The intrinsic curvature of a space is a subtle concept; it took a lot of hard work to formulate, clarify and crystallize its most prominent measure, the Riemann curvature tensor. Now that that work is done, of course, you can learn about it even in some undergraduate courses. Perhaps you start with a rigorous definition, then see a geometric interpretation, and compute the curvature tensor for a few example spaces. You may get a little mystified in the beginning, but as you come across it in various contexts, read about alternative definitions and interpretations, and see theorems where it plays a fundamental role, the mystery slowly disappears—or appears to. Perhaps at some point, you start to think it doesn’t seem that scary anymore.

Here are two quotes from famous geometers—two people who probably have as intimate an understanding of curvature as anybody—to help you keep your guard up.

The  curvature tensor of a Riemannian manifold is a little monster of (multi)linear algebra whose full geometric meaning remains obscure.

–Mikhail Gromov, Sign and Geometric Meaning of Curvature

Do not despair if the curvature tensor does not appeal to you. It is frightening for everybody.

–Marcel Berger, A Panoramic View of Riemannian Geometry

I came across the first quote recently, but had seen the second one a while ago, while browsing Berger’s delightful book (which is great for leisure reading, if you are into the subject—lots of informal and intuitive comments, puzzles, history, all written in a fun and refreshing style).

Symmetries of the curvature tensor “Milnor’s octahedron”

Berger also has a similar (and intriguing) comment about the Levi-Civita connection in his book. I had a fun email exchange with him about that comment once, but let me leave that to another time.

(See also: 1, 2)

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The video above shows Myrmegraph, a screen-based artwork by Scott Snibbe. The picture, by Victor Juhasz, is from The Mind’s I, a book by D. Hofstadter and D. Dennett.

From Snibbe’s page:

As you move the mouse over Myrmegraph’s screen, you release a stream of simulated ants and simulated pheromone: the chemicals ants use to communicate. These ants obey a simple set of rules to follow the pheromone gradients stored invisibly in the image. From moment to moment, ants can change their heading to better pursue the trail of pheromone; and, like real ants, they sometimes lose their way and wander off.

Among the four combinations of the form <building blocks><emergent structure>,

(1) nonsentient → nonsentient
(2) nonsentient → sentient
(3) sentient → nonsentient
(4) sentient → sentient,

the possibilities (1) and (3) do not cause much cognitive dissonance, and to some degree, one learns to live with (2), however uncomfortably. The possibility (4), though, is rather unfamiliar, and confusing/fascinating in its own unique way.

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Weyl and Huxley on the abstract and the concrete

Weyl, Hermann (1885-1955)

Important though the general concepts and propositions may be with which the modern industrious passion for axiomatizing and generalizing has presented us, . . . I am convinced that the special problems in all their complexity constitute the stock and core of mathematics; and to master their difficulties requires on the whole the harder labor. . . . The general theories are shown here as springing forth from special problems[.]

–Hermann Weyl, The Classical Groups


Oh, no, no, no. I think fiction, and biography and history, are the forms. I think one can say much more about general abstract ideas in terms of concrete characters and situations, whether fictional or real, than one can in abstract terms. . . . And I must say I think that probably all philosophy ought to be written in this form; it would be much more profound and much more edifying. It’s awfully easy to write abstractly, without attaching much meaning to the big words. But the moment you have to express ideas in the light of a particular context, in a particular set of circumstances, although it’s a limitation in some ways, it’s also an invitation to go much further and much deeper.

Aldoux Huxley, interviewed by Raymond Fraser and George Wickes, The Paris Review

I have collected a few other quotes along these lines over the years on my pedagogy page, including one by Michael Atiyah, who expresses great admiration for Weyl and his book. Here is another, beautiful bit from the latter’s preface:

The stringent precision attainable for mathematical thought has led many authors to a mode of writing which must give the reader an impression of being shut up in a brightly illuminated cell where every detail sticks out with the same dazzling clarity, but without relief. I prefer the open landscape under a clear sky with its depth of perspective, where the wealth of sharply defined nearby details gradually fades away towards the horizon.

In the same preface, we also find a complaint:

The gods have imposed upon my writing the yoke of a foreign tongue that was not sung at my cradle.

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