systems

    18

  • 11.3. SOURCES 19

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

  • Chapter 12

    Dolphins (M. C. Escher)

    Dolphins also known as a Dolphins in PhosphorescentSea is a woodcut print by the Dutch artist M. C. Es-cher. This work was rst printed in February, 1923. Es-cher had been fascinated by the glowing outlines of oceanwaves breaking at night and this image depicts the out-lines made by a school of dolphins swimming and breach-ing ahead of the bow of a ship. The glow was created bybioluminescent dinoagellates.

    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 rst printed in January 1948. It depicts a sheetof paper out of which, from wrists that remain at onthe page, two hands rise, facing each other and in theparadoxical act of drawing one another into existence.Although Escher used paradoxes in his works often, thisis one of the most obvious examples.It is referenced in the book Gdel, Escher, Bach, byDouglas Hofstadter, who calls it an example of a strangeloop. It is also used in Structure and Interpretation ofComputer Programs by Harold Abelson and Gerald JaySussman as an allegory for the eval and apply functions ofprogramming 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 mediawork by the Dutch artist M. C. Escher completed in June1952. It was rst printed as a black-and-white lithographand then coloured by hand in watercolour.It depicts a nonconvex regular polyhedron known as thesmall stellated dodecahedron. Each facet of the gure hasa trapezoidal doorway. Out of these doorways protrudethe heads and legs of twelve turtles without shells, who areusing the object as a common shell. The turtles are in sixcoloured pairs (red, orange, yellow, magenta, green andindigo) 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 Reecting Sphere

    Hand with Reecting Sphere also known as Self-Portraitin Spherical Mirror is a lithograph print by Dutch artistM. C. Escher, rst printed in January 1935. The piecedepicts a hand holding a reective sphere. In the reec-tion, most of the room around Escher can be seen, andthe hand holding the sphere is revealed to be Eschers.Self-portraits in reective, spherical surfaces are com-mon in Eschers work, and this image is the most promi-nent and famous example. In much of his self-portraitureof this type, Escher is in the act of drawing the sphere,whereas in this image he is seated and gazing into it. Onthe walls there are several framed pictures, one of whichappears to be of an Indonesian shadow puppet.

    15.1 Popular cultureFrank O'Connor, the manager of the Halo video gameseries, made an illustration that references this work. Itappears in the Halo Graphic Novel.In Disneys TRON: Legacy, Je Bridges Character, CLU,is seen holding a reective apple in which he sees his ownreection. This may be in homage to Escher, as there aretwo octahedra on a nearby shelf, and much of the digitalworld is made up of tessellations, a subject largely focusedon 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 ofStairs (disambiguation).

    House of Stairs is a lithograph print by the Dutch artistM. C. Escher rst printed in November 1951. This printmeasures 18" 9". It depicts the interior of a tallstructure crisscrossed with stairs and doorways.A total of 46 "wentelteefje" (imaginary creatures createdby Escher) are crawling on the stairs. The wentelteefjehas a long, armored body with six legs, humanoid feet, aparrot-like beak and eyes on stalks. Some are seen to rollin through doors, wound in a wheel shape and then unrollto crawl up the stairs, while others crawl down stairs andwind up to roll out. The wentelteefje rst appeared ear-lier the same month in the lithograph Curl-up. Later thatmonth, House of Stairs was extended to a vertical lengthof 55" in a print titled House of Stairs II by repeatingand 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. Forother uses of Magic mirror, see Magic mirror.

