Where Does A Plane Touch A Sphere?

Recently my dear love, the professional philosopher, got to thinking about a plane that just touches a sphere, and wondered: where does the plane just touch the sphere? I, the mathematician, knew just what to call that: it’s the “point of tangency”, or if you want a phrasing that’s a little less Law French, the “tangent point”. The tangent to a curve is a flat surface, of one lower dimension than the space has — on the two-dimensional plane the tangent’s a line; in three-dimensional space the tangent’s a plane; in four-dimensional space the tangent’s a pain to quite visualize perfectly — and, ordinarily, it touches the original curve at just the one point, locally anyway.

But, and this is a good philosophical objection, is a “point” really anywhere? A single point has no breadth, no width, it occupies no volume. Mathematically we’d say it has measure zero. If you had a glass filled to the brim and dropped a point into it, it wouldn’t overflow. If you tried to point at the tangent point, you’d miss it. If you tried to highlight the spot with a magic marker, you couldn’t draw a mark centered on that point; the best you could do is draw out a swath that, presumably, has the point, somewhere within it, somewhere.

This feels somehow like one of Zeno’s Paradoxes, although it’s not one of the paradoxes to have come down to us, at least so far as I understand them. Those are all about the problem that there seem to be conclusions, contrary to intuition, that result from supposing that space (and time) can be infinitely divided; but, there are at least as great problems from supposing that they can’t. I’m a bit surprised by that, since it’s so easy to visualize a sphere and a plane — it almost leaps into the mind as soon as you have a fruit and a table — but perhaps we just don’t happen to have records of the Ancients discussing it.

We can work out a good deal of information about the tangent point, and staying on firm ground all the way to the end. For example: imagine the sphere sliced into a big and a small half by a plane. Imagine moving the plane in the direction of the smaller slice; this produces a smaller slice yet. Keep repeating this ad infinitum and you’d have a smaller slice, volume approaching zero, and a plane that’s approaching tangency to the sphere. But then there is that slice that’s so close to the edge of the sphere that the sphere isn’t cut at all, and there is something curious about that point.