# Mc Escher Relativity Descriptive Essay

### #25: Relativity by M. C. Escher

Relativity, by M. C. Escher, 1953

This analysis copyright Scott M. McDaniel, 2010.

## The Image

Click here to see a larger version.

Man, I love M. C. Escher’s stuff! I know I’m hardly alone in that, but I thought I’d get it out of the way right in the beginning. And Relativity is one of my favorites. In college I had a poster of it, and I’d bring in dorm-mates after they’d been drinking too much and sit them down in front of it saying, “Look at this, man!” OK, college humor.

A few years back the National Gallery of Art in Washington DC had a showing of Escher’s work. Something that really caught me by surprise was how much you could see on the actual pieces that doesn’t show up even in the good quality, coffee table books of Escher’s work. Sure, he’s known for the perspective tricks, tessellation, and space warping, but he’s also a great artist and draftsman. I love Relativity because of the wonder and imagination, but it’s also a great example of things like lighting and defining mass and form. We’ll look at all of these things in this analysis.

## The Perspective

Escher uses three point perspective for Relativity. That’s not too unusual. When you’ve got three vanishing points they form a triangle. Usually when we use three point perspective two of the vanishing points are on the horizon and the third is either above (the zenith) or below (the nadir).

In Relativity Escher played with this concept. Because he set up the vanishing points as an equilateral triangle, it meant he could build a structure in perspective that didn’t look distorted if you changed the horizon from one side of the triangle to the others. Here’s a preparatory drawing that he did to work out the vanishing points. (From The Magic Mirror of M. C. Escher by Bruno Ernst.)

In the actual version, these vanishing points are about two meters outside the picture’s borders. The first perspective most of us notice is the one with the vanishing point triangle’s horizon at the bottom and the zenith directly above. It’s only natural, that’s how we usually see the world. The other two orientations, though, treat the other lines of the triangle as the horizon. Let’s spin the picture to see what it looks like when each of the other sides serves as the horizon.

Rotating the basic triangle counter-clockwise we have:

Rotating it clockwise we have:

The building, stonework, and doors all appear to make visual sense from any of the three orientations. It may not make **logical** sense to put a door in the floor, but visually we can see that it doesn’t look distorted. The three points are just three points. Assigning “up” and “down” happens in our heads – at least when it comes to the building. One other thing to mention – my first assumption was that the triangle of stairs at the core of Relativity’s composition was one of those impossible shapes that Escher uses so well in other pieces. It’s not. It doesn’t need to be because the structure isn’t an impossible one. To prove the point, if you haven’t already seen it, here’s a link to Andrew Lipson’s Lego version of Relativity.

When it came to the people, Escher had to go ahead and choose an up and down for each one. Regardless of their orientation, though, none of the figures appear stretched or pinched. They make visual sense too. This is all possible because the vanishing points form an equilateral triangle.

## The Lighting

What initially fascinates us about this picture is the illusion of three “ups” and the way they’re all mixed in together. Working out the perspective isn’t enough, though. If Escher hadn’t given the scene a sense of depth, mass, and solidity Relativity wouldn’t have nearly the impact it does. Values and shading are critical to this sense, but how do you handle light sources in a scene with three different horizons? He could have used sculptural lighting – a general non-defined light source, kind of like being outside on a cloudy day. Instead, though, Escher chose the three outside areas as light sources and then added two others to help with the composition.

Here are the light sources in Relativity.

The light coming from the outside makes sense. In fact, we even see the sun itself (or a sun symbol, anyway) through one of the windows.

There’s another light source that I find curious – it’s the most enigmatic in the whole picture. See the guy who has just come up the staircase from a basement carrying a sack? He’s lit from the front and is about to walk off down a hall, heading into the light. Where is that light coming from? What’s down that hall? The space down there would seem to be a tunnel beneath a sidewalk or perhaps a chimney above the sun. I’ve indicated that light source and one other with yellow in the graphic above to show that those sources are more ambiguous than the daylight coming through the three windows/porches.

One of the effects of that mystery light is to create a nice curve of sharp contrast in the upper center of the picture. Our eyes can follow that arch in either direction and we end up traveling around the picture in many loops. That curve of contrast, though, is critical to getting us started, and it depends upon an unseen light source that seems to come from some other dimension.

## The Lines

Now that we know where the light sources are we can look at the shading Escher used to create forms that look solid. He uses lots of cross hatching and feathering in both the figures and the structures. In just about every case we can see that the direction of those shading lines acts as a contour map that indicates a 3D shape. Here are some examples.

