ℝolliℵg M∀th Thr∑a∂

https://www.youtube.com/watch?v=sdlWiDgvjAY

sometimes things work out well:

I taught a course in computational algebraic geometry this summer.

To high school students.

Here's how it went.https://t.co/fCASlEoExx

— Bill Shillito (@solidangles) August 3, 2021

Good stuff, especially leading up to this:

In short, we need to stop conflating logical foundation with pedagogical foundation.

I think this is good but I also think it is super unclear to me that "pedagogical strategies that work well in a small group of student selected specifically as self-motivated high math achievers" has THAT much to say about what K-12 should be doing generally. And of course the idea that school math should involve playing around, discovery, "why do we do it this way? what if we did it this other way? if that doesn't work, what went wrong?" rather than "follow these rules to get the answer" is exactly the aspect of Common Core that was hated by legislators and parents and to be frank lots of kids (though I still can't help feeling there has to be a way to do it right.)

Fair enough.

I want to look at his divide by zero thing later.

never divide by zero, it always ends badly

and yes, it absolutely depends on this being a schoolroom of the self-selectedly curious in a territory where the outcome stakes aren't especially high yet (in a bridge-will-fall-down car-will-catch-fire sense)

never divide by zero, it always ends badly

trying to vibe at the pole of a reimann sphere, just not happening man

I was trying to work out myself how to prove that pi is a constant, that all circles have the same ratio between diameter and circumference. I got stuck -- I couldn't figure out a way to prove that if there exists the same angle-side-angle values for two triangles, they are congruent. I looked it up in Euclid's Elements, but Euclid's proof only works when that applies for ANY given side. Is there a simple proof of this without going into the law of sines?

Found it, Book I Proposition 26 in Elements.

To me the definition of a circle implies that all circles are similar to each other so pi has to be constant.

I suppose using Euclidean methods you could use dilation and then superposition.

here’s one way to prove it using integral calculus

take any circle x^2 + y^2 = r^2

circumference is the arc length of the curve, which is equal to 2r times an integral not involving t (as here https://math.stackexchange.com/a/1049390)

not involving r*

Okay, someone take the Terence Tao masterclass and tell me what he says.

I never heard the term The Martians before.

I'm laughing so hard at this slide a friend sent me from one of Geoff Hinton's courses;

"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it." pic.twitter.com/nTakZArbsD

— Robbie Barrat (@videodrome) June 10, 2018

Lol

Old enough to remember when the thing was to watch Tom Banchoff rotate a tesseract to help see the fourth dimension.

https://www.youtube.com/watch?v=90olwwLdEYg

Hinton's a real one

who knows how to visualize a complex or even quaternionic one

Anyway I see what he is saying.

Category theorists are always wilding out on my timeline

I have just learned that it may not be related to category theory and is likely a byproduct of Twitter’s new recommendation features (I assume it’s inferring an interest based on following math people and following computer science people).

pic.twitter.com/maySTJd1wA

— depths of wikipedia (@depthsofwiki) April 18, 2022

What does Snopes have to say?

Surely there is a Medium article or two addressing this. I believe I may have gotten an email alert about one this morning iirc.

https://www.newyorker.com/magazine/2022/05/16/the-mysterious-disappearance-of-a-revolutionary-mathematician

https://www.youtube.com/watch?v=uqwC41RDPyg

a tweet linking to an essay i can't now find: explaining how, yes, in 4D a klein bottle does not intersect itself but NO, in 4D a klein bottle does NOT have "only one side"

bcz "sides" don't really exist in 4D

(viz -- i believe -- bcz the concept of side is topologically incoherent in 4D: as eg the concept of edge is topologically incoherent in 3D)

anyway i've probably explained the stuff in brackets wrong or used the wrong terms or something lol and i can't find the piece that explained it but it was a good ACTUALLY so i did remember it to post here, eventually)

I don't know how to search for this thread so I didn't post this at the time that it appeared on redactle:

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Isn't this trivially true, since every real root is also a complex root (with imaginary part equal to zero)?

he concept of edge is topologically incoherent in 3D

doesn't a mobius strip have only one edge?

It’s talking about the the coefficients, not the roots, the roots may still be complex.

but (let's say) if a polynomial (with complex coefficients) were to have one single real root, it would therefore have 'at least one complex root'?

It's trivially true but not for the reason you're saying - it's saying that every polynomial with real coefficients is automatically a polynomial with complex coefficients. Real roots are a red herring.

i found the relevant thread and (as i kind of knew) i got the argument wrong lol (nothing to do with edges as analogy):

It's common to hear the following facts about Klein bottles:
1) To avoid self-intersections, it must live in at least 4-dimensional space.
2) Like a Möbius band, it's a one-sided surface.
The problem is, a Klein bottle living in 4-dimensional space is NOT one-sided! Why not? 1/13 pic.twitter.com/ertVreTkRx

— Dave Richeson (@divbyzero) August 23, 2022

xp ok, well i get what you're saying - i don't really get why that wikipedia sentence isn't talking meaningfully about the roots but whether that's down to me or the article, either way i'll let it pass.

Toby otm

TIL John Truss is Liz Truss’ father.

Meanwhile, what are people working on?

I realize I haven’t written since I moved from academia to industry. I’ve been working with a colleague on a paper about the use of transformers for root-finding. It’s been remarkable to see the learning capability of these models. We started simply by learning the root-finding method numerically (i.e., from coefficients) but we’re starting to encode symbols (i.e., the parse tree of a polynomial). It’s way too hypothetical to discuss elsewhere but I think it’s possible we’ll be able to use these transformers to discover entirely new and useful generalized functions.

Working on helping kid with freshman calculus homework.

How’s that going? I liked calculus. I fondly remember Riemann sums (I believe this was my first experience writing a proof) and especially loved some of the functional analysis (e.g., curve sketching).

I am enjoying the review. We won’t see integration until next term I assume.

Nice. In retrospect, I think it would’ve been nice to spend an entire term on functions and differentiation. Was your freshman a “new math” kid? Have you noticed any impact on their intuition?

This Freshman went to Catholic school so it didn’t seem to be the New Math. She is also dyslexic so some things take longer to sink in even though she is very bright, but it is also fun to see her pick things up. My other Freshman has never needed my help with any homework as far as I can recall.

But came to post that I learned a new thing from this: the AC method of factoring a quadratic.

Helping with chemistry homework too. Just learned that if I type “Avogadro’s Number” into chrome it brings up a little calculator with the number already entered for my convenience.

(how are you ever meant to find this thread?)

https://www.mirror.co.uk/news/weird-news/impossible-maths-question-rishi-sunak-28870041

for some (very low) value of 'impossible'. there's a spoiler at the end so don't scroll down beyond the diagram if you want to try it yourself. took me literally 2 minutes using nothing more complicated than multiplication.