The h Bar

obe

00:00

I used kittel's solid state physics.

FenderLesPaul

solid state is awesome though

0celo7

do QM dammit

get your shit together

obe

right, fail.

FenderLesPaul

especially once you get into modern condensed matter

skull patrol

^

0celo7

00:00

your mom is a modern condensed matter

FenderLesPaul

fite me irl

0celo7

I'm down

you know where I'm at

FenderLesPaul

that's true I do

0celo7

the building with the centipedes, 8AM

FenderLesPaul

but can you come here instead

0celo7

00:01

or 9AM

FenderLesPaul

I don't have a car

0celo7

me neither

right now

next year then

great Windows defender tells me I have malware but won't tell me where it's coming from

I haven't even done anything shady

should I be worried?

I'm new to this PC thing

FenderLesPaul

"haven't done anything shady" he says

snickers

eats snickers

obe

Download a trial of an antivirus.

0celo7

why won't it tell me where I downloaded it

HDE 226868

00:08

Does anybody know if Reimer's law is the current accepted mass loss relation in astronomy?

0celo7

@obe just ask your questions here

there's no shame

obe

I will.

0celo7

ask away then

obe

I just started ch3.

Why is the christoffel symbol defined the way it is?

FenderLesPaul

pretty sure he explains that

in good detail

obe

00:13

Right, I didn't read that far yet.

0celo7

because babies

+10 for reference get

FenderLesPaul

@obe read his explanation and see if it makes sense

if it doesn't then we can try to explain

obe

What page is it explained in?

NeuroFuzzy

@0celo7 jeez, what book are you talking about now?

obe

Whatever I'll find it myself.

0celo7

00:15

@NeuroFuzzy Relativity on Curved Manifolds

it has some stuff that HE was unclear on

FenderLesPaul

@obe hmm he doesn't seem to explain it the way I would like for someone learning GR for the first time

he basically does it the way Wald does it

which is by defining covariant derivatives abstractly

but that doesn't necessarily offer much physical insight

0celo7

the best way to define the covariant derivative is Straumann

NeuroFuzzy

@0celo7 Oh, that looks really good.

0celo7

@NeuroFuzzy It's hard.

obe

@FenderLesPaul I'm on pg95 and so far it's ok.

FenderLesPaul

00:18

@obe basically the point is that you need a way to compare vectors that live in different tangent spaces

in flat space this means we need to transport a vector $V = V^i e_i$ from one point to another in some coordinate system

obe

in 3.6 that is for a (1,1) tensor

what would it be for a different type of tensor?

Physics Meta

0

Q: How to suggest an edit to fix a 1-character typo

I tried to make a suggested edit for a post1 that was otherwise great, but the OP had made a single-character typo. When I tried to correct the typo, I was told that the edit needed to be 6 characters. As it was, the OP had an image there with the default [image description goes here] text, so I...

FenderLesPaul

but $e_i$ can vary with the coordinates so the $\Gamma^i_{jk}$ take this into account as you consider $DV/Dt$ along some curve $\gamma(t)$

NeuroFuzzy

@0celo7 Yeaaah and I'm not adding anything else GR to my todo list. But it does look fun.

obe

what do you make "takes this into account"?

FenderLesPaul

00:21

crap my internet is spazzing out

NeuroFuzzy

BRB, installing ooboontoo.

erm, ooh-bun-two

0celo7

yay virus free

FenderLesPaul

@obe it might be easier to just read it in a book

read section 20.4 of Hartle

obe

ok.

I understand 3.8 but not 3.7

0celo7

@FenderLesPaul what is wrong with sect. 15.2 in Straumann

obe

00:23

why is there a second $V^{\nu}$ term?

FenderLesPaul

@0celo7 :p

0celo7

oh that's a fighting face

FenderLesPaul

raises fists

obe

fight later, help me first.

0celo7

stay outta this child

go eat your broccoli

obe

00:25

::turns head down and walks away::

0celo7

@FenderLesPaul throws Abramowitz and Stegun

FenderLesPaul

pats obe

@0celo7 throws Weinberg Vol 1

it was super effective

@obe one sec sorry

0celo7

that was low

@obe which term is confusing?

obe

3.8 makes sense to me.

3.7 does not.

0celo7

you have the first one from the partial hitting the fraction

then the second one from the second term in the first equality

obe

00:30

huh

0celo7

remember how the partial derivative of a vector transforms

the first equality. the first term turns into the first two terms in the second line, the second term in the first line is the last term on the second line

obe

i do.

that's what I don't understand

why does the first term turn into the first two terms?

specifically the second term.

it's either idk how the second term forms or idk how the first term forms

why is the first one transforming like a covariant derivative?

0celo7

because the transformation of a partial derivative acting on a vector

remember that gives you two terms

(Zee makes a big deal out of this)

obe

hmm

though i remember from 2.18

that it's only one term.

0celo7

no you need the one acting on a vector

I thought we did that yesterday

2.38

obe

00:34

right.

fail.

2.18 isn't a vector?

FenderLesPaul

failure is the road to success

and success is the road to Insomnia cookies

mmmm Insomnia cookies

0celo7

no that's how the operator transforms

well partials are vectors

ok Carroll is shit

I would like to say Straumann defines this all very nicely using parallel transport

ACuriousMind

@FenderLesPaul You have a problem. ;)

obe

I think I sort of understand it.

0celo7

@ACuriousMind maybe you can explain this to the lad

obe

00:37

It's ok I'm re-reading after ch3 to fix my issues.

@0celo7 to me?

0celo7

unless you understand it of course

ACuriousMind

@0celo7 My "explanation" of Christoffels is that they are a $\mathfrak{gl}(n)$ gauge field :P

obe

idk if I do

FenderLesPaul

@ACuriousMind this is true

0celo7

@ACuriousMind well yes

obe

00:39

@ACuriousMind I'm drowning now.

0celo7

I need a good fiber bundle book!!!

FenderLesPaul

dives in to save obe

0celo7

Halp!!!

FenderLesPaul

dies

skill patrol

-__-

obe

00:40

@skillpatrol Sorry pal if you're patrolling for skill here, now is not a good time as their is no evidence of skill in my questions.

bolbteppa

Christoffel symbols are very nicely explained here books.google.ie/…

skill patrol

@obe :D

0celo7

"viewing limit"

how is that calculated

ACuriousMind

@0celo7 Just reload

skill patrol

^

0celo7

00:42

wtf

that's so dumb

ACuriousMind

Yep

skill patrol

^

0celo7

GTM Fibre Bundles should be interesting

ACuriousMind

@0celo7 nice!

That probably took a while to draw in TeX

skill patrol

beautiful

0celo7

00:51

> 7. Functorial Description of the Homotopy Classification of Vector Bundles

ok wrong book

all I want is a good description of how the connection form is intrinsically defined

@ACuriousMind ?

bolbteppa

Can you help me fix the following awesome Lie-representation interpretation of solving the Klein-Gordon equation, based off pages 438 and 441 of Woit
http://www.math.columbia.edu/~woit/QM/qmbook.pdf
?:

ACuriousMind

@0celo7 Have you read the Wikipedia article for an Ehresmann connection

bolbteppa

I give you a manifold (Minkowski spacetime) and it's Lie group (Poincare group). I find it's lie algebra (the Poincare algebra), which I view as a representation of the Lie group, build it's universal enveloping algebra (representation), and construct a Casimir operator (an element of the center of the UEA, i.e. the Simplest Group-Invariant Differential Operator).

0celo7

@ACuriousMind I think so

I'll read it again

bolbteppa

This Casimir is called the Klein-Gordon operator, and setting up an eigenvalue equation for this Casimir give us irreducible representations of the group, which are lie algebra elements. By going to dual space we can turn this Casimir eigenvalue equation into a PDE which can be solved normally, giving us irreducible representations of the dual Poincare group.

Now the functions $\psi(x), \pi(x)$ and their commutation relations provide a unitary representation of a function on the dual poincare group. By combining those $\psi(x), \pi(x)$ representations of the dual Poincare group in a certain way and integrating over all space we find a representation of the original Poincare group (this is the Hamiltonian), and this has something to do with Intertwining operators (i.e. equivalent representations).

Now for some reason the Hamiltonian (representing the original) is in the same space
as the $\psi(x), \pi(x)$ which represent the dual.