This is the guy at the bottom of the picture in it’s normal orientation. He’s our gateway into the image and our viewpoint character. He’s lit most strongly from the front right, although the window with the sun and tree provides secondary lighting. Let’s look at the hatching on the figure’s legs, back, and head. They all clearly follow the figure’s form like a contour map or wood grain, which helps us see it as a rounded shape rather than a flat one.

Now notice the white outline on the left side of his legs. That lines up well with the secondary light source and keeps the form of the legs crisp. As the lines wrap around the legs their number and thickness drop away until we have a highlight on the right side of the legs corresponding to the strongest lighting. There is a thin outline to the figure on that side – most noticeable on the arm and head. The line their symbolizes an edge. On the darker side of the head and arm, though, there is no outline. Just an edge, chiaroscuro style. Escher follows this general approach on all of the figures, using both lighting and the direction of the hatching lines to define the form.

Now let’s look at the the guy in the center with the sack, emerging from the basement.

This time let’s focus on the texture and shading of the walls. Escher uses a basic cross-hatch to give them a mid-value grey. Going “down” that stairwell we see that the lines get thicker to produce a darker tone. There aren’t more lines, though, just thicker ones. For the most part the hatch lines point to one of the three vanishing points. Despite the fact that they’re the same grey tone it’s easy to tell which way the walls are going. In the arch in the lower corner of the picture, we can see some hatch lines pointing to a vanishing point down and to the left. On the side of the arch, though, we see the hatch lines curve to follow the arch’s form. They start off headed toward the zenith but then bend around to the left where they thicken into darkness.

Now we’re in the upper left corner of the picture. Here’s an interesting lighting choice. Why are the walking figures so much lighter than the one with his hands on the balcony? We can see from the shadow that the light is coming from the couple’s right side. Logically I’d think that the shadows on them should be nearly as strong on them as on the other figure, but he’s much darker. I think the main reason is that Escher wanted to push them into the background detail and bring the other figure forward. A second reason could be that there’s reflected light from the building on the walking couple but not on the single figure.

Here’s a final detail of the couple at the table in the right of the picture. Once again, see how the shading and lines wrap around the forms.

I believe this shading and linework is just as important to the illusion as the perspective. Without the impression of solidity and mass we’d get the idea about the perspective intellectually but it wouldn’t seem as real and powerful as this version does.

## The Elements

To wrap up we’ll go through Lee Moyer’sElements of a Successful Illustration.

**Focus**: The initial focus is just about in the center – it’s the guy walking vertically up the wall carrying the sack. After that, the strong contrast curve above him begins to move our eye around the picture where it is supposed to wander and take in the contradictory details.

**Composition and Design**: Like most of Escher’s work, it’s got a strong theoretical and geometric design. The core pattern is a triangle of stairs that’s a flipped version of the unseen triangle of vanishing points. (Unseen because they are beyond the border of the drawing.)

**Palette**: Greyscale, defined through hatching.

**Value**: Escher carefully controls the light source and value to enhance the illusion of 3D, solid figures moving around a spacious interior.

**Mass**: Defined through value and line direction.

**Texture**: Escher uses the hatching to produce a clear texture formed from cross-hatching in the walls and structures. He wraps the lines around the figures to produce a wooden texture on them.

**Symbolism**: Why do you suppose that Escher didn’t give the figures individual identities? Perhaps he didn’t want to distract us from the main idea of the painting. Maybe, from a certain perspective, people are interchangeable.

**Micro/Macro**: Escher chose not to draw every stone in every wall. The texture takes care of most of that. There are stones in the archways over the doors, however. For the most part, though, lots of detail would distract us from looking at the perspective paradox.

**Ornament**: There is only minimal ornament. The figures wear a loincloth of sorts, but you won’t see knobs on the stair rails or other decorations. They’re not the point.

**Narrative**: A story isn’t the point here, but there are a few hints. The walking couple have their arms around each other. There seems to be a servant or waiter. Is the guy with the sack on his shoulder stealing something or just retrieving it. We don’t know the answers, but we have enough information to pose the questions.

**Juxtaposition**: The main juxtaposition here is the perspective itself.

**Stylization**: Normally I talk here about how people handle lines and shapes. Like Magritte, though, a key part of Escher’s style is how he thinks and presents us with an image. We see something that, try as we might, we just cannot take for granted. While he often did lithography, prints, and woodcuts, Escher’s most identifiable style is a sense of awe and wonder.

**Character**: Like the narration comment above, we see only enough about these figures to make some assumptions and ask questions.

**Tension**: The tension here is a perceptual one. We know that there can’t be more than one “up” in a single scene, yet there we see it before us. We feel compelled to resolve the problem.

**Line**: Discussed above.

**Research/Reference**: I showed the preparatory drawing above, and I got to see more of them for other works at the Escher exhibition at the National Art Gallery a few years back. Rest assured that he did not produce these images from the air on his first try.

**Vignette**: While the lighting is critical, there is not a single silhouette that is key to the picture. The core of it, however, is the triangle of stairs in the center.

**Perspective**: There are three vanishing points set in an equilateral triangle. Pick any two and you will see evidence of the line between them as a horizon line. It’s the foundation of the drawing.

That’s it for this time. Next time we’ll venture into multimedia and symbolism with David Mack.

**Maurits Cornelis Escher** (Dutch pronunciation:[ˈmʌurɪts kɔrˈneːlɪs ˈɛsxər]; 17 June 1898 – 27 March 1972), or commonly **M. C. Escher**, was a Dutch graphic artist who made mathematically inspiredwoodcuts, lithographs, and mezzotints.

His work features mathematical objects and operations including impossible objects, explorations of infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had no mathematical ability, he interacted with the mathematicians George Pólya, Roger Penrose, Harold Coxeter and crystallographerFriedrich Haag, and conducted his own research into tessellation.

Early in his career, he drew inspiration from nature, making studies of insects, landscapes, and plants such as lichens, all of which he used as details in his artworks. He traveled in Italy and Spain, sketching buildings, townscapes, architecture and the tilings of the Alhambra and the Mezquita of Cordoba, and became steadily more interested in their mathematical structure.

Escher's art became well known among scientists and mathematicians, and in popular culture, especially after it was featured by Martin Gardner in his April 1966 Mathematical Games column in Scientific American. Apart from being used in a variety of technical papers, his work has appeared on the covers of many books and albums. He was one of the major inspirations of Douglas Hofstadter's Pulitzer Prize-winning 1979 book *Gödel, Escher, Bach*.

Despite wide popular interest, Escher was for long somewhat neglected in the art world; even in his native Netherlands, he was 70 before a retrospective exhibition was held. In the twenty-first century, he became more widely appreciated, with exhibitions across the world.

## Early life

Maurits Cornelis^{[a]} Escher was born on 17 June 1898 in Leeuwarden, Friesland, the Netherlands, in a house that forms part of the Princessehof Ceramics Museum today. He was the youngest son of the civil engineerGeorge Arnold Escher and his second wife, Sara Gleichman. In 1903, the family moved to Arnhem, where he attended primary and secondary school until 1918.^{[1]}^{[2]} Known to his friends and family as "Mauk", he was a sickly child and was placed in a special school at the age of seven; he failed the second grade.^{[3]} Although he excelled at drawing, his grades were generally poor. He took carpentry and piano lessons until he was thirteen years old.^{[1]}^{[2]}

In 1918, he went to the Technical College of Delft.^{[1]}^{[2]} From 1919 to 1922, Escher attended the Haarlem School of Architecture and Decorative Arts, learning drawing and the art of making woodcuts.^{[1]} He briefly studied architecture, but he failed a number of subjects (due partly to a persistent skin infection) and switched to decorative arts,^{[3]} studying under the graphic artist Samuel Jessurun de Mesquita.^{[4]}

## Study journeys

In 1922, an important year of his life, Escher traveled through Italy, visiting Florence, San Gimignano, Volterra, Siena, and Ravello. In the same year, he traveled through Spain, visiting Madrid, Toledo, and Granada.^{[1]} He was impressed by the Italian countryside and, in Granada, by the Moorish architecture of the fourteenth-century Alhambra. The intricate decorative designs of the Alhambra, based on geometricalsymmetries featuring interlocking repetitive patterns in the coloured tiles or sculpted into the walls and ceilings, triggered his interest in the mathematics of tessellation and became a powerful influence on his work.^{[6]}^{[7]}

Escher returned to Italy and lived in Rome from 1923 to 1935. While in Italy, Escher met Jetta Umiker – a Swiss woman, like himself attracted to Italy – whom he married in 1924. The couple settled in Rome where their first son, Giorgio (George) Arnaldo Escher, named after his grandfather, was born. Escher and Jetta later had two more sons — Arthur and Jan.^{[1]}^{[2]}

He travelled frequently, visiting (among other places) Viterbo in 1926, the Abruzzi in 1927 and 1929, Corsica in 1928 and 1933, Calabria in 1930, the Amalfi coast in 1931 and 1934, and Gargano and Sicily in 1932 and 1935. The townscapes and landscapes of these places feature prominently in his artworks. In May and June 1936, Escher travelled back to Spain, revisiting the Alhambra and spending days at a time making detailed drawings of its mosaic patterns. It was here that he became fascinated, to the point of obsession, with tessellation, explaining:^{[4]}^{[8]}

It remains an extremely absorbing activity, a real mania to which I have become addicted, and from which I sometimes find it hard to tear myself away.

