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Anyone who subscribes via my PayPal button will see that I uploaded a banner image onto the payments page. Unfortunately, PayPal seems to have trouble preserving resolution. It asked me to upload an image that was 1200×800, but then scaled it to 616×200 in a particularly horrible way. This is my original:

And this is the image as PayPal prefers to display it. Its fuzziness is annoying, but there is one good thing. It gives me an excuse to introduce the topics I'll probably write most of my articles about: art, meaning, maths, and cartoons. I'll start with the need for more "intelligence" in image processing, and then explain how this relates to the objects depicted in the image.

Artists often talk about "the power of line". For example, here is the blurb for a recent volume of essays, The Power of Line: Linea III:

There are no lines in nature — they are always the creation of humans, manifestations of human action, perception, and design. Lines can divide or connect, may be static or full of movement, and represent and create forms in space and time. And in many cultures, lines take center stage in science, art, writing, drawing, and construction. Employing a vast array of academic perspectives, this fascinating collection delves into the phenomenon of the line, as well as the power it holds for us.

From the history of art and science to philosophy, the essays in The Power of Line elucidate the semantic and conceptual depth of the line in European, Asian, and Islamic cultures. As they trace the continuity and transformation of the line over the course of centuries, the authors not only reveal it to be a constitutive element in architecture, art, and writing, but also uncover its importance as a medium of expression in both choreography and the scientific and technological fields. With copious full-color images, The Power of Line is a captivating exploration of the line as an essential artistic and cultural means of expression.

But what do artists mean by this? I know few who have described their own reactions to line, and many non-artists who look puzzled when I discuss it. In this essay, I set out my own experiences. Some of these may arise from quirks and cross-connections of brain wiring that we might not expect to find if our brains had had a rational designer.

We can easily show that drawings are about more than just the objects they depict. Here are recent or current logos of Labour, the Liberal Democrats, and the Conservatives, England's three main political parties:

This month, I'll turn from connotation to denotation, looking at the surprisingly varied ways in which the lines are used to convey geometry, surface quality, and tone. I'll show that the lines in drawing are an invention. They give the artist a chance to collaborate with the original scene in new and surprising ways, inventing or adapting a graphic language into which to translate them. I'll look at a few examples of such languages. I'll also argue that it is not wrong to omit or exaggerate information; in fact, the constraints of the language mean doing so is essential.

Line drawings can be very simple:

In my introduction, I said that category theory has been used in some impressive mathematical approaches to semiotics. Taking the view that the relations between mathematical objects are more important than the objects themselves, this branch of mathematics provides conceptual tools for working with networks of objects linked by relations, and for relating these to other networks. I'm going to explain how I applied this to a mathematical description of what artistic style is.

Let's start by imagining a car driver who can magically jump to a mirror-image Earth where everything is reflected, including roads. In mirror-England, people drive on the right and overtake on the left, while roundabouts go anti-clockwise: