Conversation

Notices

1. Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 13:27:43 UTC Robert McNees

   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

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 13:31:06 UTC Robert McNees

     in reply to

     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].

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 13:33:29 UTC Robert McNees

     in reply to

     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.

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 13:35:57 UTC Robert McNees

     in reply to

     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.

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 14:00:56 UTC Robert McNees

     in reply to

     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.

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 14:06:46 UTC Robert McNees

     in reply to

     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.

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 14:09:49 UTC Robert McNees

     in reply to

     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.

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 14:12:58 UTC Robert McNees

     in reply to

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

     🎃Happy Halloween!🎃

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 15:21:18 UTC Robert McNees

     in reply to

     • Doctor Memory

     @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.

   • Doctor Memory (memory@m.blank.org)'s status on Tuesday, 31-Oct-2023 15:21:19 UTC Doctor Memory

     in reply to

     @mcnees (holy god I am not a mathematician, this question may in fact be infinitely, recursively stupid) was this the first fractal?

   • Patrick Hadfield (patrickhadfield@mastodon.scot)'s status on Tuesday, 31-Oct-2023 15:22:34 UTC Patrick Hadfield

     in reply to

     @mcnees they'd have loved fractals, then!

   • Aaron (hosford42@techhub.social)'s status on Tuesday, 31-Oct-2023 15:22:34 UTC Aaron

     in reply to

     • Patrick Hadfield

     @patrickhadfield @mcnees Isn't it one? Looks self-similar at different scales to me!

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 15:22:34 UTC Robert McNees

     in reply to

     • Aaron

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

   • ConTeXt group (context@fosstodon.org)'s status on Tuesday, 31-Oct-2023 15:42:23 UTC ConTeXt group

     in reply to

     @mcnees
     Thank you for the nice t(h)rea(d|t)!

     Our “chief mathematician” has this on a poster, generated with Mathematica and ConTeXt MkII (in 2006). 0 < a < 1, 0 < x < 1
     Discussion how to do it in MetaPost is ongoing 😉

   • Seán Fobbe (seanfobbe@fediscience.org)'s status on Tuesday, 31-Oct-2023 17:10:17 UTC Seán Fobbe

     in reply to

     @mcnees Now this is a glorious thread. Kudos!

   • Robert McNees (mcnees@mastodon.social)'s status on Tuesday, 31-Oct-2023 17:10:17 UTC Robert McNees

     in reply to

     • Seán Fobbe

     @seanfobbe Thanks!

   • Dana Fried (tess@mastodon.social)'s status on Tuesday, 31-Oct-2023 17:51:00 UTC Dana Fried

     in reply to

     @mcnees I now understand the provenance of the first line of this song:

     https://www.youtube.com/watch?v=6tsutU92rrE

   • Spooky SteveP 🇨🇦 (stevecantsmell@mastodon.social)'s status on Tuesday, 31-Oct-2023 22:56:29 UTC Spooky SteveP 🇨🇦

     in reply to

     @mcnees ugh. and here I thought I wouldn't be sleepless tonight

   • Mark Hoemmen (mhoemmen@c.im)'s status on Wednesday, 01-Nov-2023 00:23:52 UTC Mark Hoemmen

     in reply to

     @mcnees an excellent Halloween post; thank you!

   • Robert McNees (mcnees@mastodon.social)'s status on Wednesday, 01-Nov-2023 00:23:52 UTC Robert McNees

     in reply to

     • Mark Hoemmen

     @mhoemmen thanks!

   • Robert McNees (mcnees@mastodon.social)'s status on Wednesday, 01-Nov-2023 01:42:25 UTC Robert McNees

     in reply to

     • Chaw

     @Chawarma Thanks!

   • Chaw (chawarma@piaille.fr)'s status on Wednesday, 01-Nov-2023 01:42:26 UTC Chaw

     @mcnees it's perfect. Thank you for your posts !

   • Robert McNees (mcnees@mastodon.social)'s status on Thursday, 31-Oct-2024 15:28:44 UTC Robert McNees

     in reply to

     ☝️The most terrifying Halloween story I know.