^{[8]}

The sketches he made in the Alhambra formed a major source for his work from that time on.^{[8]} He also studied the architecture of the Mezquita, the Moorish mosque of Cordoba. This turned out to be the last of his long study journeys; after 1937, his artworks were created in his studio rather than in the field. His art correspondingly changed sharply from being mainly observational, with a strong emphasis on the realistic details of things seen in nature and architecture, to being the product of his geometric analysis and his visual imagination. All the same, even his early work already shows his interest in the nature of space, the unusual, perspective, and multiple points of view.^{[4]}^{[8]}

## Later life

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 ideals other than the expressions of his own concepts through his own particular medium, but he was averse to fanaticism and hypocrisy. When his eldest son, George, was forced at the age of nine to wear a Ballila uniform in school, the family left Italy and moved to Château-d'Œx, Switzerland, where they remained for two years.^{[9]}

The Netherlands post office had Escher design a semi-postal stamp for the "Air Fund" in 1935^{[10]}, and again in 1949 he designed Netherlands stamps. These were for the 75th anniversary of the Universal Postal Union; a different design was used by Surinam and the Netherlands Antilles for the same commemoration.^{[11]}^{[12]}

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 (Ukkel), a suburb of Brussels, Belgium.^{[1]}^{[2]}World War II forced them to move in January 1941, this time to Baarn, Netherlands, where Escher lived until 1970.^{[1]} Most of Escher's best-known works date from this period. The sometimes cloudy, cold, and wet weather of the Netherlands allowed him to focus intently on his work.^{[1]} After 1953, Escher lectured widely. A planned series of lectures in North America in 1962 was cancelled after an illness, and he stopped creating artworks for a time,^{[1]} but the illustrations and text for the lectures were later published as part of the book *Escher on Escher*.^{[13]} He was awarded the Knighthood of the Order of Orange-Nassau in 1955;^{[1]} he was later made an Officer in 1967.^{[14]}

In July 1969 he finished his last work, a large woodcut with threefold rotational symmetry called *Snakes*, in which snakes wind through a pattern of linked rings. These shrink to infinity toward both the center and the edge of a circle. It was exceptionally elaborate, being printed using three blocks, each rotated three times about the center of the image and precisely aligned to avoid gaps and overlaps, for a total of nine print operations for each finished print. The image encapsulates Escher's love of symmetry; of interlocking patterns; and, at the end of his life, of his approach to infinity.^{[15]}^{[16]}^{[17]} The care that Escher took in creating and printing this woodcut can be seen in a video recording.^{[18]}

Escher moved to the Rosa Spier Huis in Laren in 1970, an artists' retirement home in which he had his own studio. He died in a hospital in Hilversum on 27 March 1972, aged 73.^{[1]}^{[2]} He is buried at the New Cemetery in Baarn.^{[19]}^{[20]}

## Mathematically inspired work

Further information: Mathematics and art

Escher's work is inescapably mathematical. This has caused a disconnect between his full-on popular fame and the lack of esteem with which he has been viewed in the art world. His originality and mastery of graphic techniques are respected, but his works have been thought too intellectual and insufficiently lyrical. Movements such as conceptual art have, to a degree, reversed the art world's attitude to intellectuality and lyricism, but this did not rehabilitate Escher, because traditional critics still disliked his narrative themes and his use of perspective. However, these same qualities made his work highly attractive to the public.^{[21]}

Escher is not the first artist to explore mathematical themes: Parmigianino (1503–1540) had explored spherical geometry and reflection in his 1524 *Self-portrait in a Convex Mirror*, depicting his own image in a curved mirror, while William Hogarth's 1754 *Satire on False Perspective* foreshadows Escher's playful exploration of errors in perspective.^{[22]}^{[23]} Another early artistic forerunner is Giovanni Battista Piranesi (1720–1778), whose dark "fantastical"^{[24]} prints such as *The Drawbridge* in his *Carceri* ("Prisons") sequence depict perspectives of complex architecture with many stairs and ramps, peopled by walking figures.^{[24]}^{[25]} Only with 20th century movements such as Cubism, De Stijl, Dadaism, and Surrealism did mainstream art start to explore Escher-like ways of looking at the world with multiple simultaneous viewpoints.^{[21]} However, although Escher had much in common with, for example, Magritte's surrealism, he did not make contact with any of these movements.^{[26]}

