![]() To illustrate what’s involved, we may imagine two extremely different attitudes toward the relationship between words, reality, and meaning. We now know (and indeed have known for over a century) that their efforts were doomed to failure, but to see why, one needs to shift one’s view off to the side. (The Greek word “axiom” has connotations of worthiness and value.) Modern writers usually replace Euclid’s version of the parallel postulate with John Playfair’s streamlined version, which features one line and one point and then one more line: “Given a line L and a point P not lying on L, there exists one and only one line through P parallel to L.” Many smart people labored to derive the parallel postulate from the first four but were unable to do it. ![]() In comparison with the sleek first four, Euclid’s fifth axiom, featuring three lines and a whole bunch of angles, was an ugly thing - it seemed like something that should be derived from more basic notions, not elevated to the status of an axiom. I really did find myself typing “postule” rather than “postulate” as I typed the preceding sentence, and I guess part of me was thinking of the word “pustule”, which would be appropriate because for centuries mathematicians picked at the parallel postulate the way you might pick at a skin-growth. The most notorious of Euclid’s axioms is the ungainly fifth, known as the parallel postule postulate. Later on mathematician David Hilbert realized that Euclid’s system is missing a few axioms conveying properties of the plane that were as invisible to Euclid as water is to a fish, but since it took many centuries for anyone to notice the gaps let’s not dwell on that. But how did we get to the modern state of affairs, where these three concepts are locked in a very specific sort of embrace? The pseudosphere is part of what got us here.Įuclid based his theory of plane geometry on five axioms (such as Axiom 2: “Any straight line segment can be extended indefinitely in a straight line”). And my real themes today are the imagination, math, and meaning. I’m not going to talk about curvature today, or the exact shape of the pseudosphere, because to a large extent the pseudosphere has been displaced by better ways of getting a handle on hyperbolic geometry (such as Escher’s sketches), better suited to the human imagination. ![]() (It didn’t help that Euclidean, Riemannian, and Lobachevskian geometries were also called parabolic, elliptic, and hyperbolic, respectively, even though there were no parabolas, ellipses, or hyperbolas in sight.) I also remember the book mentioned something called curvature, which could be positive (for the sphere), zero (for the plane), or negative (for the pseudosphere), and that confused me even more. Euclidean (think: vanilla) geometry was the familiar geometry of the plane, Riemannian (chocolate) geometry was the geometry of the sphere, and Lobachevskian (strawberry-ripple-crunch) geometry was the geometry of the pseudosphere. The author described a trinity of two-dimensional geometries: Euclidean, Riemannian, and Lobachevskian. I saw my first pseudosphere in the early 1970s in some higher-math-for-the-public book, in a discussion of non-Euclidean geometry. But if the pseudosphere were merely the most famous example of a mathematical surface that looks like a wizard’s cap, I wouldn’t have picked it. After all, wizards are a staple of fantasy literature, and these essays are about math as a form of magic, taking place in a domain where the sorts of miracles of transformation that occur in fantasy actually happen and actually matter. I’ve partly answered this question already by calling the shape a wizard’s cap. So why did I choose it to visually represent what this blog is about? I don’t study the pseudosphere in my research, and I can’t say I have a lot of intuition about it in fact I don’t especially like the thing. The wizard’s-cap graphic that appears at the top of my blog as part of the logo is a piece of an infinite mathematical surface called the pseudosphere.
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