Henri Poincare was the first to understand the terrible thing Weierstrass had done. He called it "an outrage against common sense.”
Somehow, Weierstrass had stitched together a pathological function made out of pointed teeth and sharp corners, and brought it to life.
"Weierstrass's Monster," as Poincare called it, is a function that is [lightning flashes outside the window] everywhere continuous but nowhere differentiable [huge crash of thunder].
Here are three plots showing the first 1000 terms in the sum (over [-2,2], [-0.2,0.2], and [-0.02,0.02] ) using the values a=1/3 and b=21 from Weierstrass's original proof.
[a wolf howls in the distance]
As you study the plots, a hideous truth reveals itself.
No matter how far you zoom in it's all sharp corners. Any finite stretch along the x-axis becomes a terrible mouth lined with infinite teeth, every pointed sliver opens up into a thousand little razors.
@memory This is definitely an example of a fractal curve. Peano and Hilbert gave their fractal curves about 20 years later, and Koch was 20th century. So it may be first with what we’d call a formal mathematical definition, though there’s probably a much older history of self-avoiding curves in art.
Weierstrass was very pleased with himself. The fools at The Academy had dismissed his ideas as the ravings of a madman. But after the 1872 presentation many mathematicians discovered that they, too, could bring monsters to life.
His colleagues began to publish similarly vile constructions.
@mcnees What about continuous functions with Hausdorff dimensionality of 1? Are they always at least piecewise-differentiable? I'm curious what the minimal constraints are to ensure differentiability.
@mcnees But like the other monsters, you (almost) never actually see one in action. (I've a vague notion that you can do analysis in a way where all definable functions are differentiable, but I also suspect that quite a lot of baby goes out with the bathwater if you do that.)
@mcnees loved this one, thank you! (And I did not know! Somebody may have told me during my math lectures but it obviously did not stick then - it will now because MONSTERS 😅)
@vicgrinberg@mcnees We had some simpler examples in basic analysis courses, but - as I remember it 40 years later - it was the measure theory course that went to town, devoting its final lecture (of 16 or 24) to examples of things like uncountable but measure zero and unmeasurable (if axiom of choice). You'll need these in the exam (not said, but obvious).
@mcnees I’ve long wanted analysis from a mathematician’s standpoint of an equation I discovered nearly 25 years ago through an artistic process, creative coding. It bears a striking resemblance to the Weierstrass, which I just learned today from your post: https://www.desmos.com/calculator/orxkgmveq3
Our “chief mathematician” has this on a poster, generated with Mathematica and ConTeXt MkII (in 2006). 2 < a < 3, 0 < x < 1
Discussion how to do it in MetaPost is ongoing 😉
@mcnees thank you for reminding me of this beautiful monster! It's such a nice example, and really helps understand the difference between being merely continuous and differentiable.
@mcnees I spent a while looking at this and then I realised: this is just octave summation used for procedural noise in games. He's just describing a coastline, or a mountain range. Everywhere continuous but nowhere differentiable (due to its fractal nature).
@mcnees This is why funding for arts and funding for science should have equally low requirements for practical applications. Scaring children into studying their differentiable trigonometry functions and identities always has societal value.
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