Tuesday, December 9, 2008

Evolving Polygons,DNA, and the Renaissance

I would like to dedicate this to my friend Doctor Q at Q-optical in Boston.

This morning my rss feed from Slash Dot led me to this very interesting post by Roger Alsing in which he demonstrates the generation of a copy of the Mona Lisa comprised of 50 polygons. I can not pretend to understand the technical aspects of his process, but what I find interesting 
is the similarity to calculus. From the point of view of computer science this may be stating the obvious, but what I suspect is less obvious to those who have not studied the development of pictorial perspective is the way all this relates to the process of creating a perspective image on a two dimensional surface. I am speaking specifically of drafted, or measured perspective
This process can be very complicated geometrically and is without question a "scientific method."
This image was drawn by Paulo Uccello, a florentine artist of the late 14th- early 15th century. It is particularly good for illustrating the way that the volume of the object is reduced to many polygons. I will try to explain briefly what you are seeing here.  The various circles which in fact form the shape of the chalice - it's foot, the rings, the stem, the rim etc. have been drawn as circles on a separate drawing and then projected into ellipses by the method shown in the wikipedia article linked above. But this only gives the broad outlines. Say for instance a decorative pattern  circles the chalice, how do you locate the intervals of the pattern as they turn around the circumference? This is very likely the exact problem Uccello was addressing in this study. 

The way one locates the points turning around the circumference is to return to the plan view (the view from directly above), of the actual circles, and divide their circumference into the number units required. The circles of the plan are easily drawn with a compass and they are easily divided with a protractor. These constructions are often called "Projections" in this case we are creating a "perspective projection." Refer to the article, this is a "One point perspective projection" The term "Projection" is used because what we do at this point is to project, using the vanishing point, those divisions of the circumference of the circle in plan onto the circle drawn "In Perspective" (the ellipse.) The trickiest part of this is the fact that when viewed from above those divisions are even, but when projected they diminish as the approach the edge.
I have done a quick drawing to show this.  (forgive me, but the Uccello image jumped right into this post where I wanted it and my own drawing is stuck at the top of the post out of sequence- sorry) 

The circle in my drawing is divided into 12 equal segments. Slices of a pie- A pie chart, another connection- perhaps this is also used to convert pie charts into graphs in power point- all these things are done on the same principle. Above the "Pie chart" and centered exactly is the ellipse that represents the circle receding in space. What is very interesting is that the ellipse acquires the appearance of a receding circle only by it's context- it is of course a two dimensional ellipse but we understand it as a three dimension circle receding on a plane by comparing it to the circle. I have raised lines (this is the projection) from the points on the circumference to the analogous points (analogous-algorithm) on the ellipse. Notice the way that the space between the lines diminishes as they move away from the center. The Uccello drawing is doing this same thing in an incredibly elaborate way. Imagine this process turned 90 degrees and dealing with rectangles and you have the method of representing a checkerboard pattern receding in space; again, the way the squares diminish is determined by a geometric projection.

What now become obvious (at least to me) is the connection between the studies of Uccello, Brunelleschi, and Leonardo on the one hand, with the method of calculating surface area employed in the calculus on the other.

Of course that diminishing of the divisions can be calculated using algorithms, which is what calculus and computers do, and thereby we have a direct connection between the studies of Uccello and Roger Alsing - is it a reducio ad absurdum to propose that they have done the same thing, just used different tools?

As another student of this art, L.B. Alberti said "Art without science is nothing."


No comments: