Mathematics - 2021-03-16

Semiclassical

00:00

@leslietownes great, so i never have to pay attention to pi day ever again

(i don't mind Pi Day as an excuse to eat pie, but I find the focus on Pi a bit silly)

Balarka Sen

What is $\text{GL}_2(\Bbb R)/\text{GL}_2(\Bbb Z)$? It's the unit tangent bundle to the modular curve $X(1) = \Bbb H^2/\text{PSL}_2(\Bbb Z)$, I think.

Fun fact, $\text{GL}_2(\Bbb Q_p)/\text{GL}_2(\Bbb Z_p)$ (p-adics) is the $p+1$-regular tree. This is the "unit tangent bundle" on the vertices of the $p+1$-regular tree, I guess.

OK, I'm gone

leslie townes

i really hated pi day when i was an academic. it seemed like it only really began to catch on after 2000. drove me crazy. now i am one degree removed. it's silly to focus on pi. i have identified more interesting constants in my published work.

Edward Evans

TREE(3) day

PM 2Ring

00:19

Some pi trivia. 22/7 is an ok approximation, but it's even better if you subtract 0.04%. That is, (22/7)(1-4/10000). Of course, 355/113 is much better, but that denominator isn't much fun when you're doing mental arithmetic.

leslie townes

when i'm doing what, now? :)

i was on a call the other day where three people spent about ten minutes computing the volume of a very small chamber. the stumbling block seemed to be once you've cubed everything, you're getting thousands of cubic meters, which aren't liters. i remained silent and hoped that nobody remembered my background. it was a success.

i may have opened this window and chatted about something else while it was playing itself out.

PM 2Ring

I must admit that my mental arithmetic skills aren't as sharp as they were a few decades ago. But I still like to do some stuff mentally, to keep those brain circuits functional.

leslie townes

and when i say about ten minutes, i mean, at least ten minutes. it was a long time. i floated the possibility of suggesting that people compute a volume offline, but abandoned it because i was one of five or six conference participants and didn't want to disrupt the flow of the meeting.

i think i have only gotten better with mental arithmetic over time, but i still suck.

i did a finance interview where they asked me to estimate the number of gallons of gas sold in the US in a day. my estimate was right, to the number of digits. they didn't hire me anyway. it's all a game.

PM 2Ring

One good thing about going to school during the era of slide rules & log tables: you got pretty good at keeping track of powers of 10.

leslie townes

i did mention this. i said "we all agree it's 1.32 times something and we just need to pin down the power of 10." it was ignored.

PM 2Ring

00:29

I'm slightly disappointed that I got no responses to this:

Mar 8 at 20:46, by PM 2Ring

Quick Fermi problem: What's the square root of the ocean? That is, what volume is $v$, such that $n$ is the number of molecules of water in $v$ and $n^2$ is the number of water molecules in the ocean, IOW, the volume of the ocean is $nv$.

leslie townes

also it's not going to be a large power of 10, or a small power of 10. same as when i used to go to the grocery store and buy apples, i know if i buy five apples it's not going to be $25 or $0.50.

i hate fermi problems. they take me back to finance interviews, which i generally performed well in until the non-fermi part of the interview. and finance guys are d*cks anyway.

and by guys i do mean guys. i did not have a single female interviewer.

PM 2Ring

@leslietownes Exactly. Kids who depend on calculators don't seem to see the benefit in doing rough estimates just to make sure you haven't screwed up a decimal place somewhere.

leslie townes

it's really that simple. i'm not buying a new car for $2000 or $200,000. i might buy a car for $20,000.

oh, mathjax, where would i be without you.

geocalc33

$ ds^2=c^2dt^2-dx^2-dy^2-dz^2 $ anybody know how to write this in null coordinates?

Akiva Weinberger

\\$

leslie townes

00:33

geocalc, no time travelling. we go positive t only. none of this goofy business.

Akiva Weinberger

@PM2Ring 5 mL

or a teaspoon

leslie townes

one of my officemates in grad school was notoriously bad at practical math. to the point where he had a collection agency going after him for missed cell phone payments. another officemate, completely useless at practical calculation, off by powers of a thousand. but she could spot errors in arguments. she had a superhuman ability.

PM 2Ring

@AkivaWeinberger Very good. I think it's closer to 8 mL, but the water volume I used includes all the water in the atmosphere, and groundwater in the crust. And it's probably a fair bit more if we include the water "dissolved" in the mantle.

Akiva Weinberger

Yeah I'll take it as being within the margin of error

PM 2Ring

@geocalc33 Mr Wiki knows.

Light-cone coordinates

In special relativity, light-cone coordinates is a special coordinate system where two of the coordinates, x+ and x− are null coordinates and all the other coordinates are spatial. Call them x ⊥ {\displaystyle x_{\perp }} . Assume we are working with a (d,1) Lorentzian signature. Instead of the standard coordinate system (using Einstein notation) d s 2 = − d t...

Akiva Weinberger

00:39

1-2 tsp

PM 2Ring

@AkivaWeinberger Sure. If you're within an order of magnitude or two, you're fine.

Akiva Weinberger

Another way to phrase your question is, what's the geometric mean of the oceans and a molecule of water

$\sqrt{ab}$

PM 2Ring

Yep

We had a similar sort of question recently on Astronomy.SE. If you could bring all of the stars from the Milky Way together into one giant ball, without changing their density, how big would that ball be. There's somewhere between 100 & 400 billion stars in the galaxy.

geocalc33

$\delta_{ij}dx^idx^j$ what does this expression mean?

leslie townes

it would seem to not care about the quantity unless i = j, in which case it does care. maybe an integral involving coordinates labeled with superscripts.

PM 2Ring

00:49

1

Q: How big would the Milky Way be if all the stars were emptied into one sphere?

Neglecting the fact that this ball of gas would just collapse on itself - I'm curious if there's an agreed-upon measurement that takes into account the volume of all the ~100 billion stars to predict the diameter of a milky way where all the gas is in one orb. If not, is it safe to just use the v...

milky-way

leslie townes

the sun is a mass of incandescent gas. some wise sage said that

PM 2Ring

@geocalc33 What leslie townes said.

Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = { 0 if i ≠ j ,...

Ted Shifrin

@PM2Ring yup!

PM 2Ring

Also see en.wikipedia.org/wiki/Levi-Civita_symbol

geocalc33

so $ds^2=dudv+dr^2+dw^2$?

PM 2Ring

00:55

One of my science teachers had a telescopic cylindrical slide rule, with helical scales. It only had a few scales on but the scale length was 60".

geocalc33

actually I think it's $ds^2=dudv-dr^2-dw^2$

leslie townes

my high school calculus teacher had a box full of damaged or otherwise inoperable slide rules. he let me keep one. it had something that let it do trig functions, or would have let it do that, but the indicator had slid away and it was just two marked pieces of plastic. it needed the third piece of plastic.

PM 2Ring

@geocalc33 That looks better. Assuming the u & v are the null coords, and r & v are the spacelike ones.

Standard slide rules had sin & tan scales, usually on the back of the slide. Having cos as well would've been a waste of space.

Oops. I meant *and r & w are the spacelike ones. Sorry about that.

I still have a small 5" slide rule. But it's packed away in storage, under the house.

leslie townes

01:15

our storage space is above our garage. we did not inspect when we evaluated the house. when we moved in it was clear that somebody had been growing weed there. at least until it became legal. there were newspapers from the 1980s. who knows what happened.

there were all these housings for lights and fans and homemade cardboard ductwork to put skunky aromas out of the garage.

it's legal now, you can walk literally 5 minutes away and get anything. probably ruined the point of all the ductwork.

Ted Shifrin

@PM2Ring I had a prized circular slide rule I bought in high school. I bequeathed it to the UGA Math Club when I retired.

PM 2Ring

01:31

I had a circular one too, as well as a standard one. The circular slide rules were great, due to the continuous scale. But they were a bit too big to fit into a pocket. :)

leslie townes

i want a circular one.

also want a pocket antikythera mechanism. and as long as we're wanting, a pony. for my daughter.

someone should investigate the invisible link between children and horses.

Ted Shifrin

There were tiny circular ones (3” diameter), but mine was respectable!

PM 2Ring

Yeah, I've got a 3" circular one, embeded into a large rectangular thing, but it's practically useless, so I didn't mention it earlier.

leslie townes

ebay is overpricing these things. it's marked plastic.

no sale.

Ted Shifrin

I bought a couple of planimeters on ebay. One ended up photographed for my multivariable book.

PM 2Ring

01:44

I've just been looking at slide rules on Ebay. They have a few circular ones, but they're mostly pretty pricey. I did see one for 4 bucks, though.

leslie townes

i'm afraid if i buy the four buck one i get a color printout of the ebay listing.

PM 2Ring

I'm almost tempted to write an SVG file for printing a slide rule...

leslie townes

my daughter is absolutely losing her mind right now. it's mealtime. she's throwing dishes around the kitchen. terrible twos.

Ted Shifrin

Maybe you should be in there and not here.

leslie townes

yeah, probably. you remind me of my wife's therapist. that's a positive thing. she seems nice. she argues my point of view and i'm not even there.

Ted Shifrin

01:49

Go.

leslie townes

i'm on it. thanks.

Ted Shifrin

Bye :)

Wolgwang

02:03

@robjohn Please justify the construction :-)

leslie townes

02:13

daughter is now in my home office, yelling at the cat. this counts as parenting.

my daughter was asked to explain her actions and said: "it's a mystery." flashbacks to A Serious Man.

another ready-baked answer she has for why she did something is: "for some reason." as long as you did it for some reason, it's unassailable. check mate.

user 85795

in the UK, they punish their kids for being terrible when they were two, by putting them in preschool when they turn three

leslie townes

i've investigated when my daughter becomes the problem of the state. it seems like no earlier than 5.

in the meantime it's just all of this nonsense. i secretly love it and will miss it when it's gone. but it's a pain in the ass.

we spent 20 minutes after she got home from (paid, expensive) day care feeding a stuffed mouse to a bird puppet.

she really loves the idea of animals eating other animals. again, at some point this becomes the state's problem.

PM 2Ring

02:39

A Mandelbrot pi calculator in Sage.

robjohn

03:20

The Tangent Secant Theorem says that $PT^2=PM^2=PA\cdot PB$

and the Right Triangle Altitude Theorem says that $PD^2=PC\cdot PB$

since $\triangle CDB$ is a right triangle

Edward Evans

Pair of pants (mathematics)

In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms between 1-dimensional...

Wolgwang