# Discrete and Continuous

Another aspect of the two paintings posted in Why Art is that they illustrate the difference between discrete and continuous, at least in terms of color.

In Keith Haring’s painting, every point is either yellow, red, green, or purple: four discrete possibilities. In Wayne Thiebaud’s painting, there are many continuous transitions from one color to another. Notably, for me, a field that changes from gray to brown to indigo. If you look at the body of work of these artists, you would see that they each stay pretty much in one color camp. Haring in the discrete camp, Thiebaud in the continuous.

In mathematics the discrete and continuous camps exist too. People typically pick one or the other. What camp they choose is an individual matter probably much as it was for Haring and Thiebaud. Is one camp “easier” or “simpler” than the other? Not necessarily. It’s tempting to think that the discrete camp is easier – all whole numbers, no messy extended decimals. No Zeno’s Paradox.

However, the continuum can be comforting because it is filled with solutions. We know, for example, that two non-parallel lines in the plane definitely intersect, which means that the system of equations defining two such lines always has a solution. Ask a similar question in the discrete world – does a given line intersect any lattice points in the plane? And that question is harder.

(By lattice points in the plane, I mean points with whole-number coordinates in the X-Y plane, such as (2,1) and (3, 56). These points form a regular grid of dots.)

Then there is the famous Diophantine equation (love that name, Diophantine, makes me think of an equation in a flowing white dress with a long trunk):

In the continuum are many solutions. But if you ask about discrete solutions, you get Fermat’s Last Theorem, which stumped mathematicians for centuries.

On the other hand, the continuum is a very mysterious place. That’ll have to be another post.

# Why art?

I feel a need to say more about A Study in Scarlet (Rectangles). What place does a painting have in a math club?

I think it’s worth pointing out that the kinds of thinking people use when they think about art are similar to mathematical thinking. In fact, from my perspective, it’s all just thinking.

# Color

Before jumping in to some obvious observations about colors and numbers let’s look at two paintings. I think color is important in both of them:

25_89 by Keith Haring

One of Wayne Thiebaud’s landscapes

When I look at paintings, I ask myself, what is this painting about?
This is similar to what happens when I look at math problems. A math problem can seem impossible, or like I’m stumped, until I start to ask myself, what is this about? Is it about numbers or geometry? etc.

Back to the paintings, Keith Haring’s painting uses four distinct colors. Two colors are allowed to drip over the other colors. I think the bold colors express strong or bold feeling, and the dripping shows us different relationships between colors; red next to yellow is different from red next to green.

Wayne Thiebaud’s painting shows he put a lot of thought and care in mixing colors. Shadows and outlines are highlighted in slightly nonrealistic colors, lots of purples and greens, like reflections in abalone shell. When I see this use of color it is both familiar and surprising to me; yes, there are purples and turquoises in the shadows, but no, I don’t always see them.

Mixing colors is well known to be challenging. Just rendering something like a green sweater is not a simple matter of green paint and maybe gray for the shadows. Maybe the lighter parts are more yellow, maybe the dark parts more brown. Like numbers, colors can be “added together”. But unlike numbers, which have one dimension, a size, colors are said to have three dimensions: hue, saturation and value.

It’s easy to visualize the set of all numbers as a line. What would the set of all colors look like? Would it be fully three dimensional like a cube? Would it extend in three directions, or have boundaries? Would parts of the color space fold in on itself? (There are many ways to make gray, for example.)

If you look at a mostly monochromatic painting like the one in A Study in Scarlet (Rectangles) you can see how the interplay of value and saturation can play a big role if the hue is mostly left alone.

Here is more about Keith Haring.

# A Study in Scarlet (Rectangles)

Professor Bear’s exponent, Marjorie, saw this painting in a museum in France. It’s a beautiful example of a (mostly) monochromatic painting. Red. And it is a study in rectangles. It’s interesting to see where the artist broke lines and boundaries and symmetries, and where the artist continued them.

The painting has both representative and abstract features. It looks like a scene in traffic, and achieves light and depth. Small touches of contrasting colors, greens and blues, make the light very interesting. In places there is strong three-dimensional feeling, and in other places it’s more two-dimensional.

Enjoy.