    Magic Mirror is a lithograph print by the Dutch artist M.C. Escher rst printed in January, 1946.It depicts a mirror standing vertically on wooden supportson a tiled surface. The perspective is looking down atan angle at the right hand side of the mirror. There isa sphere at each side of the mirror. The main focus ofthe image is a procession of small grin (winged lion)sculptures that emerge from the surface of the mirror andtrail away from it in single le. Both the angular reec-tion of the tiles and the oset between the reection ofthe sphere in front of the mirror and the sphere behind itprove it is a mirror. Yet the reection of the grin pro-cession continues to emerge from behind the mirror. Thegrin processions of both sides loop around to the frontand 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 artistM. C. Escher which was rst printed in May, 1937. Thispiece measures 19.5 by 90.8 centimetres (7.7 in 35.7in) and is printed on two sheets.The concept of this work is to morph one image intoa tessellated pattern, then gradually to alter the outlinesof that pattern to become an altogether dierent image.From left to right, the image begins with a depiction ofthe coastal Italian town of Atrani (see Atrani, Coast ofAmal). The outlines of the architecture then morph toa pattern of three-dimensional blocks. These blocks thenslowly become a tessellated pattern of cartoon-like guresin 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 artistM. C. Escher. It was created between November, 1939and March, 1940. This print measures 19.2 by 389.5centimetres (7.6 in 153.3 in) and was printed from 20blocks on 3 combined sheets.Like Metamorphosis I, the concept of this piece is tomorph one image into a tessellated pattern and thenslowly alter that pattern eventually to become a new im-age.The process begins left to right with the word metamor-phose (the Dutch form of the word metamorphosis) ina black rectangle, followed by several smaller metamor-phose rectangles forming a grid pattern. This grid thenbecomes a black and white checkered pattern, which thenbecomes tessellations of reptiles, a honeycomb, insects,sh, birds and a pattern of three-dimensional blocks withred tops.These blocks then become the architecture of the Italiancoastal town of Atrani (see Atrani, Coast of Amal). Inthis image Atrani is linked by a bridge to a tower in thewater, which is actually a rook piece from a chess set.There are other chess pieces in the water and the waterbecomes a chess board. The chess board leads to a check-ered 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 artistM. C. Escher created during 1967 and 1968. Measuring19 cm 680 cm (7 268 inches - 22'4), this is Escherslargest print. It was printed on thirty-three blocks on sixcombined sheets and mounted on canvas. This print waspartly coloured by hand.It begins identically to Metamorphosis II, with theword metamorphose (the Dutch form of the wordmetamorphosis) forming a grid pattern and then becom-ing a black-and-white checkered pattern. Then the rstset of new imagery begins. The angles of the checkeredpattern change to elongated diamond shapes. These thenbecome an image of owers with bees. This image thenreturns to the diamond pattern and back into the check-ered pattern.It then resumes with the Metamorphosis II imagery un-til the bird pattern. The birds then become sailing boats.From the sailing boats the image changes to a second shpattern. Then from the sh to horses. The horses then be-come a second bird pattern. The second bird pattern thenbecomes black-and-white triangles, which then becomeenvelopes with wings. These winged envelopes then re-turn to the black-and-white triangles and then to the orig-inal bird pattern. It then resumes with theMetamorphosisII print until its conclusion.

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

    20.2 External links Images of Metamorphosis III and other well knownworks at mcescher.com

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

    28

  • Chapter 21

    Print Gallery (M. C. Escher)

    Print Gallery (Dutch: Prentententoonstelling) is alithograph printed in 1956 by the Dutch artist M. C. Es-cher. It depicts a man in a gallery viewing a print of a sea-port, and among the buildings in the seaport is the verygallery in which he is standing. In the book Gdel, Es-cher, Bach, Douglas Hofstadter explains it as a strangeloop showing three kinds of in-ness": the gallery is phys-ically in the town (inclusion); the town is artistically inthe picture (depiction); the picture is mentally in theperson (representation).Eschers signature is on a circular void in the center of thework. In 2003, two Dutch mathematicians, Bart de Smitand Hendrik Lenstra, reported a way of lling in the voidby treating the work as drawn on an elliptic curve overthe eld of complex numbers. They deem an idealizedversion of Print Gallery to contain a copy of itself, ro-tated clockwise by about 157.63 degrees and shrunk by afactor of about 22.58.[1]

    Print Gallery has been discussed in relation to post-modernism by a number of writers, including SilvioGaggi,[2] Barbara Freedman,[3] Stephen Bretzius,[4] andMarie-Laure Ryan.[5]

    21.1 References[1] de Smit, B. (2003). The Mathematical Structure of Es-

    chers Print Gallery. Notices of the American Mathemat-ical Society 50 (4): 446451.

    [2] Gaggi, Silvio (1989). Modern/Postmodern: A Study inTwentieth-Century Arts and Ideas. University of Pennsul-vania Press. pp. 4445. ISBN 0-8122-8154-3.

    [3] Freedman, Barbara (1991). Staging the gaze: postmod-ernism, psychoanalysis, and Shakespearean comedy. Cor-nell University Press. pp. 124126. ISBN 0-8014-9737-X.

    [4] Bretzius, Stephen (1997). Shakespeare in theory: the post-modern academy and the early modern theater. Universityof 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 me-dia. Johns Hopkins University Press. p. 165. ISBN 0-8018-6487-9.

    21.2 External links HarryCarry5 (Jul 26, 2009). Eschers Print Gallery

    Explained. YouTube. Artful Mathematics: The Heritage of M. C. Escher,by Bart de Smit and Hendrik Lenstra

    29

  • Chapter 22

    Puddle (M. C. Escher)

    Puddle is a woodcut print by the Dutch artist M. C. Es-cher, rst printed in February 1952.Since 1936, Eschers work had become primarily focusedon paradoxes, tessellation and other abstract visual con-cepts. This print, however, is a realistic depiction of asimple image that portrays two perspectives at once. Itdepicts an unpaved road with a large pool of water in themiddle of it at twilight. Turning the print upside-downand focusing strictly on the reection in the water, it be-comes a depiction of a forest with a full moon overhead.The road is soft and muddy and in it there are two dis-tinctly dierent sets of tire tracks, two sets of footprintsgoing in opposite directions and two bicycle tracks. Es-cher has thus captured three elements: the water, sky andearth.

    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

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

    Regular Division of the Plane is a series of drawingsby 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 interlockcompletely to cover a surface or plane.The inspiration for these works began in 1936 with avisit to the Alhambra, a fourteenth-century Moorish cas-tle near Granada, Spain. Escher had visited the Alhambraonce before in 1922 but in this visit he had spent severaldays studying and sketching the ornate tile designs there.In 1958 Escher published his book The Regular Divi-sion of the Plane. This book included several woodcutprints to demonstrate the concept, but the series of draw-ings continued until the late 1960s, ending at drawing#137. While not Eschers most artistically importantworks, some of these patterns are among Eschers mostfamous, having been used for a number of commercialproducts, including neckties.

    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: Visionsof Symmetry Harry N. Abrams, Inc. ISBN 0-8109-4308-5.

    31

  • Chapter 24

    Relativity (M. C. Escher)

    Relativity is a lithograph print by the Dutch artist M. C.Escher, rst printed in December 1953.It depicts a world in which the normal laws of gravity donot apply. The architectural structure seems to be thecentre of an idyllic community, with most of its inhab-itants casually going about their ordinary business, suchas dining. There are windows and doorways leading topark-like outdoor settings. All of the gures are dressedin identical attire and have featureless bulb-shaped heads.Identical characters such as these can be found in manyother Escher works.In the world of Relativity, there are three sources of grav-ity, each being orthogonal to the two others. Each in-habitant lives in one of the gravity wells, where normalphysical laws apply. There are sixteen characters, spreadbetween each gravity source, six in one and ve each inthe other two. The apparent confusion of the lithographprint comes from the fact that the three gravity sourcesare depicted in the same space.The structure has seven stairways, and each stairway canbe used by people who belong to two dierent gravitysources. This creates interesting phenomena, such as inthe top stairway, where two inhabitants use the same stair-way in the same direction and on the same side, but eachusing a dierent face of each step; thus, one descendsthe stairway as the other climbs it, even while moving inthe same direction nearly side-by-side. In the other stair-ways, inhabitants are depicted as climbing the stairwaysupside-down, but based on their own gravity source, theyare 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 belowthe parks. Though physically possible, such basementsare certainly unusual and add to the surreal eect of thepicture.This is one of Eschers most popular works and hasbeen used in a variety of ways, as it can be appreciatedboth artistically and scientically. Interrogations aboutperspective and the representation of three-dimensionalimages in a two-dimensional picture are at the core of Es-chers work, and Relativity represents one of his greatestachievements in this domain.

