The Impossible World of M. C. Escher

  • :: 3 Works Cited
  • Length: 2079 words (5.9 double-spaced pages)
  • Rating: Excellent
Open Document

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Text Preview

More ↓

Continue reading...

Open Document

The Impossible World of M. C. Escher


Something about the human mind seeks the impossible. Humans want what they don’t have, and even more what they can’t get. The line between difficult and impossible is often a gray line, which humans test often. However, some constructions fall in a category that is clearly beyond the bounds of physics and geometry. Thus these are some of the most intriguing to the human imagination. This paper will explore that curiosity by looking into the life of Maurits Cornelis Escher, his impossible perspectives and impossible geometries, and then into the mathematics behind creating these objects.

The works of Escher demonstrate this fascination. He creates worlds that are alien to our own that, despite their impossibility, contain a certain life to them. Each part of the portrait demands close attention.

M. C. Escher was a Dutch graphic artist. He lived from 1902 until 1972. He produced prints in Italy in the 1920’s, but had earned very little. After leaving Italy in 1935 (due to increasing Fascism), he started work in Switzerland. After viewing Moorish art in Spain, he began his symmetry works. Although his work went mostly unappreciated for many years, he started gaining popularity started in about 1951. Several years later, He was producing millions of prints and sending them to many countries across the world. By number of prints, he was more popular than any other artist during their life times. However, especially later in life, he still was unhappy with all he had done with his life and his art—he was trying to live up to the example of his father, but he didn’t see himself as succeeding (Vermeleun, from Escher 139-145).

While his works of symmetry are ingenious, this paper investigates mostly those that depict the impossible. M. C. Escher created two types of impossible artwork— impossible geometries and impossible perspectives. Impossible geometries are all possible at any given point, and also have only one meaning at any given point, but are impossible on a higher level. Roger Penrose (the British mathematician) described the second type—impossible perspectives—as being “rather than locally unambiguous, but globally impossible, they are everywhere locally ambiguous, yet globally impossible” (Quoted from Coxeter, 154).

Need Writing Help?

Get feedback on grammar, clarity, concision and logic instantly.

Check your paper »

