Maurits Cornelis Escher

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Maurits Cornelis Escher


«Mathematical Art Escher»

Dutch painter Moritz Kornilis Asher, born in 1898 in Leeuwarden has created unique and charming work, which used or displayed a wide range of mathematical ideas.


When he was in high school, my parents had planned that he would become an architect, but ill health prevented Maurice complete their education, he became an artist. Until the early 50's, he was not widely known, but after a series of exhibitions and articles in American magazines (Time, etc.) he receives worldwide recognition. Among his enthusiastic admirers were mathematicians, who saw in his work an original visual interpretation of some mathematical laws. It is more interesting in that he had no special Asher mathematics education.


In its work, he found the idea of ​​mathematical papers, which focuses on the mosaic tilings of the plane, the projection of three-dimensional shapes on a plane and Euclidean geometry, as will be described below. He was fascinated by all sorts of paradoxes, including the "impossible figures." Paradoxical idea of ​​Roger Penrose's been used in many works of Escher. The most interesting to study the ideas of Asher are all possible partitions of the plane and the logic of the three-dimensional space.





Mosaic Escher

Interested in mosaics Asher began in 1936 during a trip to Spain. He spent much time in the Alhambra, drawing Arab mosaics, and later said that it was for him "a rich source of inspiration." Later in 1957, in his essay on the mosaics Asher wrote: «In mathematical work regular division of the plane is considered theoretically ... Does this mean that the issue is a purely mathematical? Mathematicians have opened the door leading to another world, but they enter into this world did not dare. They are more interested in the way in which the door is, than a garden that lies behind it.»






Sketch from the Alhambra

Escher interested in all kinds of mosaics - regular and - and also introduced its own species, which he called "metamorphosis", where the figures change and interact with each other, and sometimes change, and the plane itself.


«Day and night»





Polyhedra Escher


Correct geometric solids - polyhedra - had a special fascination for Escher. In his many works polyhedra are the main figures and even more work they occur as secondary elements.


The engraving "Four bodies" Escher main intersection of regular polyhedra, located on one axis of symmetry, in addition polyhedra appear translucent, and through any of them can see the rest.


"Four bodies"

A number of different polyhedra can be obtained by combining the regular polyhedra, as well as the transformation of a polyhedron into a star. To convert into a star polyhedron must replace each of its faces a pyramid whose base is the face of. Elegant example of star dodecahedron can be found in the "Order and Chaos". In this case, the star-shaped polyhedron is placed inside a glass sphere. Austere beauty of this design contrasts with randomly scattered on the table debris. Note also that the analysis of the picture you can guess the nature of the source of light for the entire composition - is a window that reflects the upper left part of the sphere.


Order and Chaos

The figures obtained by the union of the regular polyhedra, can be found in many works of Escher. The most interesting of these is the engraving of the "Stars", where you can see the body, obtained by combining tetrahedra, cubes and octahedrons. If Escher in this paper are only different versions of polyhedra, we would never know about it. But for some reason, he placed inside the central figure of the chameleon to obstruct our perception of the whole figure. Thus, we need to escape from the usual perception of the picture and try to look at it with fresh eyes, to present it in full. This aspect of this painting is another admiration of mathematicians work of Escher.




Form of space Escher


Among the most important works of Escher from a mathematical point of view are paintings that operate with the nature of space itself. Lithograph "Three intersecting planes" - a good example for the start of the review of these paintings. This example demonstrates the artist's interest in the space dimension and the brain's ability to recognize three-dimensional images on two-dimensional drawings. As will be lower, Escher later used this principle to create amazing visual effects.


«Three intersecting planes»

Under the influence of images in the book Mathematics H. Coxeter Escher created many illustrations of hyperbolic space. One example can be seen in "the circle III». Here is one of two kinds of non-Euclidean space, described by the French mathematician Poincare. To understand the features of this space, imagine that you are inside the picture itself. As you move from the center of a circle to its edge will decrease your height as well as the decrease of fish in this picture. Thus the path that you have to go to the circle will seem endless. In fact, being in such prostarnstve you at first glance do not notice anything unusual in it, compared with the usual Euclidean space. For example, to reach the boundaries of the Euclidean space you also need to go through an endless path. However, if you look carefully, you will notice some differences, for example, all have similar triangles in this space the same size, and you will not be there to draw a figure with four right angles, connected by straight lines, as in this area there are no squares rectangles. Strange place, is not it?

