The mathematical art of Moritz Escher February 28th, 2014
Original taken from imit_omsu in The Mathematical Art of Moritz Escher
“Mathematicians opened the door leading to another world, but they themselves did not dare to enter this world. They are more interested in the path on which the door stands than in the garden that lies behind it.”
(M.C. Escher)
Lithograph "Hand with a mirror sphere", self-portrait.
Maurits Cornelius Escher is a Dutch graphic artist known to every mathematician.
The plots of Escher's works are characterized by a witty understanding of logical and plastic paradoxes.
He is known primarily for his works in which he used various mathematical concepts - from the limit and the Möbius strip to Lobachevsky geometry.
Woodcut "Red Ants".
Maurits Escher did not receive any special mathematical education. But from the very beginning creative career was interested in the properties of space, studied its unexpected sides.
"Bonds of Unity"
Escher often dabbled with combinations of the 2-dimensional and 3-dimensional world. 
Lithograph "Drawing hands".
Lithograph "Reptiles".
Tessellations.
A tiling is a partition of a plane into identical figures. To study this kind of partition, the concept of symmetry group is traditionally used. Let's imagine a plane on which some tessellation is drawn. The plane can be rotated around an arbitrary axis and shifted. The shift is determined by the shift vector, and the rotation is determined by the center and angle. Such transformations are called movements. They say that this or that movement is symmetry if after it the tiling turns into itself.
Let us consider, for example, a plane divided into equal squares—an infinite sheet of a checkered notebook in all directions. If such a plane is rotated 90 degrees (180, 270 or 360 degrees) around the center of any square, the tiling will turn into itself. It also transforms into itself when shifted by a vector parallel to one of the sides of the squares. The length of the vector must be a multiple of the side of the square.
In 1924, geometer George Pólya (before moving to the USA, György Pólya) published a paper on tessellation symmetry groups, in which he proved a remarkable fact (although already discovered in 1891 by the Russian mathematician Evgraf Fedorov, and later happily forgotten): there are only 17 groups symmetries that include shifts in at least two different directions. In 1936, Escher, interested in Moorish patterns (from a geometric point of view, a variant of tiling), read Pólya’s work. Despite the fact that, by his own admission, he did not understand all the mathematics behind the work, Escher was able to capture its geometric essence. As a result, based on all 17 groups, Escher created more than 40 works.
Mosaic.
Woodcut "Day and Night".
"Regular tiling of plane IV".
Woodcut "Sky and Water".
Tessellations. The group is simple, generating: sliding symmetry and parallel transfer. But the paving tiles are wonderful. And combined with the Mobius Strip, that's it. 
Woodcut "Horsemen".
Another variation on the theme of the flat and volumetric world and tessellations. 
Lithograph "Magic Mirror".
Escher was friends with physicist Roger Penrose. In his free time from physics, Penrose spent his time solving mathematical puzzles. One day he came up with the following idea: if we imagine a tessellation consisting of more than one figure, would its group of symmetries be different from those described by Pólya? As it turned out, the answer to this question is in the affirmative - this is how the Penrose mosaic was born. In the 1980s it turned out that it is associated with quasicrystals ( Nobel Prize in Chemistry 2011).
However, Escher did not have time (or perhaps did not want) to use this mosaic in his work. (But there is an absolutely wonderful mosaic by Penrose, “Penrose’s Chickens”, they were not painted by Escher.)
Lobachevsky plane.
Fifth in the list of axioms in Euclid's Elements in Heiberg's reconstruction is the following statement: if a straight line intersecting two straight lines forms internal one-sided angles less than two right angles, then, extended indefinitely, these two straight lines will meet on the side where the angles are less than two right angles . IN modern literature prefer an equivalent and more elegant formulation: through a point that does not lie on a line, there passes a line parallel to the given one, and, moreover, only one. But even in this formulation, the axiom, unlike the rest of Euclid’s postulates, looks cumbersome and confusing - which is why for two thousand years scientists have been trying to derive this statement from the other axioms. That is, in fact, turning a postulate into a theorem.
In the 19th century, mathematician Nikolai Lobachevsky tried to do this by contradiction: he assumed that the postulate was incorrect and tried to discover a contradiction. But it was not found - and as a result, Lobachevsky built a new geometry. In it, through a point that does not lie on a line, there passes an infinite number of different lines that do not intersect with the given one. Lobachevsky was not the first to discover this new geometry. But he was the first who decided to declare it publicly - for which, of course, he was laughed at.
The posthumous recognition of Lobachevsky's work took place, among other things, thanks to the appearance of models of his geometry - systems of objects on the ordinary Euclidean plane that satisfied all of Euclid's axioms, with the exception of the fifth postulate. One of these models was proposed by mathematician and physicist Henri Poincaré in 1882 - for the needs of functional and complex analysis.
Let there be a circle, the boundary of which we call the absolute. The “points” in our model will be the inner points of the circle. The role of “straight lines” is played by circles or straight lines perpendicular to the absolute (more precisely, their arcs falling inside the circle). The fact that the fifth postulate does not hold for such “direct” lines is almost obvious. The fact that the remaining postulates are fulfilled for these objects is a little less obvious, however, this is so.
It turns out that in the Poincaré model you can determine the distance between points. To calculate the length, the concept of a Riemannian metric is required. Its properties are as follows: the closer a pair of “straight line” points is to the absolute, the greater the distance between them. Angles are also defined between the “straight lines” - these are the angles between the tangents at the point of intersection of the “straight lines”.
Now let's return to tilings. What will they look like if the Poincaré model is divided into identical regular polygons (that is, polygons with all equal sides and angles)? For example, polygons should become smaller the closer they are to the absolute. This idea was realized by Escher in the series of works “The Limit Circle”. However, the Dutchman did not use regular partitions, but their more symmetrical versions. The case where beauty turned out to be more important than mathematical accuracy.
Woodcut "Limit - Circle II".
Woodcut "Limit - Circle III".
Woodcut "Heaven and Hell".
Impossible figures.
Impossible figures are usually called special optical illusions - they seem to be an image of some three-dimensional object on a plane. But upon closer examination, geometric contradictions are revealed in their structure. Impossible figures are of interest not only to mathematicians; psychologists and design specialists also study them.
The great-grandfather of impossible figures is the so-called Necker cube, a familiar image of a cube on a plane. It was proposed by the Swedish crystallographer Louis Necker in 1832. The thing about this image is that it can be interpreted in different ways. For example, the corner indicated in this figure by a red circle can be either the closest to us of all the corners of the cube, or, conversely, the farthest.
The first real ones impossible figures as such were created by another Swedish scientist, Oskar Rutersvärd, in the 1930s. In particular, he came up with the idea of assembling a triangle from cubes, which cannot exist in nature. Independently of Ruthersward, the already mentioned Roger Penrose, together with his father Lionel Penrose, published a paper in the British Journal of Psychology entitled “ Impossible objects: Special type optical illusions"(1956). In it, the Penroses proposed two such objects - the Penrose triangle (a solid version of Ruthersward's design of cubes) and the Penrose staircase. They named Maurits Escher as the inspiration for their work.
Both objects - the triangle and the staircase - later appeared in Escher's paintings.
Lithograph "Relativity".
Lithograph "Waterfall".
Lithograph "Belvedere".
Lithograph "Ascent and Descent".
Other works with a mathematical meaning:
Star polygons: 
Woodcut "Stars".
Lithograph "Cubic division of space".
Lithograph "Surface covered with ripples".
Lithograph "Three Worlds"
This work by Escher depicts a paradox - the falling water of a waterfall drives a wheel that directs the water to the top of the waterfall. The waterfall has the structure of an "impossible" Penrose triangle: the lithograph was created based on an article in the British Journal of Psychology.
The structure is made up of three crossbars stacked on top of each other at right angles. The waterfall in the lithograph works like a perpetual motion machine. Depending on the movement of the eye, it alternately appears that both towers are identical and that the tower on the right is one floor lower than the left tower.
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Notes
Links
- Official website: (English)
Excerpt characterizing the Waterfall (lithograph)
- There is none; orders for battle have been made.Prince Andrei headed towards the door from behind which voices were heard. But just as he wanted to open the door, the voices in the room fell silent, the door opened of its own accord, and Kutuzov, with his aquiline nose on his plump face, appeared on the threshold.
Prince Andrei stood directly opposite Kutuzov; but from the expression of the commander-in-chief’s only seeing eye it was clear that thought and concern occupied him so much that it seemed to obscure his vision. He looked directly at the face of his adjutant and did not recognize him.
- Well, have you finished? – he turned to Kozlovsky.
- Right this second, Your Excellency.
Bagration, short, with an oriental type of hard and motionless face, dry, not yet old man, went out to get the commander-in-chief.
“I have the honor to appear,” Prince Andrei repeated quite loudly, handing over the envelope.
- Oh, from Vienna? Fine. After, after!