How to Cite this Page

MLA Citation:
"The Impossible World of M. C. Escher." 123HelpMe.com. 23 Apr 2018
    <http://www.123HelpMe.com/view.asp?id=28639>.
Title Length Color Rating  
M.C. Escher Essay - M.C. Escher M.C. Escher was a Dutch graphic artist, most recognized for spatial illusions, impossible buildings, repeating geometric patterns (tessellations), and his incredible techniques in woodcutting and lithography. · M.C. Escher was born June 1898 and died March 1972. His work continues to fascinate both young and old across a broad spectrum of interests. · M.C. Escher was a man studied and greatly appreciated by respected mathematicians, scientists and crystallographers yet he had no formal training in math or science....   [tags: Visual Arts Paintings Art] 918 words
(2.6 pages)
Strong Essays [preview]
Essay on The World Working Together - is it Impossible? - The World Working Together – is it Impossible. The world is today at a very peculiar balance, a balance at which it has never been before. Statistics reveal that the richest fifth of the population use up over 80% of the world’s resources. This makes an important question and a very central topic for debates worldwide. In contemporary times, as the richest countries have so much wealth, it is general consensus that some should be given to poorer countries, as a form of assistance, to help them build a well-functioning society and to improve the quality of life in poorer areas....   [tags: wealth, conflict, corruption] 1011 words
(2.9 pages)
Strong Essays [preview]
Did World War II Make World War III Impossible? Essay - “I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.”-Albert Einstein Fifty-two million souls perished in the storm of World War II. The actions engaged after World War I from the Treaty of Versailles became the ultimate cause of World War II. After World War II, the United States procured countless undertakings to insure that no greater cataclysmic event would propel the people of the world into the grasp of a one-world government. Prior to World War II, no one had the power to destroy mankind....   [tags: world war, versailles treaty, atomic bomb] 1016 words
(2.9 pages)
Strong Essays [preview]
M.C. Escher Essay - M.C. Escher occupies a unique spot among the most popular artists of the past century. While his contemporaries focused on breaking from traditional art and its emphasis on realism and beauty, Escher found his muse in symmetry and infinity. His attachment to geometric forms made him one of modernism’s most recognizable artists and his work remains as relevant as ever. Escher’s early works are an odd mix of cubism and traditional woodcut. From these beginnings, one could already note Escher’s fondness for repetition and clean shapes....   [tags: Art] 1121 words
(3.2 pages)
Strong Essays [preview]
The Genius of M.C. Escher Essay - The Genius of M.C. Escher Mathematics is the central ingredient in many artworks. While notions of infinity and parallel lines brought “perspective” to the artistic realm in creating realistic representations of depth and dimension, mathematics has influenced art in a more definite way – by actually becoming art. The introduction of fractal geometry and tessellations as creative works spawned the creation of new and innovative genres of art, which can be exemplified through the works of M.C Escher....   [tags: Biography]
:: 4 Works Cited
1242 words
(3.5 pages)
Better Essays [preview]
M.C. Escher and Salvador Dahlia Essay - M.C. Escher and Salvador Dahlia Maurits Cornelis Escher and was born on June the 17th, 1898 in Leeuwarden Netherlands. Escher was not encouraged to be an artist at a young age. He was encouraged to learn carpentry and other craft skills by his father. At school, he was an average student generally, but showed obvious artistic talent early in his schooling. Escher's was fascinated by the art of structure and this is shown in a lot of his work. His early work however, tended towards realistic portrayals of the landscape and architecture observed during his travels....   [tags: Papers] 363 words
(1 pages)
Strong Essays [preview]
Essay on Salvador Dali and M.C. Escher - Salvador Dali and M.C. Escher The artists that I am comparing in my paper come from two different backgrounds, yet in some ways, the deep psychological and philosophical message that their works reflect, together with their shared fascination with the insect-world, bring them together. Salvador Dali, a poor farmer’s son (1904-1989) was born in Spain, and throughout his childhood, according to him, he was treated like royalty by his parents because they thought he was the incarnation of his dead brother, who died nine months before he was born....   [tags: Art] 570 words
(1.6 pages)
Strong Essays [preview]
Art And Mathematics:Escher And Tessellations Essay - Art And Mathematics:Escher And Tessellations On first thought, mathematics and art seem to be totally opposite fields of study with absolutely no connections. However, after careful consideration, the great degree of relation between these two subjects is amazing. Mathematics is the central ingredient in many artworks. Through the exploration of many artists and their works, common mathematical themes can be discovered. For instance, the art of tessellations, or tilings, relies on geometry....   [tags: Math Artistic Papers]
:: 8 Works Cited
2039 words
(5.8 pages)
Powerful Essays [preview]
Impossible War Essay - But, in a larger sense, we cannot dedicate -- we can not consecrate -- we can not hallow -- this ground. The brave men, living and dead, who struggled here, have consecrated it, far above our poor power to add or detract. The world will little note, nor long remember what we say here, but it can never forget what they did here. It is for us the living, rather, to be dedicated here to the unfinished work which they who fought here have thus far so nobly advanced. It is rather for us to be here dedicated to the great task remaining before us -- that from these honored dead we take increased devotion to that cause for which they gave the last full measure of devotion -- that w...   [tags: U.S. History ]
:: 15 Works Cited
1698 words
(4.9 pages)
Powerful Essays [preview]
Essay on Escher - Escher For my art piece I chose M.C. Escher’s “Eight Heads” from 1922. It depicts eight different heads that all form from each other. One of Escher’s many styles was to make images that form other images inside themselves. “Eight Heads” show 2 faces that could be considered evil or the devil. It has four different women in the piece and the pattern of position of the heads is more prevalent here than with any other head. The last two figures are the heads of two men wearing hats of the style worn at the time....   [tags: essays research papers] 517 words
(1.5 pages)
Strong Essays [preview]

Related Searches






Several examples of impossible perspectives are Convex and Concave, High and Low, Relativity, and House of Stairs.