    32

  • Chapter 25

    Reptiles (M. C. Escher)

    Reptiles is a lithograph print by the Dutch artist M. C.Escher rst printed in March 1943.It depicts a desk on which is a drawing of a tessellatedpattern of reptiles. The reptiles at one edge of the draw-ing come to life and crawl around the desk and over theobjects on it to eventually re-enter the drawing at its op-posite edge. The desk is littered with ordinary objects, aswell as a metal dodecahedron that the reptiles climb over.Although only the size of small lizards, these reptiles ap-pear to have tusks and the one standing on the dodecahe-dron blows smoke from its nostrils.Like many of Eschers works, this image was intended todepict a paradoxical and slightly humorous concept withno real philosophical meaning. There were, however,many popular misconceptions about the images mean-ing. Once a woman telephoned Escher and told himthat she thought the image was a striking illustration ofreincarnation". The most common myth revolves arounda small book on the desk with the letters JOB printed onit. 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 rockband Mott the Hoople as the sleeve artwork for itseponymous rst 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 artistM. C. Escher rst printed in June 1938.The basis of this print is a regular division of theplane consisting of birds and sh. Both prints havethe horizontal series of these elementstting into eachother like the pieces of a jigsaw puzzlein the middle,transitional portion of the prints. In this central layer thepictorial elements are equal: birds and sh are alternatelyforeground or background, depending on whether the eyeconcentrates on light or dark elements. The birds take onan increasing three-dimensionality in the upward direc-tion, and the sh, in the downward direction. But as thesh progress upward and the birds downward they grad-ually lose their shapes to become a uniform backgroundof sky and water, respectively.According to Escher: In the horizontal center strip thereare birds and sh equivalent to each other. We associateying with sky, and so for each of the black birds thesky in which it is ying is formed by the four white shwhich encircle it. Similarly swimming makes us think ofwater, and therefore the four black birds that surround ash 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. Inthe pictures a number of visual elements unite into asimple visual representation, but separately each formsa point of departure for the elucidation of a theory in oneof these disciplines.

    26.1 See also Printmaking

    Sky and Water II

    Tessellation

    26.2 Sources M. C. EscherThe Graphic Work; Benedikt-Taschen Publishers.

    M. C. Escher29 Master Prints; Harry N. Abrams,Inc., Publishers.

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

    34

  • Chapter 27

    Sky and Water II

    Sky and Water II is a lithograph print by the Dutch artistM. C. Escher rst printed in 1938. It is similar to thewoodcut Sky and Water I, which was rst printed onlymonths earlier.

    27.1 See also Tessellation Printmaking

    27.2 Sources M. C. EscherThe Graphic Work; Benedikt-Taschen Publishers.

    M. C. Escher29 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. Es-cher rst printed in July 1969.It depicts a disc made up of interlocking circles that growprogressively smaller towards the center and towards theedge. There are three snakes laced through the edge ofthe disc.Snakes has rotational symmetry of order 3, comprising asingle wedge-shaped image repeated three times in a cir-cle. This means that it was printed from three blocks thatwere 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 andblack. In several earlier works Escher explored the limitsof innitesimal size and innite number, for example theCircle Limit series, by actually carrying through the ren-dering of smaller and smaller gures to the smallest possi-ble sizes. By contrast, in Snakes, the innite diminutionof size and innite increase in number is only sug-gested in the nished work. Nevertheless, the print showsvery clearly how this rendering would have been carriedout to the limits of human visibility.This was Eschers 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 Eschers 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 artistM. C. Escher rst printed in October 1948, depictingtwo chameleons in a polyhedral cage oating throughspace.[1][2]

