Everywhere-Nowhere: Weierstrass Meets Borges

A circle whose center is everywhere and circumference nowhere (Borges)

A function that's continuous everywhere and differentiable nowhere (Weierstrass)

I sort of randomly recalled those two objects, noted the similarity of the diction, and decided there had to be a reason both ideas fit so snugly into the same sentence structure. I think I encountered the Borges idea years before Weierstrass's, but I'd been exposed to both concepts for a long time before my mind noticed any connection. And I know this connection is probably spurious, but it's fun to think about. So here goes.

Imagine a circle with a designated center and a radius that separates its circumference from the center. No such circle, constructed from this model, but whose center is everywhere and circumference nowhere, exists as an object or has ever been defined, as far as I know. However, it seems like any center located inside such a space, could achieve this impossibility by emanating and reaching (or becoming) its edges. By straining itself toward and finally through its limits with enough power, it could conceivably obliterate its own borders. And if it were emanating in every direction with roughly the same force and speed, its limits would look like a circumference—assuming one were to stop it in its progress at some definite point in time to examine the shape it made.

[Actually, the surface of a sphere fits Borges's description remarkably well without being the least bit paradoxical. Think of Riemann's one-point compactification of the (complex) plane. All it takes is w = 1/z to bring infinity to its knees. But back to the paradox.]

It's almost like the circumference, at every turn, is threading itself through the center—the center straining outward and the circumference inward. So that each point of the circumference is sliced by the shards to which it is itself converting the center. Both are slicing into each other and splintering each other so that the slivers get thinner and thinner—ever thinner, advancing to zero. But that only explains the circumference's being nowhere. The center's supposed to be everywhere, not nowhere too. What gives?

Weierstrass's function was a contrived one. The function I remember off the top of my head is x cos (1/x), but I'm not sure that's the right one. Its derivative is cos(1/x) + (1/x) sin (1/x), where—if we sought a limit as x approached infinity—we'd improve the situation a very little. So we'd have to pull a L'Hospital to extricate ourselves from that situation. But it'd be enough. 

But actually, the real Weierstrass's function (above) is a little bit more complicated (understatement) and truly undifferentiable. (Clearly I differentiated the above: that and my unreliable memory were my first clues that I had probably got it wrong.)

I got this visual conceptualization from the text book from which I first learned this:

from Understanding Analysis, by Stephen Abbott, p. 146

That h equals |x| which simplifies the oscillation of cos(x) a little bit. So h itself is fails to be differentiable only at one point, but since |x| looks like /\, the 2^n factor causes a proliferation of those spikes. No matter how much you zoom, the graph looks the same amount of spiky.

So what does that fractal have in common with our circle, for which (apart from Riemann) we still haven't figured out how to visualize its center everywhere?

Well, if every time the circumference and center slice through each other, they create shards that are being shaved down to a zero magnitude (gradually, but certainly), then that means the interstices or gaps between represent the nowhere contrasting to which the infinitude of points of (eventually) zero magnitude represent the everywhere. And since the center and circumference share an inverse relationship (since each threads through the other), when the center achieves location everywhere, the circumference achieves location nowhere.

Of course, that also means if we can locate the circumference everywhere, the center would be nowhere. But that probably defines the usual relationship, as centers have zero dimension.