### Tessellation

Further information: Tessellation

In his early years, Escher sketched landscapes and nature. He also sketched insects such as ants, bees, grasshoppers, and mantises,^{[27]}, which appeared frequently in his later work. His early love of Roman and Italian landscapes and of nature created an interest in tessellation, which he called *Regular Division of the Plane*; this became the title of his 1958 book, complete 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 wrote, "Mathematicians have opened the gate leading to an extensive domain".^{[28]}

After his 1936 journey to the Alhambra and to La Mezquita, Cordoba, where he sketched the Moorish architecture and the tessellated mosaic decorations,^{[29]}, Escher began to explore the properties and possibilities of tessellation using geometric grids as the basis for his sketches. He then extended these to form complex interlocking designs, for example with animals such as birds, fish, and reptiles.^{[30]} One of his first attempts at a tessellation was his pencil, India ink, and watercolour *Study of Regular Division of the Plane with Reptiles* (1939), constructed on a hexagonal grid. The heads of the red, green, and white reptiles meet at a vertex; the tails, legs, and sides of the animals interlock exactly. It was used as the basis for his 1943 lithograph *Reptiles*.^{[31]}

His first study of mathematics began with papers by George Pólya^{[32]} and by the crystallographer Friedrich Haag^{[33]} on plane symmetry groups, sent to him by his brother Berend, a geologist. He carefully studied the 17 canonical wallpaper groups and created periodic tilings with 43 drawings of different types of symmetry.^{[c]} From this point on, he developed a mathematical approach to expressions of symmetry in his artworks using his own notation. Starting in 1937, he created woodcuts based on the 17 groups. His *Metamorphosis I* (1937) began a series of designs that told a story through the use of pictures. In *Metamorphosis I*, he transformed convex polygons into regular patterns in a plane to form a human motif. He extended the approach in his piece *Metamorphosis III*, which is four metres long.^{[8]}^{[35]}

In 1941 and 1942, Escher summarized his findings for his own artistic use in a sketchbook, which he labeled (following Haag) *Regelmatige vlakverdeling in asymmetrische congruente veelhoeken* ("Regular division of the plane with asymmetric congruent polygons").^{[36]} The mathematician Doris Schattschneider unequivocally described this notebook as recording "a methodical investigation that can only be termed mathematical research." She defined the research questions he was following as

(1) What are the possible shapes for a tile that can produce a regular division of the plane, that is, a tile that can fill the plane with its congruent images such that every tile is surrounded in the same manner?

(2) Moreover, in what ways are the edges of such a tile related to each other by isometries?

### Geometries

Further information: Perspective (geometry) and Curvilinear perspective

Although Escher did not have mathematical training—his understanding of mathematics was largely visual and intuitive—his art had a strong mathematical component, and several of the worlds that he drew were built around impossible objects. After 1924, Escher turned to sketching landscapes in Italy and Corsica with irregular perspectives that are impossible in natural form. His first print of an impossible reality was *Still Life and Street* (1937); impossible stairs and multiple visual and gravitational perspectives feature in popular works such as *Relativity* (1953). *House of Stairs* (1951) attracted the interest of the mathematician Roger Penrose and his father, the biologist Lionel Penrose. In 1956, they published a paper, "Impossible Objects: A Special Type of Visual Illusion" and later sent Escher a copy. Escher replied, admiring the Penroses' continuously rising flights of steps, and enclosed a print of *Ascending and Descending* (1960). The paper also contained the tribar or Penrose triangle, which Escher used repeatedly in his lithograph of a building that appears to function as a perpetual motion machine, *Waterfall* (1961).^{[37]}^{[38]}^{[39]}^{[40]}

Escher was interested enough in Hieronymus Bosch's 1500 triptych *The Garden of Earthly Delights* to re-create part of its right-hand panel, *Hell*, as a lithograph in 1935. He reused the figure of a Mediaeval woman in a two-pointed headdress and a long gown in his lithograph *Belvedere* in 1958; the image is, like many of his other "extraordinary invented places",^{[41]} peopled with "jesters, knaves, and contemplators".^{[41]} Thus, Escher not only was interested in possible or impossible geometry but was, in his own words, a "reality enthusiast";^{[41]} he combined "formal astonishment with a vivid and idiosyncratic vision".^{[41]}

Escher worked primarily in the media of lithographs and woodcuts, although 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. Integrated into his prints were mirror images of cones, spheres, cubes, rings, and spirals.^{[42]}

Escher was also fascinated by mathematical objects such as the Möbius strip, which has only one surface. His wood engraving *Möbius Strip II* (1963) depicts a chain of ants marching forever over what, at any one place, are the two opposite faces of the object—which are seen on inspection to be parts of the strip's single surface. In Escher's own words:^{[43]}

An endless ring-shaped band usually has two distinct surfaces, one inside and one outside. Yet on this strip nine red ants crawl after each other and travel the front side as well as the reverse side. Therefore the strip has only one surface.

