# Can we put the mathematics of paradoxes in visual art into perspective?

(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

(1)

Two Point Perspective

(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

(3)

-- M.C. Escher 1953 Relativity

(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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i.imgur.com/LrUt2.jpg – J. M. Nov 30 '10 at 14:48
@ J.M. : what do you want to say with that link? – Max Muller Nov 30 '10 at 14:58
Ascending and Descending depicts what are known as Penrose Stairs. – WWright Nov 30 '10 at 15:07
You've got it back asswards... Escher used mathematical ideas for his art, i.e. the mathematics already existed most of the time. I actually can't think of one Escher artwork where he actually predicted some math idea. – Raskolnikov Nov 30 '10 at 15:11
@Raskolnikov: I find your tone very impolite. I also find your comment somewhat irrelevant to the question being asked. – Qiaochu Yuan Nov 30 '10 at 15:28
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## 1 Answer

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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 @ Qioachu: you're right about Relativity, see andrewlipson.com/escher/relativity.html , only the physics is messed up. Thank you for the answer. Can you think of visual paradoxes that have a richer mathematical structure? – Max Muller Nov 30 '10 at 15:21 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: en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane – Qiaochu Yuan Nov 30 '10 at 15:29