My favorite of all pictures is Convex and Concave.


Convex and Concave

M. C. Escher suggests that the drawing be split into three vertical strips (Escher 68). The left and right side are both possible, but they are opposites. The middle part could match the left side, or it could match the right side, but it cannot match both at once. For example, the left lizard hangs from the inside corner of the ceiling, while the other lizard is on the outside corner of steps that a human could walk on. However, the two interpretations can’t both be true. The picture is ingenious and very detailed as well.

High and Low shows two towers, both identical but viewed from different angles.


High and Low

They are placed in such a way that they point in opposite directions. Curiously, the top of this picture joins to the bottom in such a way that there could be a 360 degree view of the setup if the work were curled into a ring. However, the towers wouldn’t be impossible if the figures were plain geometrical figures. However, the two boys sitting the steps contradict one another—if one could sit normally, the other would fall. We cannot tell which way is really up and which way is really down. Is the central tiled pattern the ceiling or the floor?

Relativity is based on the same gravitational impossibility.


Relativity

The many directions considered to be up are mutually contradictory—this house could be built, but it would be impossible to have guests in all these positions. The ambiguity lies in the assignment of up and down.

House of Stairs is based on a similar problem, but the actors here are alien.


House of Stairs

If the central bug doesn’t fall, then the ones just below it ought to. The whole picture looks somewhat skewed, but it is all possible. It is interesting to note the very highest and lowest creature are the same—this image could repeat vertically indefinitely, although it would lose its geometric possibility after three repetitions, due to the contained angles.

The second type of figures is geometric impossibilities. There is no mistaking what the mean on any level, but they are clearly impossible. Escher has left us with three major examples of this type of impossibility: Waterfall, Belvedere, and Ascending and Descending.

Perhaps the most famous picture by Escher is Waterfall. Escher, in a speech he wrote but never presented, described the lucky position of the miller: “The miller can keep it perpetually moving by adding every now and then a bucket of water to check evaporation” (Escher 79).


Waterfall and a tribar

The two towers are the same height, but one is a story taller. The water flows level until it drops and then flows level again, never rising. This basis for this idea is not Escher’s—he found this (along with the staircase) in an article by Roger Penrose and his son, L.S. Penrose, in 1958 (Ernst, quoted in Coxeter 125). The tribar, or Roger Penrose’s triangle, is formed of three rectangular beams. This is demonstrated several times in Waterfall. One example is the two troughs of water that are highest on the page and the four supports connecting them. Any two beams form a sensible right angle, but following the figure around reveals that the figure is illegitimate—it can’t exist in three-dimensional space. However, as critic Bruno Ernst points out, the figure doesn’t flatten into meaningless lines, but rather maintains a three-dimensional shape, declaring it to be an object, but then proves itself not to be an object, by construction (Coxeter 127).

Beldevere is a more original work (Coxeter 125). It shows a construction that has an upper level perpendicular to its lower level.


Belvedere and a Necker cube

The columns are twisted between the levels, and a ladder is both inside and outside at the same time. A boy at the bottom examines an object like the pictured Necker cube. When the edges of a cube are drawn, it is very simple to flip some edges around and get this sort of image. This idea is the basis for both the cube and the building. Escher adds people in to further invite the viewer to explore the construction with their eyes. The addition of humans softens the harsh geometry of the work and asks “What if?”

Ascending and Descending pictures individuals climbing up and down a set of stairs with no end. Of course it is impossible to climb up and up and end up at the beginning, but the flat canvas and Escher work out this feat nonetheless.


Ascending and Descending

The endlessly rising staircase intrigues me greatly. It is clearly impossible in three-dimensional space. Yet it has been drawn is two dimensions. This is a result of the large loss of information that accompanies translation into two dimensions. When we look at an object, we can perceive it from many angles and we can get a good idea of how it is put together. However, is we get only one view; we can’t necessarily construct what the entire object means. As humans see mostly surfaces, and as we see only one view at a time, so two dimensional pictures seem like a good way to depict three-dimensional worlds.