^{[43]}

The mathematical influence in his work became prominent after 1936, when, having boldly asked the Adria Shipping Company if he could sail with them as travelling artist in return for making drawings of their ships, they surprisingly agreed, and he sailed the Mediterranean, becoming interested in order and symmetry. Escher described this journey, including his repeat visit to the Alhambra, as "the richest source of inspiration I have ever tapped".^{[8]}

Escher's interest in curvilinear perspective was encouraged by his friend and "kindred spirit",^{[44]} the art historian and artist Albert Flocon, in another example of constructive mutual influence. Flocon identified Escher as a "thinking artist"^{[44]} alongside Piero della Francesca, Leonardo da Vinci, Albrecht Dürer, Wenzel Jamnitzer, Abraham Bosse, Girard Desargues, and Père Nicon.^{[44]} Flocon was delighted by Escher's *Grafiek en tekeningen* ("Graphics in Drawing"), which he read in 1959. This stimulated Flocon and André Barre to correspond with Escher and to write the book *La Perspective curviligne* ("Curvilinear perspective").^{[45]}

### Platonic and other solids

Escher often incorporated three-dimensional objects such as the Platonic solids such as spheres, tetrahedrons, and cubes into his works, as well as mathematical objects such as cylinders and stellated polyhedra. In the print *Reptiles*, he combined two- and three-dimensional images. In one of his papers, Escher emphasized the importance of dimensionality:^{[46]}

The flat shape irritates me—I feel like telling my objects, you are too fictitious, lying there next to each other static and frozen:

dosomething, come off the paper and show me what you are capable of! ... So I make them come out of the plane. ... My objects ... may finally return to the plane and disappear into their place of origin.^{[46]}

Escher's artwork is especially well-liked by mathematicians such as Doris Schattschneider and scientists such as Roger Penrose, who enjoy his use of polyhedra and geometric distortions. For example, in *Gravitation*, animals climb around a stellateddodecahedron.^{[47]}

The two towers of *Waterfall*'s impossible building are topped with compound polyhedra, one a compound of three cubes, the other a stellated rhombic dodecahedron now known as Escher's solid. Escher had used this solid in his 1948 woodcut *Stars*, which also contains all five of the Platonic solids and various stellated solids, representing stars; the central solid is animated by chameleons climbing through the frame as it whirls in space. Escher possessed a 6 cm refracting telescope and was a keen-enough amateur astronomer to have recorded observations of binary stars.^{[48]}^{[49]}^{[50]}

### Levels of reality

Escher's artistic expression was created from images in his mind, rather than directly from observations and travels to other countries. His interest in the multiple levels of reality in art is seen in works such as *Drawing Hands* (1948), where two hands are shown, each drawing the other. The critic Steven Poole commented that^{[41]}

It is a neat depiction of one of Escher's enduring fascinations: the contrast between the two-dimensional flatness of a sheet of paper and the illusion of three-dimensional volume that can be created with certain marks. In

Drawing Hands, space and the flat plane coexist, each born from and returning to the other, the black magic of the artistic illusion made creepily manifest.^{[41]}

### Infinity and hyperbolic geometry

In 1954, the International Congress of Mathematicians met in Amsterdam, and N. G. de Bruin organized a display of Escher's work at the Stedelijk Museum for the participants. Both Roger Penrose and H. S. M. Coxeter were deeply impressed with Escher's intuitive grasp of mathematics. Inspired by *Relativity*, Penrose devised his tribar, and his father, Lionel Penrose, devised an endless staircase. Roger Penrose sent sketches of both objects to Escher, and the cycle of invention was closed when Escher then created the perpetual motion machine of *Waterfall* and the endless march of the monk-figures of *Ascending and Descending*. In 1957, Coxeter obtained Escher's permission to use two of his drawings in his paper "Crystal symmetry and its generalizations".^{[51]} He sent Escher a copy of the paper; Escher recorded that Coxeter's figure of a hyperbolic tessellation "gave me quite a shock": the infinite regular repetition of the tiles in the hyperbolic plane, growing rapidly smaller towards the edge of the circle, was precisely what he wanted to allow him to represent infinity on a two-dimensional plane.^{[52]}

Escher carefully studied Coxeter's figure, marking it up to analyse the successively smaller circles^{[d]} with which (he deduced) it had been constructed. He then constructed a diagram, which he sent to Coxeter, showing his analysis; Coxeter confirmed it was correct, but disappointed Escher with his highly technical reply. All the same, Escher persisted with hyperbolic tiling, which he called "Coxetering". Among the results were the series of wood engravings *Circle Limit I–IV*. In 1959, Coxeter published his finding that these works were extraordinarily accurate: "Escher got it absolutely right to the millimeter".^{[53]}

## Legacy

Escher's special way of thinking and rich graphics have had a continuous influence in mathematics and art, as well as in popular culture.