“The circle III»

Even more strange space shown in "Snakes." Here the space goes to infinity in both directions - and toward the edge of the circle and towards the center of the circle, as shown decreasing rings. If you get into this space, what it will look like?



In addition to the features of the Euclidean and non-Euclidean geometries, Escher interested in the visual aspects of the topology. Topology studies the properties of solids and surfaces of the space, which do not change during the deformation, such as tension, compression or bending. The only thing that should not cause deformation - is to rupture. Topologists have to portray a lot of strange objects. One of the most famous is the Mobius strip, which is found in many works of Escher. This may seem strange, but this surface is only one side and one edge. If you trace the path of the ants on the lithography "Mobius loop II», you will see the ants crawling on opposite surfaces not tape, and on one and the same.


"Mobius loop II»

Another interesting lithograph called "Art Gallery", which changed both the topology and the logic of space. Asher turned the space into a ring, and it turned out that the boy is painting both inside and outside. Just as intriguing white spot in the center. Mathematicians call this spot a distinct or particular point where there is no space. There is no way to represent this part of the picture without seams or overlaps, so Asher has solved this problem by placing the center of the picture the autograph.

Escher understood that the geometry of the space defines the logic, but the logic determines the geometry of space. One of the most commonly used features of the logic of space - the play of light and shadow on the convex and concave objects. The lithograph "Cube with stripes" tabs on the tape is a visual reference of how the strips are arranged in space and how they are intertwined with the cube. And if you believe your eyes, you will never believe what is drawn in this picture.

Another aspect of the logic of space - perspective. In the figures, in which there is a perspective effect emit so-called vanishing points that tell the human eye of the infinity of space. Introducing additional vanishing points and slightly changing the elements of composition to achieve the desired effect, Escher was able to draw a picture in which the orientation of the elements varies depending on how the viewer looks at the painting. The painting " High and Low," the artist has posted just five points of extinction - in the corners and in the center of the picture. As a result, if we look at the lower part of the picture, it seems that we are looking up. If we look at the draw the top half of the picture, it seems that we are looking down. To emphasize the effect, Escher two types of the same composition.

The third type of painting with a broken logic of space - it is "impossible figures." The paradox of impossible figures based on the fact that our brain is always trying to present a paper drawn on two-dimensional images as three-dimensional. Escher created many works that addressed this anomaly. The most interesting work - lithograph "Waterfall" - is based on a figure of impossible triangle.





The central idea of self-renewal, a policy adopted by Escher, refers to the mystery of human consciousness and the ability of the human brain to process information so as not be able to handle any computer. Lithograph "Drawing Hands," and "Fish and flakes" use this idea in different ways. Self-reproduction is a course of action. Hands paint each other, creating themselves. In doing so, the hands and the process of self-reproduction are inseparable. In "Fish and flakes" concept of self-replication provides more functionality, and in this case it may be called self-similarity. In this sense, this paper describes not only the fish, and all living organisms, including humans. Of course, we are not a member of the small copies of themselves, but every cell in our body carries information about the body in the form of DNA.


Drawing Hands

«Fish and Scales»

We have examined only a small part of the work of hundreds of sketches, etchings and engravings, remaining after the death of Asher in 1972. There is still much to be said and has been said about the significance and importance of his work. Every year there are more and more books, which highlights the artist's work, analyzed the various aspects of his work. This is particularly impressive because Asher has not received higher education.

For me personally, creativity Escher particularly important, because I am very interested in the topic itself, the device space of the universe and of human perception. Abstract vision of the mathematical and physical laws of space, helps to understand the fundamental discoveries. Asher artist-scientist, a genius. He is a great example of how close can be such a seemingly distant things as science and art.

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