However, I wasn’t satisfied with this answer, and I decided to make an impossible staircase of my own to better understand the creation. I’ll outline the steps I took to demonstrate what goes into the creation of such objects.

The staircase needs to meet itself at a convincing angle. Thus I must determine the period of the staircase. I find that I must get the “highest” step to line up to the bottom of the “lowest” step. To do this, I pick an arbitrary point on each step, calculate the distance between these points, and find values such that the first and last points correspond. Of course, as I’m working in two dimensions, I need two equations, on for the x direction, and one for the y direction.

To set up the equations, I find that there are two important factors—the size of the steps in each direction (I’m making them all the same size for simplicity, although Escher used angled steps at the corners and his steps rotated as the went along) and the number of steps on every side—notice that the opposite sides do not contain the same number of steps. Given these quantities, we can make two equations. I have Chosen nine variables in order to properly describe the staircase. These are described in the figures below.


Staircase (Drawing by author)




Box (Drawing by author)

Here are my equations:

x equation:

c * w – x * d – c * y + z * d = 0

y equation:

w * (h + a) + x * (h + b) + y * (h - a) + z * (h – b) = 0

Notice that each of the w steps increases x by c—thus we have c * w. Similarly, each of the w steps increases y by h + a. The rest of the equations follow from similar logic. Keep in mind that the changes in x and y from all four sides must be 0 so that the picture lines up.

Finding values that solve these equations is no trick (the easiest solution is that they all equal zero) but it is harder to find reasonable values. The number of stairs must be integers—if one step was half as big as the rest, it would look pretty silly. The dimension should be integers for easy graphing. The numbers must also be positive. The values for the step dimensions must make a convincing box, and the values for the numbers of steps in each flight must be somewhat close to each other to create a convincing illusion. Finally, they should all be somewhat small for ease of drawing.

For my box values, I choose h = 1 because h is pushing the staircase up on the page, while I want the stair case to meet itself. I considered making c = d and a = b, but I came out with an exact copy of a staircase made by L. S. Penrose himself. I choose, after some consideration, to set my values at a = 4, b = 1, d = 4, c =2. This made my equation:

x equation:

2 * w – x * 4 – y * 2 + z * 4 = 0

y equation:

w * 5 + x * 2+ y * -3 = 0

After juggling some numbers, I obtained acceptable values: w = 3, x = 3, y = 7, z = 5. Now it was merely a matter of drawing the figure out on graph paper.

By showing calculations, I hope that I’ve shown how much work goes into this sort of nonsense. This is because it is grounded into reality. The departure from reality occurs on a very high level. This is why Escher pieces seem so alive—they are carefully constructed to follow the rules of our reality carefully enough to fool us into wondering if what is portrayed is really impossible at all.

Impossible figures are visual paradoxes. Is the staircase’s lowest step the one closest to us? No, there is a lower step. This can’t be the lowest either. This continues around until we conclude that no lowest step exists. This is isomorphic to Epimenides Paradox. Is it true or false? If it is, the sentence we see that the sentences declaration of falsehood is true. But this makes it false, but then the declaration is false. The process can continue indefinitely, so we see that this question has no answer either.

However, the mystery of the impossible is what makes it so intriguing. It forces us to take a step partway out of our limited system and to envision other systems. Escher’s gift was the ability to make people think from new perspectives.

Bibliography

Escher, M. C. Escher on Escher. With a contribution by J. W. Vermeulen. : Harry N. Abrams, 1986.

Hofstadter, Douglas. Gödel Escher Bach: an Eternal Golden Braid. Basic books. 1979.

International Congress on M.C. Escher. M.C. Escher, art and science: proceedings of the International Congress on M.C. Escher, Rome, Italy, 26-28 March, 1985. Edited by H.S.M. Coxeter ... [et al.]. 2nd ed. Elsevier Science Pub. Co. Amsterdam: 1987, c1986.


Return to 123HelpMe.com