Visualizing Math

Lynet writes about how she tried explaining the Poincaré Conjecture to a literary theorist and a historian. The bone of contention wasn’t really the conjecture, which the media dumbed down just enough so that non-mathematicians could understand a statement vaguely resembling what Perelman actually proved. Rather, it was how mathematicians could visualize a four-dimensional world.

“Can mathematicians actually picture four dimensional space?”

“Roger Penrose says he did it – briefly – once,” I said, grinning [1].

“No, but are there people out there who can actually…”

“Not that I know of.”

My historian friend was relieved. My literary theorist friend was confused. “If you can’t picture it,” he asked, “how could you have any intuition about it? I mean, you could just say whatever you wanted about it and no-one would be able to refute you.”

I’ll leave it up to you to make an appropriate snarky comment about literary theorists. I’d reply to Lynet’s friend by noting that at least the simpler intuitions, namely, those relating to the vector space structure, are easily generalizable.

I can view points in the plane as pairs of coordinates (x, y), and work out things like angles between lines, lengths of lines, tangent lines to curves, functions on the plane, and so on. Regarding the point (x, y) as a vector from (0, 0) to (x, y), I can add vectors pointwise by (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2) and multiply them by scalars by k(x, y) = (kx, ky). I can look at linear transformations of the plane, or even affine transformations.

All of those are intuitively thought of using very concrete notions: length and angle are measurable concepts; tangent lines touch curves at only one point; vector addition consists of walking from (0, 0) to (x1, y1) and then along the same direction and for the same distance as from (0, 0) to (x2, y2); linear transformations are combinations of rotations, reflections, shears, stretches, and compressions, while affine transformations add translations.

But in higher mathematics, they’re considered abstractly, in order to generalize as much as possible. Length is defined using Pythagoras’s theorem, and angle is defined using inner products. Linear transformations are defined by the more easily generalized property that T(v1 + v2) = T(v1) + T(v2) and T(kv) = kT(v), which coincides with the more concrete definition in two dimensions.

Not coincidentally, for two or three years of university, students only ever see algebraic arguments, which don’t involve visualizing anything. Later some geometric aspects return, but even they are typically schematic; people who draw a line with a loop in it in algebraic geometry only look at very general aspects, like having a point with two tangent lines and not being decomposable into two lines.

16 Responses to Visualizing Math

  1. muppt says:

    it’s not difficult to visualize a 4 dimensional object, just use time as the 4th dimension.

  2. muppt says:

    it’s called animation, btw

  3. Ran Halprin says:

    Some people are more visual then others. Some do math by visualization (me included), so do it with pure abstract thought. I think the later are more rare, its not very natural.

  4. muppt: Or colour. Or some other simultaneously visualizable parameter. Still doesn’t give you an at-all-times visual of what’s going on.

    For a core example: you can visualize the tesseract by just looking at animated sequences of slices of the tesseract; but this won’t give you an intuition as to how that picture changes if you start rotating the entire thing.

  5. Maya Incaand says:

    More remarkable is that anyone considers 4D(spatial) to be anything other than a mathematical construct.

    You cannot rotate something that does not exist.

  6. Ran Halprin says:

    Nothing really exists. Things exist only in our minds, models of the perceptual reality. Mathematicians take this one step further and construct models which are not derived entirely from perception, but rotating a 4D hypercube in your head is no different in nature then rotating a 3D cube in your head – except that the later happens to be easier because most of us have a vivid image of a 3D cube, and do not have a vivid image of a 4D cube.

  7. D. Eppstein says:

    I mean, you could just say whatever you wanted about it and no-one would be able to refute you.”

    The first step on the road to understanding what mathematical proofs are good for. Because if all we had to rely on was visual intuition, then that would be an accurate description.

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