My father could not get through a restaurant meal without pulling a propelling pencil from his jacket pocket and drawing, on either napkins or the paper tablecloth. Typically he would begin an explanation by drawing two cartesian axes, intersecting at zero with a small overhang to the netherworld of negative values. The lead was HB, so lines were medium gray, giving his marks the feel of a polite suggestion that could be rubbed off, not a definitive ink statement. He was a skilled draughtsman, so each line had the exact combination of straightness and swing that gave it a bespoke look, like the slight hesitations in the seam of a hand-sewn suit.
To this day, there is nothing more reassuring to me than seeing someone draw two axes, certain in the knowledge that an insight into the link between two things will soon follow. When my father did it, it could be temperature and pressure, with the added bonus of a state of matter at the triple point that was neither solid, liquid nor gas. It could be two intersecting lines of some economic law I forgot soon after, relating prices and profits. It could be some terrifying relation between the load-bearing capacity of a spar and its length, which I still think about six decades later when I see an airliner wing flexing in turbulence.
He also told me about dimensions. Zero dimension: a point. One and two dimensions: a line and a plane. I have since tended to see things as dimensions. For example, until I visit and prove its existence to myself beyond reasonable doubt, a place is a zero-dimensional point with no extension. This works especially with places that are known to be confined, e.g. islands. If I book a flight to Rhodes, all the time I vaguely wonder whether there will be room to accommodate me and my carry-on in that zero-dimensional dot.
My father used to really enjoy being on the most one-dimensional thing of all, the beach. Beaches combine all the pluses and minuses of low dimensionality. They are easily crowded, which happens when you project points from a plane to a line. But oh, the joys afforded to one-dimensional man! You can sit there and be practically sure you will see everyone go past, usually twice, when they get restless on their coordinate and go for a walk. You can stroll to the end of the beach and enjoy the linear succession of establishments and in-between deserts in both directions. It seems to be a law of nature that owners of beach bars always extend the line to infinity by naming them after another beach: Copacabana in St. Tropez and, I assume, vice versa.
One-dimensional architecture has changed the world. We owe many of the major technological inventions since WWII to the foresight of Bell Labs management, who arranged offices in a linear corridor with only one exit, reasoning that enforced encounters on the way to the toilets or cafeteria would generate ideas. The rabbit warren of some universities, the circular path at Apple, and the two-dimensional arrangement of cubicles on a big engineering floor will never be as good. Entire civilizations are essentially one dimensional: California; the Baltic Republics (three different beaches), pushed to the water by barbarians behind; Chile, the only country where the airline map looks like a single train track from desert to fjord.
Two dimensions allow the mapping imagination to work. Talk to anyone who has lived in Manhattan. Their mental map is like those tediously arbitrary maps shown in the frontispieces of fantasy novels, only for real, and they use coordinates to describe it. Maps are scale replicas of things and encourage things-within-things, as in Little Italy and Chinatown. When I was a child at French school, they used to teach us to draw the map of France as a hexagon, with the coastline on each of the six sides drawn from memory, and to populate that hexagon with all of God’s bounty (coal, forests, wine), the aim being to prove that France had it all and was, in fact, a scale model of the world at large, a sort of acupuncture ear map of the planet.
Those of us who are not topologists live comfortably in two dimensions. The third dimension has something unnatural and frightening about it. Think of those embossed plastic wall maps of mountain ranges in relief, which look well-behaved from a distance but up close turn out to be rough, as if the map had developed a crusty skin disease. Look at one of Bela Julesz’s random-dot stereograms and watch your brain slowly compute a ghostly third dimension from an initial disparity in your visual fields to a rock-stable image in less than a minute. I never really got used to three dimensions, and to this day, I love Tintin, woodcuts, lithographs, and heroic Art Deco frescoes that flat-pack things with no assembly required. At sunset, sometimes the light flattens things well enough to fit in my low-dimensional head.
Two special things that come to mind about two domensions: in conformal field theory, it is only in the 2D case that you have an infinite number of conformal transformations available, which leads to tight constraints on your physical theory that you can leverage. The other thing is that in two spatial dimensions (and one temporal), gravity is an entirely topological theory. That doesn't make it trivial, but things don't depend on your Riemannian metric anymore.
Mr. Turin, with your scientific background and immense life of mind, I wonder what you would think of Amber Jobin’s work? Her Aether Arts is a fascinating brand (more like wearable art, quite experimental). I especially enjoy her Exobotany series (based on things like atmospheric readings of exoplanet Gliese-667e, imagined event horizons, etc.). Fun stuff! IAO-award-winning, too. I’d be most interested in your thoughts, whatever they turn out to be. 😀