Geometry, Topology and Art

Is Antarctica so long compared to Russia?�NO, then why does the map indicate so?

Why do many ancient paintings seem 2 dimensional? �This is certainly not how the world appears to our eyes!

How did humans manage to make the art of painting more realistic eventually?

• Weather it is a map or a painting, it is a model to represent a particular fragment of reality as we perceive it.
• Maps act as guides in travelling, so they must preserve the ratio of distances. In our world map, ratio of distance between horizontal ends of antarctica to that of Russia wasn’t proportional to the real world. This can be misleading to a 17 th century ship using this map to travel!
• Paintings often capture a moment in time. To describe that moment more accurately, it should have a sense of depth. This lacked in the Egyptian painting, hence it was less realistic.
• We will try to define these problems mathematically and try to understand concept of geometry and topology by using them.

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Defining the problems mathematically

Turns out the problems that we faced with both the maps and paintings can be made mathematically precise.

Key mathematical concepts used to study these problems-

• Topological Spaces and homeomorphism
• Isometry
• Projective Geometry

Topological Spaces

• Sphere, cube, polygons,circle,Line, point are examples of topological spaces.
• Sin(x), cos(x), polynomials are well know functions on real numbers. These are continuous functions. There are discontinuous functions also, like in the following diagram

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Homeomorphism(Similarity for us)

• Take topological space X and Y.
• If there is a continuous map F from X to Y, so that two different points in X go to different points in Y and for every point y in Y, we have a point x in X that goes to y ie F(x)=y, then we say X and Y are homeomorphic.
• e.g. Take continuous function x --> x/(1+x) from real numbers to interval (-1,1) satisfies above mentioned properties.

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• So for a topologist, interval (-1,1) and line are same things.
• Ever heard the joke that for a topologist, coffee mug and donut are same things!
• Now, if we want a map to be good analogue of real world we want it to be similar to real world.
• Our first priority would be – the map and the original object should be homeomorphic! 
• So interval (-1,1) is a good map of real line . Fun to think- if our world was one dimensional, our map could have been the interval
• Sin and cos take values in (-1,1), but they can't help us get a map of 1-D world. Think why! (hint-  sin(180)=sin(0))

Think about the joke! Donut and coffee mug are same to a topologist.

    Circle and Sqaure are same to us:

• In the following diagram, take center of the cercle, now define a map from circle to sqaure as follows – for point x on the circle, consider line outward from origin and passing through point x. 
• It will intersect the sqaure in exactly one point ( say y)
• Send all such x

   To corresponding y.

Isometry-

• We already discussed that if we want our map/painting to be perfect, it should be similar(homeomorphic) to the real life object it represents.
• We say a function between topological spaces X and Y is isometry if it preserves distance. E.g.- just take the identity map on real numbers- x goes to x for every real number. This map is isometry. If such a map exists, we say X and Y are isometric
• Since a map will guide us through our voyage, we want it to tell the distances correctly
• Now in real life, our map can't be isometric to real world. Otherwise our map will be as big as the real world
• So we use a weaker concept- weak isometry. It mean that even though distances are not preserved, ratio of distances is preserved.
• e.g. Turns out the homeomorphism real line to (-1,1) is indeed a weak isometry. So it's kind of a perfect map!

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Then why didn’t we draw a better map?

Then why didn’t we draw a better map?

Our map clearly wasn’t weakly isotopic to real world, since its ratio of distances wasn’t same to the real world. (This can be noticed from the antarctica)

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Turns out that we can't

• We will state following  theorems in topology and geometry that kind out give us idea that why we can't have perfect maps of our world.
• Sphere is compact manifold and Real plane a sheet is not, hence they cannot be homeomorphic.  (note that compact and manifold are technical terms, but one just needs to know that if spaces X and Y are similar (homeomorphic) and X is compact then so is Y.)
• There is no distance preserving continuous function from sphere to plane

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• Now from the first theorem we can never draw a map similar to sphere on a plane sheet.
• From second theorem, even if we allow little discontinuity (like in our graph, north pole and south pole are not single points instead they are lines, so not similar), we can't get isometry.
• The image shows how we made our map. Take vertical axis, and draw lines radially outward. Every point on the sphere except the poles go to unique point on cylinder.
•  One can see that the north and south poles go to top and bottom circles.
• In the end, cut one vertical line of cylinder to get the map.

Cylinderical Projection

What about paintings?�

• We have a similar problem for paintings.

First lets define a euclidian plane. In one dimension we have line, plane in 2D, our  world in 3D. Every point on line is represented by a real number, every point on plane is given by 2 real numbers. So we define n-dimensional euclidian plane to consist of points of form

(a1,a2,...an) where all a1,a2,…,a_n are real numbers

• In mathematical language we have following theorem-

    Any n – dimensional euclidian plane is homeomorphic to m- dimensional euclidian plane then n=m. 

In other world different dimensional worlds are not same!

• So 2 d plane is not same as 3d world, so we can never make a precise painting of an object in the real world on a 2d paper.
• Which is justified because for eg if one is painting a person , they don’t expect to paint design on the back of his/her dress!

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But turns out in paintings, we can give a sense of depth and preserve distances

In other words, we can find nice relations between 2D  and 3D euclidian planes. (Such a relation did not exist between plane and sphere so there was less hope for maps)

 This is where projective geometry comes into the picture!

• If one wants to paint a moment in the real word, for painting to be realistic, it should be similar to what human eyes detect.
• In Durer's machine, after pivoting the rope, join the other end of the rope the small area of the object to be painted. The rope will intersect the vertical paper in some small portion. Draw the small object in the corresponding part of the paper.
• The Durer's machinery explains how it can be done. Consider the point at which the rope is pivoted to the wall. If human eye observes the room from that point, the image detected by the eye will be exactly same as the one that appears on the screen.

Another illustration of Durer's method

Grids to draw specific parts of the subject of interest 

Making things mathematically Precise

• The point of the position of human eye is called perspective.
• Theorem- Consider two planes X and Y in real world, choose a point P which is not in either of X to Y. Then projection map from X to Y with respect to perspective P is a weak isometry.
• As shown in the diagram every P on X goes to unique Q in Y and this map is weak isometry. 
• In particular, in Durer's picture, X is the plane where the object is kept and Y is the vertical paper  .

• Now as mentioned earlier, object behind subject is not visible and we do not expect to paint it.
• This phenomenon is also observed in projective geometry. If you think about it, in projective geometry, a line passing through the perspective is a point. (YAY, we are in a new mysterious world!) This happens since any colinear points go to same point on the plane on which they are drawn. (In fact projective plane is a weird topological space)
• e.g. in following figure, P and Q go to 

    Same point A.

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Durer's Method indeed gives sense of depth and makes paintings more Realistic

• Suppose we have to paint a symmetric object like a ball or a flower plot. We obviously want the drawing to be proportional and symmetric. To be more precise, we wish the midpoint of object of interest to go to midpoint in painting. In Durer's methods, it can be observed as follows-
• Fundamental theorem of Projective geometry-

      A,B,C be colinear points, and A',B',C' be colinear points.

     Then midpoints of segments AA', BB', CC' are colinear.

• Hence, we know Durer's method preserves midpoints.

References�

• Albrecht Dürer - Wikipedia
• Cundy, H. Martyn. "Geometry: Euclid and beyond." (2001): 368-370.
• n-23 (polsl.pl)370.
• Art of ancient Egypt - Wikipedia
• Pinterest

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