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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).
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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.
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:
c * w – x * d – c * y + z * d = 0
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:
2 * w – x * 4 – y * 2 + z * 4 = 0
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.
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.