### In art collections

The Escher intellectual property is controlled by the M.C. Escher Company, while exhibitions of his artworks are managed separately by the M.C. Escher Foundation.^{[e]}

The primary institutional collections of original works by M.C. Escher are the Escher Museum in The Hague; the National Gallery of Art (Washington, DC);^{[56]} the National Gallery of Canada (Ottawa);^{[57]} the Israel Museum (Jerusalem);^{[58]} and the Huis ten Bosch (Nagasaki, Japan).^{[59]}

### Exhibitions

Despite wide popular interest, Escher was for a long time somewhat neglected in the art world; even in his native Netherlands, he was 70 before a retrospective exhibition was held.^{[41]}^{[g]} In the twenty-first century, major exhibitions have been held in cities across the world.^{[62]}^{[63]}^{[64]} An exhibition of his work in Rio de Janeiro attracted more than 573,000 visitors in 2011;^{[62]} its daily visitor count of 9,677 made it the most visited museum exhibition of the year, anywhere in the world.^{[65]} No major exhibition of Escher's work was held in Britain until 2015, when the Scottish National Gallery of Modern Art ran one in Edinburgh from June to September 2015,^{[63]} moving in October 2015 to the Dulwich Picture Gallery, London.^{[60]} The exhibition moved to Italy in 2015–2016, attracting over 500,000 visitors in Rome and Bologna,^{[64]} and then Milan.^{[66]}^{[67]}^{[68]}

### In mathematics and science

Doris Schattschneider identifies 11 strands of mathematical and scientific research anticipated or directly inspired by Escher. These are the classification of regular tilings using the edge relationships of tiles: two-color and two-motif tilings (counterchange symmetry or antisymmetry); color symmetry (in crystallography); metamorphosis or topological change; covering surfaces with symmetric patterns; Escher's algorithm (for generating patterns using decorated squares); creating tile shapes; local versus global definitions of regularity; symmetry of a tiling induced by the symmetry of a tile; orderliness not induced by symmetry groups; the filling of the central void in Escher's lithograph *Print Gallery* by H. Lenstra and B. de Smit.

The Pulitzer Prize-winning 1979 book *Gödel, Escher, Bach* by Douglas Hofstadter^{[69]} discusses the ideas of self-reference and strange loops, drawing on a wide range of artistic and scientific sources including Escher's art and the music of J. S. Bach.

The asteroid4444 Escher was named in Escher's honor in 1985.^{[70]}

### In popular culture

Main article: M. C. Escher in popular culture

Escher's fame in popular culture grew when his work was featured by Martin Gardner in his April 1966 "Mathematical Games" column in *Scientific American*.^{[71]} Escher's works have appeared on many album covers including The Scaffold's 1969 *L the P* with *Ascending and Descending*; Mott the Hoople's eponymous 1969 record with *Reptiles*, Beaver & Krause's 1970 *In A Wild Sanctuary* with *Three Worlds*; and Mandrake Memorial's 1970 *Puzzle* with *House of Stairs* and (inside) *Curl Up*.^{[h]} His works have similarly been used on many book covers, including some editions of Edwin Abbott's *Flatland*, which used *Three Spheres*; E. H. Gombrich's *Meditations on a Hobby Horse* with *Horseman*; Pamela Hall's *Heads You Lose* with *Plane Filling 1*; Patrick A. Horton's *Mastering the Power of Story* with *Drawing Hands*; Erich Gamma et al.'s *Design Patterns: Elements of Reusable Object-oriented software* with *Swans*; and Arthur Markman's *Knowledge Representation* with *Reptiles*.^{[i]} The "World of Escher" markets posters, neckties, T-shirts, and jigsaw puzzles of Escher's artworks.^{[74]} Both Austria and the Netherlands have issued postage stamps commemorating the artist and his works.^{[12]}^{[75]}