It's basically putting formal mathematical words on each step of working out the Free Klein-Gordon in QFT

ACuriousMind

01:00

@bolbteppa Your terminology seems a bit off - irreducible representations are not "Lie algebra elements" (although they may be classified by their weights, which are duals of the Cartan subalgebra), what is a "unitary representation of a function on the dual poincare group" meant to say, and how "is" a Hamiltonian a representation?

0celo7

what are those arrows

ACuriousMind

@0celo7 Something that appears because MathJax doesn't play nice with multi-line posts.

bolbteppa

@ACuriousMind well on page 438 Woit says Field operators $\psi(x), \pi(x)$ and commutation relations between them provide a representation of a Heisenberg Lie algebra, a "unitary representation of a Heisenberg Lie algebra on $M \oplus R$, where M is the dual of the space of solutions of the Klein-Gordon equation", so $\psi(x) = \dfrac{1}{(2 \pi)^{3/2}}\int a(p)e^{i \vec{p} \cdot x} + ..$ (p.438, eq. 41.11) is a representation of a function over the dual of the Poincare group right?

NeuroFuzzy

Yessss for once in my miserable little life a dual OS install went off without a hitch and for once I have all my data backed up properly.

how I feel files.doobybrain.com/wp-content/uploads/2009/08/…

0celo7

@NeuroFuzzy ::runs 2 OSs::

welcome to the club

ACM15sok00L is an "excellent" password for 1Password

NeuroFuzzy

01:14

I just moved everything to an ssd so my ubuntu boot time is a few seconds :D I should time it.

FenderLesPaul

classes start in a week

omgbbq

I can't wait to make new frieeends aaand play hide n seeek

aaaand eat lots of ice cream

0celo7

@ACuriousMind that wiki article is familiar to me

skill patrol

is this your final undergrad year?

FenderLesPaul

@skillpatrol yep!

and then I can pursue my career in music

skill patrol

nice

0celo7

01:18

what I really want is an explicit construction of the connection form I guess

do mathematical music

write a book on GR topology in music

skill patrol

the music of the spheres :P

0celo7

lol

that might be before his time

FenderLesPaul

I don't get the reference :(

0celo7

sucks to be you

bolbteppa

Have you guys looked at this book amazon.com/The-Topos-Music-Geometric-Performance/dp/3764357312 he does music with topology and sheaves, wtf...

skill patrol

01:20

Musica universalis

Musica universalis (lit. universal music, or music of the spheres) or Harmony of the Spheres is an ancient philosophical concept that regards proportions in the movements of celestial bodies—the Sun, Moon, and planets—as a form of musica (the Medieval Latin term for music). This "music" is not usually thought to be literally audible, but a harmonic and/or mathematical and/or religious concept. The idea continued to appeal to thinkers about music until the end of the Renaissance, influencing scholars of many kinds, including humanists. == HistoryEdit == The Music of the Spheres incorporates the...

0celo7

you should have been no-lifing this place for longer then

:2354266 are you jk or do you not get it either

FenderLesPaul

ohhh

0celo7

that's not it

@ACuriousMind can you use your chat memory + 20k powers pretty please??

FenderLesPaul

The topos of music

I want that to be the title of my EP

skill patrol

EP?

FenderLesPaul

01:22

track 1, song 1: Simply disconnected

bolbteppa

I'll just say most of your favourite albums were done with Pro Tools, fact...

FenderLesPaul

@skillpatrol like a debut album

skill patrol

right

FenderLesPaul

1

Q: Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

In local quantum field theory one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum observables of the theory. This structure is commonly referred as a presheaf. Why are the states over the open sets not a sheaf structure?

quantum-field-theory mathematical-physics

askin the real questions

0celo7

screw that guy

he could be doing things like curing cancer

physics nerds are so selfish

user54412

01:24

@0celo7 At the largest state schools, how else are they going to teach half the incoming freshmen some basic subject? Often the prof lectures in one hall and they pipe the video into other lecture halls. Of course these days they might just say "don't go to class; download the lecture here."

0celo7

Feb 21 at 18:08, by ACuriousMind

I want to listen to the hypersphere. That is my goal. We can hear sums. So if I add up every single point on the sphere I can hear it. Those graphs are the 3d one. The equation to number 5 is shown above. — user1698948 21 mins ago

skill patrol

@ChrisWhite but, 5,000 really?

0celo7

@ChrisWhite I guess my school isn't that big, only 27,000. Aren't some upwards of 100,000?

the army of TAs lol

FenderLesPaul

idk about 100,000

0celo7

"you take this 100"

FenderLesPaul

01:25

but Arizona state has like 60k

0celo7

"oh these guys look smart"

FenderLesPaul

5000 in one class?

skill patrol

that is the claim

0celo7

hmm I thought some were a lot larger than that

ACuriousMind

@bolbteppa Hm, the $\psi$ is an element of $M\oplus R$, I think. The $M$ is the sum of duals of representations of the Poincare group (since it's the sum of duals of solutions to the KG eq.), so $\psi$ is also an element of the representions, not a function on it, I think, but I would have to read more of that thing to retrace your thoughts. I guess we're quibbling over terminology, anyway.

0celo7

01:27

Ivy Tech Community College has 100,000

skill patrol

2 hours ago, by Chris White

Perhaps some prof linked to it in a class with 5000 students?

user54412

@HDE226868 I actually haven't heard of it, and it seems to have neither a Wikipedia entry nor any mention in the big orange tome of astrophysics. I'm not saying it's not a thing, just that it's niche.

0celo7

1,400 pages!

skill patrol

:O

0celo7

$170!

FenderLesPaul

01:30

What the heck is ivy tech community college?

0celo7

Not as bad as MTW I guess...

FenderLesPaul

sounds like 3 different colleges in one

0celo7

@FenderLesPaul Baby Ivy League.

user54412

@HDE226868 What I can tell you is that stellar winds are hard. You can come up with basic models that get basic results (for instance predicting velocities at infinity to be about the escape velocity from the photosphere, in I think the optically thick limit), but reality is much more complicated.

FenderLesPaul

Reality is an enigma

skill patrol

01:31

Enigma is reality

user54412

@HDE226868 One of the biggest problems is that many winds are line-driven -- the momentum transfer is due pretty much entirely to resonant absorption by e.g. iron species. Such winds depend sensitively on the strengths of the lines and the ionization state of the gas, in addition to the usual Rayleigh-Taylor instability and other (M)HD concerns.

skill patrol

@FenderLesPaul nytimes.com/2000/11/17/nyregion/…

0celo7

you sure do know a lot about this space stuff

Does anyone here use a password manager to generate and store secure passwords?

My father uses 1Password.

user54412

My critical passwords are known only in my head. Everything else is written in plaintext files or on scattered pieces of paper.

user54412

(don't tell the IT people who run the clusters)

0celo7

01:40

::closes gmail::

Ok...

bolbteppa

@ACuriousMind I'd like to quibble over terminology haha, and I'm most likely wrong, but what I mean by calling it a function is that it looks like the Fourier expansion of a function, i.e. a continuous linear combination of irreducible representations of the dual, in the way that functions $e^{ax}$ form a representation of the additive group of the reals & a fourier integral takes the place of this integral, e.g. $\mathbb{R}$ is the space of solutions $e^{iax}$ just like $M$ is for Poincare

user54412

Haha -- unless you sent from a known address and encrypted the email with a trusted public scheme, they probably wouldn't read it anyway. Those people live in their own little world.

0celo7

They won't read an email from a univ.edu account? Really?

For real, how do sheaves and presheaves help anybody?

ACuriousMind

@bolbteppa Hmmm...I guess $\psi$ is a "function" (or rather, distribution) on $M\oplus R$, since the $a(p)$ and $a^\dagger(p)$ are elements of $M$, if I understand this correctly, and they are Fourier transformed here. As a distribution, it acts linearly on $M\oplus R$, and hence from the commutation relation it is the representation of an element of a Heisenberg algebra acting on a representation of duals of the Poinvare group.

0celo7

bah, distributions

bolbteppa

01:51

Sheaves are things you do algebraic geometry on just like manifolds are things you do differential geometry on, but on a manifold you define a global manifold and then go down to local open sets, while sheaves let you define local stuff and string it all together to define something global, e.g. analytic continuation and stuff

ACuriousMind

@0celo7 They are a very general way to understand what "attaching local information to a space" means, and they appear naturally in almost every geometrical setting (e.g. every complex manifold carries the sheaf of analytic functions), and many statements about "finding a global section" or somesuch can be seen to be cohomological statements through them.

In a way, they also are the proper generalization of line and vector bundles

0celo7

what is "somesuch" in German?

I've never seen anyone besides you use that.

ahnliches?

with umlaut

ACuriousMind

@0celo7 derartiges, I think

It's an English word, though: en.wiktionary.org/wiki/somesuch

0celo7

I know it is, but not common.

indeed you are the most frequent user in this chat

bolbteppa

@ACuriousMind thanks yeah I think that's what's going on, will think about this overnight, thanks for the help

user54412

01:56

@ACuriousMind Those definitions! "some such thing" "some such"

0celo7

It seems somesuch surged in popularity in 1986 and is now declining.

"some such" was on a plateau but is now declining.

skill patrol

It's a cool word.

user54412

pics or it didn't happen

user54412

(yes I'm that lazy)

ACuriousMind

@ChrisWhite Yeah, I don't know what that page gives someone who didn't know the meaning beforehand

0celo7

01:58

Overall, @ACuriousMind's spelling is disfavored.

user54412

"somesuch" sounds like "an sich" to me -- does anyone use the latter outside discussing Kant?

0celo7

"an sich" is German, no?

Er hat das an sich getan.

ACuriousMind

Yes, an sich is German, and used outside of discussing Kant here :P

@0celo7 That doesn't make sense

0celo7

@ACuriousMind well it does to me

go play with sheaves

"he did that to himself"

ACuriousMind

@0celo7 That'd be: Das hat er sich selbst angetan. "An sich" roughly means "on its own" as in "That wouldn't be bad on its own, but..." - Das wäre an sich nicht schlecht, aber...

0celo7

02:03

hmm, that makes sense to me too

I think your German is rusty, pal.

user54412

@ACuriousMind I'd prefer to keep my image of Germans always discussing Kant in every conversation.

ACuriousMind

@ChrisWhite Sometimes we also talk about Nietzsche ;)

0celo7

wow a 16 digit alphanumeric caps varying password is "good"

I'd hate to see a great password

user54412

You probably need non-printable characters for that.

Kyle Kanos

correcthorsebatterystaple

skill patrol

02:08

^

0celo7

well I put in a 16 digit password and I get the error "password must be at least 8 digits"

notallclosedformsareexact

@ACuriousMind would appreciate that one

user54412

Have you ever seen the security.SE folks get up in arms about passwords that can only draw from 60 or 70 characters? It's kind of cute how little they understand.

0celo7

how little they understand?

skill patrol

cute?

:P

user54412

@0celo7 they think that by denying them obscure characters, you deny them strong passwords, regardless of how long you let them make the passwords

ACuriousMind

02:10

@ChrisWhite Unicode for all!

0celo7

@ChrisWhite seriously?

ACuriousMind

@0celo7 I'd rephrase that as NonVanishingDeRhamCohomology ;)

0celo7

@ACuriousMind yourfaceisanonvanishingderhamcohomologyalsoscrewsheaves

ACuriousMind

@0celo7 Screwing sheaves could be fun, there even are perverse sheaves

0celo7

lol

skill patrol

02:14

for pervs :P

0celo7

I remember HE saying something about certain spacetimes being perverse

they're just so not nice

user54412

Are perverse spacetimes correlated with naked singularities?

0celo7

Good one.

ACuriousMind

Perhaps they also have weird bound states.

skill patrol

@ChrisWhite warning: do not google that

0celo7

02:17

Wow, you really are into BDSM.

Kyle Kanos

Well if you've got an 8 character long password generated from only letter, there are 26^8 possibilities. If you allow alphanumeric characters, it jump to 36^8; adding case sensitivity, you get 62^8; adding punctuation, you get 94^8 possibilities

0celo7

is that a lot

Kyle Kanos

So there is, in some sense, an argument that increasing the allowed character set would increase password possibilities

0celo7

just use 16 characters then

ACuriousMind

@0celo7 I just enjoy making your mental image of me as weird as possible

0celo7

02:19

@obe do you need anything before I either sleep or read or etc.

Kyle Kanos

But $94^8\sim6\cdot10^{15}$, which probably can be broken in a few minutes

0celo7

@ACuriousMind I'll be in the Heidelberg area in December!

Kyle Kanos

(if that)

0celo7

Maybe.

I'll be in Germany and Germany is not that big.

ACuriousMind

@0celo7 Around Christmas or earlier? (Since I'll not be here around Christmas)

0celo7

02:21

@KyleKanos How do you break a password like that if you get locked out after 3 tries?

Kyle Kanos

There is that

0celo7

@ACuriousMind I'm going for 2-3 weeks, Christmas included.

Kyle Kanos

I'm not a white hat, so I cannot say

0celo7

Where are you going over Christmas?

Kyle Kanos

Nor am I a black hat, for that matter

0celo7

02:21

It might be closer to where I am.

ACuriousMind

@0celo7 Home, to Wuppertal

0celo7

I'll be 18 so I can drive? Maybe?

IDK about German tourism driving laws.

user54412

@KyleKanos I blame the 8-character limit. No reason to restructure every level of an application just so you can include 0x07 as input or something bizarre.

0celo7

@ACuriousMind I thought that l was a ! and you were really enthusiastic about Wupperta :D

Oh I'm not driving 3 hours to see you.

On the other hand Hberg is an hour, that I'd do.

ACuriousMind

Yeah, it's a bit far from Heidelberg (which is why I don't go home all that often)

0celo7

02:24

I'll be in the K-town area.

Also Stuttgart, Munich.

user54412

Europeans' conception of "a bit far"...

0celo7

Then excursions into Czechia, Slovakia.

ACuriousMind

Oh, Munich, you could see @Danu!

0celo7

Austria.

Frankfurt.

We have fam/friends all over.

Frankfurt is 2:30 to Wuppertal.

So if we fly in to Frankfurt, then I could drive up...that's a bit much.

ACuriousMind

@ChrisWhite We've settled our land a bit more densely than you have :P

0celo7

02:28

My dad lived 4 hours away from home while in college. Every weekend he made the trip back so he could work on the farm.

@ACuriousMind you're a slacker.

ACuriousMind

Always have been :)

skill patrol

Always will be?

ACuriousMind

Probably

0celo7

@ACuriousMind When are you going to go home? If I have exact dates, I will seriously try to make it work.

Heidelberg can be a cool day trip from K-town.

ACuriousMind

@0celo7 Hmm...probably on the 20th of December, no earlier, perhaps one or two days later

0celo7

02:30

ooh, that might work if we come in via Frankfurt.

ACuriousMind

I should be writing by master's thesis by then, no idea what kind of presence here that will require

0celo7

But if we fly in via Vienna, we'll do the Munich, former bloc stuff first.

The problem is that you'd probably intentionally mismatch your socks.

1

Q: Distinction between state space and space of functions

In Quantum Mechanics a particle is described by its wave function $\Psi : \mathbb{R}^3\times \mathbb{R}\to \mathbb{C}$. In that sense, the state of a particle at time $t_0$ is characterized by a function $\Psi(\cdot, t_0) : \mathbb{R}^3\to \mathbb{C}$. The space of all functions like that which a...

quantum-mechanics hilbert-space

@ACuriousMind I want to say Stone's theorem has something to do with it.

ACuriousMind

@0celo7 Well, one could talk about Stone-von Neumann here to say that it really doesn't matter which space you take as long as it has the canonical commutation relations on it

0celo7

Like the space of $\Psi$ is constructed by doing $\langle x|\psi\rangle$ as usual and this is an isomorphism because $\langle x|$ is unique?

wtf

I did not see those answers

that's majic

ACuriousMind

@0celo7 I rather think it is an isomorphism because it identifies $\langle x_0 \rvert$ with $\delta(x - x_0)$, so it is an isomorphism between the larger parts of the Gel'fand triple simply because it sends a generalized orthonormal basis to a generalized orthonormal basis

0celo7

02:39

Ok, whatever that means :P

I need functional analysis

@ACuriousMind huh, so this happened in the centipede building

obe

@0celo7 Right now no, thanks though.

I'm reading QM, and it's straightforward.

I'll look at that section in Straumann later today.

0celo7

later today?

you need to sleep!

$\text{no sleep}:\text{person}\mapsto\text{vampire with mismatched socks}$

obe

I will eat insomnia cookies until I finish QM.

0celo7

actually scratch de Felice from the list

his approach is really complicated

obe

Ok.

0celo7

02:56

also extend the chapters of Gougoulhon to 1, 14, 15, 16

obe

@0celo7 Frames or 3+1?

0celo7

Frames

Transcript for