    29.1 DescriptionThe print depicts a hollowed-out compound of three octa-hedra, a polyhedral compound composed of three regularoctahedra, oating in space. Numerous other polyhedraand polyhedral compounds oat in the background; thefour largest are, on the upper left, the compound of cubeand octahedron; on the upper right, the stella octangula;on the lower left, a compound of two cubes; and on thelower right, a solid version of the same octahedron 3-compound. The smaller polyhedra visible within the printalso include all of the ve Platonic solids and the rhombicdodecahedron.[3][4]

    Two chameleons are contained within the cage-like shapeof the central compound; Escher writes that they werechosen as its inhabitants because they are able to cling bytheir legs and tails to the beams of their cage as it swirlsthrough space.[5] The chameleon on the left sticks out histongue, perhaps in commentary; Coxeter observes thatthe tongue has an unusual spiral-shaped tip.[4]

    Although most published copies of Stars aremonochromatic, with white stars and chameleonson a black background, the copy in the National Galleryof Canada is tinted in dierent shades of turquoise,yellow, green, and pale pink.[6]

    29.2 InuencesThe design for Starswas likely inuenced by Eschers owninterest in both geometry and astronomy, by a long historyof using geometric forms to model the heavens, and bya drawing style used by Leonardo da Vinci. However,although the polyhedral shape depicted in Stars had beenstudied before in mathematics, it was most likely inventedindependently for this image by Escher without referenceto those studies.

    A rhombicuboctahedron drawn by Leonardo da Vinci in a simi-lar style to Escher

    Eschers interest in geometry is well known, but he wasalso an avid amateur astronomer, and in the early 1940sbecame a member of the Dutch Association for Mete-orology and Astronomy. He owned a 6 cm refractingtelescope, and recorded several observations of binarystars.[2]

    The use of polyhedra to model heavenly bodies can betraced back to Plato, who wrote in Timaeus that theconstellations were arranged in the form of a regular do-decahedron. Later, Johannes Kepler theorized that thedistribution of distances of the planets from the sun couldbe explained by the shapes of the ve Platonic solids. Es-cher, also, regularly depicted polyhedra in his artworksrelating to astronomy and other worlds.[2]

    Escher drew the octahedral compound of Stars in abeveled wire-frame style that was previously used byLeonardo da Vinci in his illustrations for Luca Pacioli'sbook, De divina proportione.[4][3][7]

    H. S. M. Coxeter reports that the shape of the centralchameleon cage in Stars had previously been described,

    37

  • 38 CHAPTER 29. STARS (M. C. ESCHER)

    with a photograph of a model of the same shape, in 1900by Max Brckner. However, Escher would not have beenaware of this reference and Coxeter writes that It is re-markable that Escher, without any knowledge of algebraor analytic geometry, was able to rediscover this highlysymmetrical gure.[4]

    29.3 AnalysisMartin Beech interprets the many polyhedral compoundswithin Stars as corresponding to double stars and triplestar systems in astronomy.[2] Beech writes that, for Es-cher, the mathematical orderliness of polyhedra depictsthe stability and timeless quality of the heavens, andsimilarly Marianne L. Teuber writes that Stars cele-brates Eschers identication with Johannes Keplers neo-Platonic belief in an underlying mathematical order in theuniverse.[8]

    Alternatively, Howard W. Jae interprets the polyhedralforms in Stars crystallographically, as brilliantly facetedjewels oating through space, with its compound poly-hedra representing crystal twinning.[9]

    However, R. A. Dunlap points out the contrast betweenthe order of the polyhedral forms and the more chaoticbiological nature of the chameleons inhabiting them.[10]In the same vein, Beech observes that the stars them-selves convey tension between order and chaos: despitetheir symmetric shapes, the stars are scattered apparentlyat random, and vary haphazardly from each other.[2] AsEscher himself wrote about the central chameleon cage,I shouldn't be surprised if it wobbles a bit.[2]