## 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)*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 Amalfi*, lithograph (1931)*Ravello and the Coast of Amalfi*, lithograph (1931)*Covered Alley in Atrani, Coast of Amalfi*, wood engraving (1931)*Phosphorescent Sea*, lithograph (1933)*Still Life with Spherical Mirror*, lithograph (1934)*Hand with Reflecting Sphere*also known as*Self-Portrait in Spherical Mirror*, lithograph (1935)*Inside St. Peter's*, wood engraving (1935)*Portrait of G.A. Escher*, lithograph (1935)*"Hell"*, lithograph, (copied from a painting by Hieronymus Bosch) (1935)*Regular Division of the Plane*, series of drawings that continued until the 1960s (1936)*Still Life and Street*(his first 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 (1939–1940)*Verbum (Earth, Sky and Water)*, lithograph (1942)*Reptiles*, lithograph (1943)*Ant*, lithograph (1943)*Encounter*, lithograph (1944)*Doric Columns*, wood engraving (1945)*Balcony*, lithograph (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)*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)*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)*Möbius Strip II (Red Ants)*, woodcut (1963)*Knot*, pencil and crayon (1966)*Metamorphosis III*, woodcut (1967–1968)*Snakes*, woodcut (1969)

## See also

## Notes

## References

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}^{l}"Chronology".*World of Escher*. Retrieved 1 November 2015. - ^
^{a}^{b}^{c}^{d}^{e}^{f}"About M.C. Escher". Escher in het Paleis. Retrieved 11 February 2016. - ^
^{a}^{b}Bryden, Barbara E.*Sundial: Theoretical Relationships Between Psychological Type, Talent, And Disease*. Gainesville, Fla: Center for Applications of Psychological Type. ISBN 0-935652-46-9. - ^
^{a}^{b}^{c}Locher, 1974. p. 5 **^**Locher, 1974. p. 17**^**Roza, Greg (2005).*An Optical Artist: Exploring Patterns and Symmetry*. Rosen Classroom. p. 20. ISBN 978-1-4042-5117-5.**^**Monroe, J. T. (2004).*Hispano-Arabic Poetry: A Student Anthology*. Gorgias Press LLC. p. 65. ISBN 978-1-59333-115-3.- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}O'Connor, J. J.; Robertson, E. F. (May 2000). "Maurits Cornelius Escher".*Biographies*. University of St Andrews. Retrieved 2 November 2015. which cites Strauss, S. (9 May 1996). "M C Escher".*The Globe and Mail*. **^**Ernst, Bruno,*The Magic Mirror of M.C. Escher*, Taschen, 1978; p. 15**^**"Aircraft over the Netherlands".*Stamp catalogue*. Colnect.com. Retrieved 2016-03-31.**^**Hathaway, Dale K. (2015-11-17). "Maurits Cornelis Escher (1898 - 1972)". Olivet Nazarene University. Retrieved 2016-03-31.- ^
^{a}^{b}

^{[5]}

^{[b]}

^{[8]}study of the same Moorish tiling in the Alhambra, 1936, demonstrates his growing interest in tessellation.

*Snakes*, 1969

*Study of Regular Division of the Plane with Reptiles*(1939). Escher reused the design in his 1943 lithograph

*Reptiles*.

*Relativity*, 1953

**^**"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.**^**The circled cross at the top of the image may indicate that the drawing is inverted, as can be seen by comparison with the photograph; the neighbouring image has a circled cross at the bottom. It is likely that Escher turned the drawing block, as convenient, while holding it in his hand in the Alhambra.**^**Escher made it clear that he did not understand the abstract concept of a group, but he did grasp the nature of the 17 wallpaper groups in practice.^{[8]}**^**Schattschneider notes that Coxeter observed in March 1964 that the white arcs in*Circle Limit III*"were not, as he and others had assumed, badly rendered hyperbolic lines but rather were branches of equidistant curves."**^**In 1969, Escher's business advisor, Jan W. Vermeulen, author of a biography on the artist, established the 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 works. In 1980, this holding was sold to an American art dealer and the Hague Museum. The Museum obtained all of the documentation and the smaller portion of the art works. The copyrights remained the possession of Escher's three sons – who later sold them to Cordon Art, a Dutch company. Control was subsequently transferred to The M.C. Escher Company B.V. of Baarn, Netherlands, which licenses use of the copyrights on all of Escher's art and on his spoken and written text. A related entity, the M.C. Escher Foundation of Baarn, promotes Escher's work by organizing exhibitions, publishing books and producing films about his life and work.^{[54]}^{[55]}**^**The poster for the exhibition is based on*Hand with Reflecting Sphere*, 1935, which shows Escher in his house reflected in a handheld sphere, thus illustrating the artist, his interest in levels of reality in art (e.g., is the hand in the foreground more real than the reflected one?), perspective, and spherical geometry.^{[23]}^{[60]}^{[61]}**^**Steven Poole comments "The artist [Escher] who created some of the most memorable images of the 20th century was never fully embraced by the art world."^{[41]}**^**These and further albums are listed by Coulthart.^{[72]}**^**These and further books are listed by Bailey.^{[73]}

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