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(Pun definitely intended.)

Dear MSE-Community,

If I were to choose one artist that has made interesting works of art not only because of their beauty but also because of their connections to Mathematics, I would choose Escher. His works have always intrigued me. Some of his paintings are mind-boggling when looked at for a long time, but I have the feeling that it can be described accurately with mathematics.

Let's compare some drawings. In the first and second drawing ((1),(2)), the artist chooses to depict some simple geometrical objects in one- and two point perspective respectively. Although the objects seem to float in space, both the images look "all right" to me. These drawing are, however, though not 'Eschers', illusory too. The brain somehow creates 3D space from 2D space, but that's more of a biological issue.

One Point Perspective

alt text (1)

Two Point Perspective

alt text (2)

Sub-question 1: How does one describe these seemingly "sound" drawings mathematically? How do (1) and (2) compare to one another?

Now, lets get to to part I find most interesting, Escher's etchings, prints and lithographs. When I look at the following pictures:

-- M.C. Escher 1960 lithograph Ascending and Descending

alt text (3)

-- M.C. Escher 1953 Relativity

alt text (4)

I recognize that these are two different types of paradoxes because Escher plays with perspective in two different ways.

Sub-question 2: How could the difference between these and other visual paradoxes of artists (mostly Escher, but I guess there are a lot more artists that mimic and extend his style) be formalized with the aid of mathematics?

Thanks,

Max Muller

P.S. I'm sorry these images are all of different sizes and some are too large. I'm a bit in a hurry so I didn't make them equally large.

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    @ Raskolnikov: I should have only asked about the depicted visual paradoxes and$how$the paradoxical thing about them could be described mathematically. I understand that you could feel a bit uneasy about the way I ask question. In these questions I try to learn about some particular branch of mathematics as much as possible through the answers. Therefore, I also included "and other visual paradoxes" because I hoped someone could give me a general reference on the topic *and* answer the (by now) narrowed version of Q2: Is it possible to explain the the paradoxical element of (3) mathematically?2010-11-30

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Question 1: this is projective geometry. In this context the horizon line is known as the line at infinity, and the drawings illustrate that "parallel lines" in the projective plane meet at the line at infinity while non-parallel lines meet at a finite point and intersect the line at infinity in different points.

Question 2: I don't know that you have to do anything sophisticated to describe what is going on in Ascending and Descending. The drawing suggests the existence of four heights $h_1, h_2, h_3, h_4$ (the heights of the corners) such that $h_1 > h_2 > h_3 > h_4 > h_1$, and this is a contradiction.

And what is paradoxical about Relativity? As far as I can tell, that room is buildable. Maybe I don't understand what you are trying to ask here.

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    Not paradoxes as such, but much of Escher's work is concerned with symmetry and there are some very famous ones that involve the absolutely beautiful subject of hyperbolic tiling: http://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane2010-11-30