Kutuzov went out with Bagration onto the porch.
“Well, prince, goodbye,” he said to Bagration. - Christ is with you. I bless you for this great feat.
Kutuzov's face suddenly softened, and tears appeared in his eyes. He pulled Bagration to him with his left hand, and with his right hand, on which there was a ring, apparently crossed him with a familiar gesture and offered him a plump cheek, instead of which Bagration kissed him on the neck.
The “Endless Staircase” was successfully used by the artist Maurits K. Escher, this time in his enchanting lithograph “Ascent and Descend”, created in 1960.
In this drawing, reflecting all the possibilities of the Penrose figure, the very recognizable "Endless Staircase" is neatly inscribed in the roof of the monastery. Hooded monks continuously move up the stairs in a clockwise and counterclockwise direction. They go towards each other along an impossible path. They never manage to go up or down.
This work by Escher depicts a paradox - the falling water of a waterfall drives a wheel that directs the water to the top of the waterfall. The waterfall has the structure of an “impossible” Penrose triangle: the lithograph was created based on an article in the British Journal of Psychology.
The structure is made up of three crossbars stacked on top of each other at right angles. The waterfall in the lithograph works like a perpetual motion machine. It also seems that both towers are the same; in fact, the one on the right is one floor below the left tower.
"Belvedere" (Italian: Belvedere). In the left foreground there is a sheet of paper with a drawing of a cube. The intersections of the edges are marked with two circles. The young man sitting on the bench holds in his hands just such an absurd semblance of a cube. He thoughtfully examines this incomprehensible object, remaining indifferent to the fact that the gazebo behind him is built in the same incredible, absurd style.
Maurits Escher is an outstanding Dutch graphic artist known all over the world for his works. In the center, in the museum, opened in 2002, and named after him "Escher in het Paleis", a permanent exhibition of 130 works by the master is open. Would you say that graphics are boring? Perhaps... perhaps this can be said about the works of graphic artists, but not about Escher. The artist is known for his unusual vision of the world and playing with the logic of space.
Escher's fantastic engravings, literally, can be perceived as graphic image theory of relativity. The works that depict impossible figures and transformations are literally mesmerizing; they are unlike anything else.
Maurits Escher was a true master of puzzles and his optical illusions show things that don't actually exist. In his paintings everything changes, smoothly flows from one form to another, staircases have no beginning or end, and water flows upward. Someone will exclaim - this cannot be! See for yourself.
The famous painting “Day and Night”
“Ascent and descent”, where people are always walking up the stairs... or down?
“Reptiles” - here alligators turn from drawn ones into three-dimensional ones...
“Drawing hands” - in which two hands draw each other.
"Meeting"
“Hand with reflective ball”
The main pearl of the museum is Escher’s 7-meter-high work “Metamorphoses”. This engraving allows you to experience the connection between eternity and infinity, where time and space are united into a single whole.
The museum is located in the former Winter Palace Queen Emma - great-grandmother of the current reigning Queen Beatrix. Emma bought the palace in 1896 and lived in it until her death in May 1934. In two halls of the museum, which are called the “Royal Rooms,” furniture and photographs of Queen Emma have been preserved, and on the curtains there is information about interior palace of those times.
On the top floor of the museum there is an interactive exhibition “Look Like Escher”. This is real magical world illusions. In the magic ball, worlds appear and disappear, walls move and change, and children appear taller than their parents. A little further there is an unusual floor that optically collapses under every step, and in the silver ball you can see yourself through Escher’s eyes.
Maurits Cornelis Escher, Dutch graphic artist
Escher Maurits Cornelis(Maurits Cornelis Escher) (June 17, 1898, Leeuwarden, the Netherlands - March 27, 1972, Hilversum, the Netherlands) Dutch graphic artist, did illustrations for books, postage stamps and frescoes, designed tapestries. Known primarily for his conceptual lithographs, wood and metal engravings, in which he masterfully explored the plastic aspects of the concepts of infinity and symmetry, as well as the peculiarities of the psychological perception of complex three-dimensional objects, he is the most prominent representative of imp art. Escher quite deliberately chose a career as an engraver rather than as an oil painter. According to Hans Locher, a researcher of his work, Escher was attracted by the opportunity to obtain many prints, which was provided by graphic techniques, since he was already in early age I was interested in the possibility of repeating images. One of the most outstanding aspects of Escher's work is his depiction of "metamorphosis", appearing in various forms in a variety of works. The artist explores in detail the gradual transition from one geometric figure to another, through slight changes in outline. In addition, Escher repeatedly painted metamorphoses occurring with living beings (birds turn into fish, etc.) and even “animated” them during metamorphoses inanimate objects, turning them into living beings. Escher produced 448 lithographs, engravings and woodcuts and over 2,000 drawings and sketches. His work continues to impress and surprise millions of people around the world. IN recent years Escher's health fails him and he practically does not work. He undergoes many operations and eventually dies in hospital from bowel cancer. Escher left behind his wonderful lithographs, paintings, drawings and three sons.
Key dates
- 1898 - Moritz Cornelis Escher was born on June 17 in Liverden (Netherland), youngest son in the family of hydraulic engineer G.A. Escher and Sarah Glichman.
- 1903 - The family moves to Arnhem.
- 1912-18 - Enters the gymnasium and fails the final exams.
- 1919 - At the request of his father, Escher begins to study architecture in Haarlem, but after a few months he transfers to the classroom graphic design under the leadership of Djeseran de Mesquite.
- 1921 - First trip to Italy. First publication in the magazine of the work “Easter Flowers” (woodcut)
- 1922 - Finishes art school and goes to travel around central Italy; makes a lot of sketches. In September he visits the Alhambra in Spain, considering it the most interesting, especially its huge mosaics of “colossal complexity and mathematical and artistic meaning.”
- 1923 - Travel to Italy; meets his future wife Jetta (Jetta Umiker). He draws from life. His first exhibition is in Siena.
- 1924 - First exhibition in The Hague, Netherlands. On June 12 he is married by Yetta in Viareggio; moves to Rome.
- 1926 - Very successful exhibition in Rome in May. Later, Escher had a permanent exhibition in Holland and received mostly positive reviews. On June 23, their first son, Georg, will be born into the Escher family. In subsequent years, Moritz Escher constantly travels (for example, to Tunisia), including on foot to Arbuzi; makes a lot of landscape and architectural sketches.
- 1928 - December 8, son Arthur is born.
- 1929 - First lithograph “View of Goriano Sicoli”, Arbuzzi
- 1931 - The first wooden engraving, but essentially it was a wooden matrix for printing invitations to an exhibition in The Hague. Escher becomes a member of the Association of Graphic Artists, and a little later - a member of the Pulchi studio. He is highly respected as a "patient, calm, cool draftsman" and his work is criticized for being "too intellectual".
- 1932 - His woodcuts are published in the almanac “XXIV Emblemata dat zijns zinnebeelden”.
- 1933 - The book “The Terrible Adventures of Scholasticism” with wood engravings by Escher is published.
- 1934 - His works at the exhibition of modern engravings (printing) “Century of Progress” in Chicago receive only positive reviews.
- 1935 - The repressive policies of Fascist Italy force Escher to move to Switzerland.
- 1936 - Trip to Spain, where he again actively worked on Moorish tile patterns (Alhambra). Redrawing them inspires Escher to create paintings in which he uses the correct periodic division of planes.
- 1938 - Another son, Jan, was born on March 6. But Escher concentrates on “internal paintings” and almost completely abandons drawing from nature.
- 1939 - Father's death at the age of 96.
- 1940 - M.C.Escher en zijn experimenten is published. His mother dies.
- 1941 - The Escher family returns to their homeland in Holland, in Baarn (B╠rn)
- 1948 Escher begins giving lectures on his work along with demonstrations of it.
- 1954 - Great Escher Exhibition on the occasion of the Great Mathematical Congress. Following it is an exhibition in Washington.
- 1955 - April 30 receives a large royal award.
- 1958 - "Regelmatige vlakverdeling" (Correct division of planes) is published.
- 1959 - “Grafik en Tekeningen” (Graphic Works) is published
- 1960 - Exhibition and lecture at the Crystallographic Congress in Cambridge, Massachusetts
- 1962 - Emergency surgery and long stay in the hospital.
- 1964 - Leaves for Canada for another operation.
- 1965 - Hilversum Art Prize. "Symmetry Aspect" is published.
- 1967 - Second Queen's Award.
- 1968 - Huge 70th anniversary retrospective in The Hague. At the end of the year, Yetta returns to Switzerland.
- 1969 - In July, Escher creates his last woodcut, "Snakes".
- 1970 - Surgery and again long hospitalization. Escher moves to the Rosa-Spier-Foundation Laaren in a home for elderly artists.
- 1971 - De werelden van M.C.Escher (Escher's World) is published.
- 1972 - M. S. Escher dies at the Lutheran Hospital in Hilversum.
