narain

johncarlosbaez, 9 months ago (edited 9 months ago) to random

What's interesting about the number 52?

1. It's the number of weeks in a year.

2. It's the number of cards in a deck, not counting jokers. This may not be a coincidence: the 4 suits correspond to the 4 seasons, there are 13 weeks per season, and the total value of all the cards is

(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13) × 4 = 364

with the joker giving 365. (The second joker helps out on leap years.)

3. There are 52 partitions of a 5-element set into disjoint nonempty subsets. So, we say 52 is the fifth 'Bell number'. They're named after Eric Temple Bell, who wrote the famous book Males of Mathematics. He did not discover these numbers.

4. In Japan, 52 chapters of the Tale of Genji have traditional symbols on the top called 'genji-mon', which correspond visibly to the 52 partitions of a 5-element set. Unfortunately the Tale of Genji has 54 chapters, so they needed to make up two extra symbols.

For more:

https://en.wikipedia.org/wiki/Bell_number

topher_batty, 9 months ago to random

I'm very excited to share our new SIGGRAPH Asia 2023 paper on improved reconstruction of explicit surface meshes from signed distance field data: "Reach for the Spheres: Tangency-Aware Surface Reconstruction of SDFs".

https://odedstein.com/projects/reach-for-the-spheres/

jonikorpi, 11 months ago to random

Fellow graphics programming nerds: any ideas for how to turn noise (I’ve got gradient, value, and cellular) into a shape with sharp ”tips”, like conifer trees?

I’ve tried:

• pow(noise, someHighNumber), but it doesn’t help with the rounded tips
• 1 - abs(noise), but it only creates long ridges, while I’d like more or less freestanding shapes

johncarlosbaez, 9 months ago to random

Here's a fun way to define df for any function f: X → ℝ on any set X:

(df)(x,y) = f(x) - f(y)

It's like calculus without the annoying division and limits. The result df is a function on X × X.

This funny derivative obeys the usual sum and difference rules:

d(f+g) = df + dg

d(f-g) = df - dg

It obeys the usual rule for multiplying by a constant c:

d(cf) = c df

It even obeys the product rule:

d(fg) = f (dg) + (df) g

But there's a catch: the last one needs to be interpreted carefully! It means

d(fg)(x,y) = f(x) dg(x,y) + df(x,y) g(y)

Note: we multiply on the left by f(x) but on the right by g(y). Check to see that this rule is true!

Needless to say, mathematicians have a way to make this idea more general and harder to understand. For any algebra A we define a 'differential'

d: A → A ⊗ A

by

da = a ⊗ 1 - 1 ⊗ a

A ⊗ A is an A-bimodule, which basically means we can multiply guys in A ⊗ A on the left by guys in A - but also on the right:

a (b ⊗ c) = ab ⊗ c
(b ⊗ c) a = b ⊗ ca

Then we get the product rule:

d(ab) = a(db) + (da)b

Let me check it. First we have

d(ab) = ab⊗1 - 1⊗ab

Now cleverly add and subtract the same thing:

d(ab) = ab⊗1 - a⊗b + a⊗b - 1⊗ab
= a(b⊗1-1⊗b) + (a⊗1-1⊗a)b
= a(db) + (da)b

Yes! 🎉

It's really just how we usually prove the product rule, with some complications removed.

I learned this from Yemon Choi, who said a lot more in fewer - but scarier - words:

https://golem.ph.utexas.edu/category/2023/08/grothendieckgaloisbrauer_theor_2.html#c062568

gregeganSF, 9 months ago to random

I'd be lost without computer algebra systems, but the fact that I can often solve equations they can't just because I know what the equations mean drives home just how far these programs remain from being able to solve every problem that (in some sense) “ought” to be easy.

christianp, 11 months ago to math

I've made a thing that helps you to make a printable template for a hexaflexagon, with your choice of pictures on each of the three faces: https://somethingorotherwhatever.com/hexaflexagon/

You can take a photo with your camera, or upload a file.

#hexaflexagon #math #RecreationalMath

narain, 1 year ago to random

If I want spherical geometry but I only have Euclidean geometry, I can get spherical geometry by restricting myself to the unit sphere.

If I want Euclidean geometry but I only have hyperbolic geometry, can I still get Euclidean geometry by restricting myself to some lower-dimensional submanifold?

narain, 9 months ago to random

Please switch to Firefox, like, yesterday.
https://arstechnica.com/gadgets/2023/09/googles-widely-opposed-ad-platform-the-privacy-sandbox-launches-in-chrome/

j_bertolotti, 9 months ago to random

Hot take: currency symbols are units, like kg or Hz. And units go AFTER the number, not before!

BartoszMilewski, 11 months ago to random

My intuition tells me that a set has the least structure (it's a discrete category). Conversely, the category of sets has the most structure, since its morphisms don't have to preserve any structure. Does it make sense?

manlius, 10 months ago to random Italian

Prigogine: “statistical mechanics and thermodynamics start where traditional dynamics (classical or quantum) breaks down.”

Maybe he's referring to non-equilibrium cases?
Otherwise 👉 quantum thermodynamics...

Ideas?

@tiago @johncarlosbaez @j_bertolotti @franco_vazza

christianp, 1 year ago to random

I poked myself in the eye with my glasses earlier (#dyspraxia) and can't reeally open my eyes. 111 said just stay at home, so I'm stuck basically immobile.
Good job I spent some time last week learning how to use BoiceOver on the work macbook. I've managed to log in and mathstodon was open, so heres a toot

gregeganSF, 1 year ago (edited 1 year ago) to random

On Tw*tter, Mike Lawler linked to a nice Physics Today article that points out that the planet whose average distance to Earth over any long period of time is the shortest is Mercury, not Venus.

I agree with their conclusions … but I wonder if there is an error in the specific formula they give for the average distance as an elliptic integral.

If we specialise to the case where one orbit is a unit circle and the other has radius r, Mathematica gives a somewhat different formula than theirs (actually two different formulas, for r>1 and r<1, blue and gold in the plot), which give close results to the Physics Today formula (green in the plot), but not exactly the same.

Numerical integration seems to confirm the formulas Mathematica gives.

Have I made some dumb mistake in the way I’ve set up the problem, or is the formula in the article wrong?

@buster and @duetosymmetry; the conventions for Mathematica’s EllipticE function and the elliptic integral function in the article are different; Mathematica’s function expects an argument that is the square of the one used in the article.]

https://pubs.aip.org/physicstoday/Online/30593/Venus-is-not-Earth-s-closest-neighbor

Breakfastisready, 11 months ago to random

There should be a mathstodon lemmy instance, for more organized discussions.

narain, 9 months ago to random

Today I learned that crocodilians are more closely related to birds than to other reptiles. WTF?
https://en.wikipedia.org/wiki/Archosaur

j_bertolotti, 10 months ago to random

The first time I ever heard the word "fortnight" was when somebody told me that one furlong per fortnight is almost exactly 1 cm per minute.
That was also the first time I was exposed to the word "furlong". You can imagine my confusion.

christianp, 10 months ago to random

Does this thing have genus 2? No way of getting from one loop to the other, no extra genuses hiding anywhere?
#topology #GenusCheck

Looking into the recess from the top, there's a tube connecting the front and back faces, making the edge of the hole. My thread is now looped around this tube.