# Ideals in Art and Mathematics: What gets us there?

Most of us begin drawings with lines.  And even though those lines may not be in the subject of a rendering, they are nonetheless perceived.  Some of the visual information we use to re-present our experience in a drawing is also used in mathematics, geometry in particular.  A difficult question to answer  but an interesting one to ask is, “What is it that allows (or inspires) us to isolate some of the visual attributes of an object (like straight or curved lines or closed figures)?”

My father taught me how to draw when I was a child.  He was very good at it and, as far as I knew, he had no training.  “Draw what you see, not what you know,” was his consistent warning to me.  And today I saw this phrase echoed in a number of web sites.  I also found a paper from members of the Department of Psychology at the University of North Carolina, Wilmington.  (The pdf can be found here) that explores my dad’s admonition.  The paper is more than ten years old, but its authors set up experiments in order to isolate the major reason most people find drawing difficult.  They concluded that things like motor coordination, the artist’s choice about what aspect of an object to represent or the artist’s misperception the drawing’s accuracy, had very little effect on the success of the drawing.  Instead, it was decided that the artist’s misperception of the object itself was the problem.

The paper sites two ways we misperceive – illusions and delusions.  But since “both poor and accomplished artists are affected by illusions,” these were not investigated with experimental trials. (It is worth noting, however, the current work on understanding illusions ).  The researchers found delusions, defined as  the memorized ideal of an object, to be the most significant factor in drawing inaccuracies.  They describe earlier work on delusions with children:

When copying an outline drawing of a table, for example, children make systematic errors that correspond to their knowledge of what a table looks like. However, when children copy outline drawings of parts of the table in isolation, they make very few copying errors (Lee, 1989). These results indicate that the children’s knowledge of the form of a table is interfering with the accuracy of their drawings.

I think this captures the significance of “draw what you see, not what you know.”  But the phrase “memorized ideal of an object” deserves some attention.  Idealizing seems to be part of our nature.  As if a visual ideal, like the table, might invite us to consider other ideals, say a mathematical ideal, like a circle, or a philosophical ideal like truth.

But our memory is not like a collection electrochemical versions of photos that we cut and paste.  Memories seem to depend on the complex interactions of many brain processes.  A study was done to record the brain’s activity during the process of drawing faces.  Participants were asked to reproduce black and white cartoons of faces.  The interesting thing:

The results show that looking at the cartoons activated visual processing areas of the brain, that are known to be responsive to faces, especially if the cartoon was displayed at the same time as they produced the drawing. But when the subjects had to wait before drawing, there was no maintained activity in these areas. This suggests that the memory of the cartoon face is transformed into a different, non-visual form.

The non-visual form is handled by the brain as spatial information.

They conclude that facial information is captured during a sequence of eye movements towards certain features of the cartoons, and the information is stored as spatial locations for subsequent eye and hand actions.

Now back to mathematics one more time. Geometries use visual information and a geometric understanding of an analytic idea will ‘visualize’ the idea (the complex plane for example).   But the generalities growing out of geometric ideas (n-dimensional spaces for example) transgress the three-dimensional limit of our visual imagination.  Perhaps the strength of their mathematical relations, and their reliability, rests, at least in part, on the way these mimic interacting brain processes that we can’t quite see yet.

As to why visual attributes are isolated and explored in the first place, I believe, like Semir Zeki, that the body is looking for the essence of things.