    29.4 Related worksA closely related woodcut, Study for Stars, completed inAugust 1948,[2][11] depicts wireframe versions of severalof the same polyhedra and polyhedral compounds, oat-ing in black within a square composition, but withoutthe chameleons. The largest polyhedron shown in Studyfor Stars, a stellated rhombic dodecahedron, is also oneof two polyhedra depicted prominently in Eschers 1961printWaterfall.[3]

    Eschers later work Four Regular Solids (Stereometric Fig-ure) returned to the theme of polyhedral compounds, de-picting a more explicitly Keplerian form in which thecompound of the cube and octahedron is nested withinthe compound of the dodecahedron and icosahedron.[10]

    29.5 Collections and publicationsStars was used as cover art for the 1962 anthology BestFantasy Stories edited by Brian Aldiss.[12]

    As well as being exhibited in the Escher Museum, copiesof Stars are in the permanent collections of the MildredLane Kemper Art Museum[13] and the National Galleryof Canada.[6]

    29.6 References[1] Locher, J. L. (2000), The Magic of M. C. Escher, Harry

    N. Abrams, Inc., p. 100, ISBN 0-8109-6720-0.

    [2] Beech, Martin, Eschers Stars", Journal of theRoyal Astronomical Society of Canada 86: 169177,Bibcode:1992JRASC..86..169B.

    [3] Hart, George W. (1996), The Polyhedra of M.C. Es-cher, Virtual Polyhedra.

    [4] Coxeter, H. S. M. (1985), A special book re-view: M. C. Escher: His life and complete graphicwork, The Mathematical Intelligencer 7 (1): 5969,doi:10.1007/BF03023010. Coxeters analysis of Stars ison pp. 6162.

    [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 theprints ofMaurits C. Escher, Scientic American 231: 90104, doi:10.1038/scienticamerican0774-90.

    [9] Jae, Howard W. (1996), About the frontispiece,Crystal Chemistry and Refractivity, Dover, p. vi, ISBN978-0-486-69173-2.

    [10] Dunlap, R. A. (1992), Fivefold symmetry in the graphicart of M. C. Escher, in Hargittai, Istvn, Fivefold Sym-metry (2nd ed.), World Scientic, pp. 489504, ISBN978-981-02-0600-0.

    [11] Locher (2000), p. 99.

    [12] Clute, John; Grant, John (1999), The encyclopedia of fan-tasy (2nd ed.), Macmillan, p. 322, ISBN 978-0-312-19869-5.

    [13] Artwork detail, Kemper Museum, retrieved 2011-11-19.

  • Chapter 30

    Still Life and Street

    Still Life and Street is a woodcut print by the Dutch artistM. C. Escher which was rst printed in March, 1937. Itwas his rst print of an impossible reality. In this artworkwe have two quite distinctly recognizable realities boundtogether in a natural, and yet at the same time a com-pletely impossible, way. Looked at from the window, thehouses make book-rests between which tiny dolls are setup. Looked at from the street, the books stand yards highand a gigantic tobacco jar stands at the crossroads.A small street in Savona, Italy, was the inspiration for thiswork.[1] Escher said it was one of his favorite drawingsbut thought he could have drawn it better.This image is a classic example of Eschers plays onperspective. In it, the horizontal plane of the table contin-ues into the distance to become the street, and the rows ofbooks on the table are seen to lean against the tall build-ings 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 artistM. C. Escher which was created in 1934.[1] The reec-tion of the mirror mingles together two completely unre-lated spaces and introduces the outside world of the smalltown narrow street in Abruzzi into internal world of thebedroom.[2] This work of Escher is closely related to hislater application of mirror eect in 1937 Still Life andStreet.[3] Escher manipulates the scale in dierent partsof the print to achieve the eect of smooth connectionbetween worlds.[4]

    31.1 References[1] Doris Schattschneider; Michele Emmer (19 September

    2005). M.C. Eschers Legacy: A Centennial Celebration: Collection of Articles Coming from the M.C. Escher Cen-tennial Conference, Rome, 1998. Springer. p. 219. ISBN978-3-540-20100-7. 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 June1984). 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. 6365. Retrieved 19 July2013.

    40

  • Chapter 32

    Still Life with Spherical Mirror

    Still Life with Spherical Mirror is a lithography print bythe Dutch artist M. C. Escher rst printed in November1934. It depicts a setting with rounded bottle and a metalsculpture of a bird with a human face seated atop a news-paper and a book. The background is dark but in thebottle can be seen the reection of Eschers studio andEscher himself sketching the scene.Self-portraits in reective spherical surfaces can be foundin Eschers early ink drawings and in his prints as late asthe 1950s. The metal bird/human sculpture is real andwas given to Escher by his father-in-law. This sculptureappears again in Eschers later prints Another World Mez-zotint (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 artistM. C. Escher rst printed in April 1946.As the title implies, it depicts three spheres resting on aat surface.The sphere on the left is transparent with a photorealisticdepiction of the refracted light cast through it towards theviewer and onto the at surface.The sphere in the center is reective. Its reection is aself-replicating image of Escher in his studio drawing thethree spheres. In the reection one can clearly see the im-age of the three spheres on the paper Escher is drawingon: in the center sphere of that image, one can vaguelymake out the reection of Eschers studio, which is de-picted in the main image. This process is implied to beinnite, recursive.The sphere on the right is opaque and diuse, i.e. neitherspecularly reective nor transparent.

    33.1 See also Still Life with Spherical Mirror Hand with Reecting 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 rst printed in December 1955.Three Worlds depicts a large pool or lake during the au-tumn or winter months, the title referring to the three vis-ible perspectives in the picture: the surface of the wateron which leaves oat, the world above the surface, observ-able by the waters reection of a forest, and the worldbelow the surface, observable in the large sh swimmingjust below the waters surface.Escher also created a picture named Two Worlds.

    34.1 See also Puddle Printmaking

    The picture is based on true optical eects, reection andrefraction. The angle of incidence is the line between thereection of the trees and refraction allowing the view ofthe sh.

    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 1928woodcut byM. C. Escher. It de-picts the Babylonians attempting to build a tower to reachGod, a story that is recounted in Genesis 11:9. God frus-trated their attempts by creating a confusion of languagesso the builders could no longer understand each other andthe work halted. Although Escher dismissed his worksbefore 1935 as of little or no value as they were for themost part merely practice exercises, some of them, in-cluding the Tower of Babel, chart the development of hisinterest in perspective and unusual viewpoints that wouldbecome the hallmarks of his later, more famous, work.In contrast to many other depictions of the biblical story,such as those by Pieter Brueghel the Elder (The Towerof Babel) and Gustave Dor (The Confusion of Tongues),Escher depicts the tower as a geometrical structure andplaces the viewpoint above the tower. This allows himto exercise his skill with perspective, but he also chose tocentre the picture around the top of the tower as the focusfor the climax of the action. He later commented:

    Some of the builders are white and oth-ers black. The work is at a standstill becausethey are no longer able to understand one an-other. Seeing as the climax of the drama takesplace at the summit of the tower which is un-der construction, the building has been shownfrom above as though from a birds eye view[1]

    35.1 See also

    Belvedere

    Waterfall

    35.2 References

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

    35.3 Notes Miranda Fellows (1995). The Life and Works of Es-

    cher. Bristol: Paragon Book Service. ISBN 0-7525-1175-0.

    44

  • Chapter 36

    Waterfall (M. C. Escher)

    Waterfall (Waterval) is a lithograph print by the Dutchartist M. C. Escher rst printed in October 1961. Itshows an apparent paradox where water from the base ofa waterfall appears to run downhill along the water pathbefore reaching the top of the waterfall.While most two-dimensional artists use relative propor-tions to create an illusion of depth, Escher here and else-where uses conicting proportions to create a visual para-dox. The waterfalls leat has the structure of two Penrosetriangles. A Penrose triangle is an impossible object de-signed by Oscar Reutersvrd in 1934, and independentlyby Roger Penrose in 1958.[1]

    36.1 DescriptionThe image depicts a village or small city with an elevatedaqueduct and waterwheel as the main