@mhoemmen thanks!

BRAINS!

@seanfobbe Thanks!

@hosford42 It is definitely a fractal curve. It was described about 20 years before Peano’s and Hilbert’s famous fractal curves.

@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.

“ ... but now that I had finished, the beauty of the dream vanished, and breathless horror and disgust filled my heart.”
– Victor Frankenstein

🎃Happy Halloween!🎃

In fact, Banach later proved that differentiable functions are a "meager set" of all continuous functions.

Differentiability is a fairy tale we tell our students to protect them from the horrible fact that most continuous functions are monsters.

Mathematically speaking, monsters are everywhere.

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.

Using accepted notions of infinite sums, limits, derivatives, and continuity, Weierstrass had assembled something unholy.

Hermite said of Weierstrass's construction: "I turn with terror and horror from this lamentable scourge of functions with no derivatives."

Though mathematicians recoiled, they had to admit Weierstrass's conclusions were sound.

Their intuition, which assumed a connection between continuity and some degree of smoothness, was now forced to accept the existence of MONSTERS.

@michael_w_busch @martinvermeer shhhhhhhhh!

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.

Mathematicians were aghast.

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.

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].

Mathematician Karl Weierstrass was born #OTD, Halloween, in 1815.

The fools at The Academy all said he was mad, but in 1872 he announced that he had succeeded in creating a ~monster~.

Image: Smithsonian Institution Libraries

A blazing skull screams in X-ray at the heart of the Perseus Cluster.

Image: A. Fabian (IoA Cambridge) et al., NASA

https://apod.nasa.gov/apod/ap001031.html

The dead comet, asteroid 2015 TB145.

Image: NAIC-Arecibo / NSF

🎃The Witch Head Nebula🎃

Image Credit: NASA / STScI Digitized Sky Survey / Noel Carboni

@deirdresm @alderson Accidentally correct! I’ll take it!

@schamschula All the driver / printer combos I tried. Generic postscript, printer-specific, etc.

Absolutely wild to me that just bringing up the print dialog in macOS will briefly beachball an M1 MBP. Any basic system-level task on this machine should be instantaneous. Un-mounting external hard drives? Good luck.