Mathematics - 2021-05-04 (page 1 of 2)

@copper.hat I was able to find a board that did 320x200 pixel graphics (that was the whole screen in those days) and wrote some 3-D graphics routines for it.

copper.hat

very nice. my dad got the pet second hand with no manuals or anything. we were total neophytes

robjohn

I may have mentioned that before. Sorry for the repetition, but that's how I make sure I remember my past.

copper.hat

same here :-)

i get the eyerolls as reminders

the next was 8085 i think

rtos stuff. i was in college then

Ted Shifrin

12:38 AM

Hi @robjohn

robjohn

I went to the 68000 family, because that is what Apple used

copper.hat

my advisor got a pl/m compiler.

robjohn

Hey, Ted!

copper.hat

that was way outside our range then :-)

the intel progression was incredible.

robjohn

The Apple II used the 6502 as well, but the Mac used the 68000 and up

copper.hat

12:39 AM

the 68k was pretty neat

but i had very limited access. the folks who had it knew i would open it up at the first opportunity

robjohn

I think it was cool. I was kind of sad when Apple moved to RISC, but it was probably the right thing to do

copper.hat

risc in incredible, but boring in some sense

a bit like circuit design has become

robjohn

It was fun to write for the 68K, not so much for the RISC processors

So I learned C and Java

copper.hat

the sense of exploration was what i enjoyed most

robjohn

as much as there is to learn about programming languages

leslie townes

12:41 AM

i was sad when 320x200 stopped being the main color graphics resolution on pc. smaller pixels don't look better.

and they were slightly non-square, which drove me mad. i thought there was something wrong with my 3d math.

but i loved it.

copper.hat

ok, here's one that will get me removed:

robjohn

The PET and Mac used square pixels, I believe. I would have gone wiggy if they didn't

copper.hat

What's the difference between light and hard?

jobs would not allow non square pixels.

robjohn

ok, what?

copper.hat

Most men can sleep with a light on.

robjohn

12:43 AM

ouch

copper.hat

sry

that will test the algorithms.

robjohn

Okay. I have two dogs who look as if they are going to eat me if we don't go out for a walk

I doubt that would be picked up by anything

copper.hat

:-)

leslie townes

you could use an undocumented mode of a vga adapter for square pixels. or some could, i mostly got weird dots and lines and snow.

copper.hat

i spent my time trying to build a d2a

robjohn

12:45 AM

cu in a 1

copper.hat

enjoy the walk :-)

robjohn

1:23 AM

wow... it's as if hadn't left

@copper.hat I did a 5 channel d2a which I used to play music on the PET.

sounded like a calliope

Ted Shifrin

You left?

I worry more about our Indian friends. Seriously scary.

robjohn

Their stats were good for so long, now the spit hits the fan

Ted Shifrin

Ours may be more fragile thanks to the Trompthugnicans.

Andrew Micallef

2:40 AM

Yo robjohn, i'm trying to solve that number of regions enclosed by a $K_n$ graph inset in a circle problem you eluded to a few weeks ago. It is a good one (dont give it away though i am no where near a solution)

Ted Shifrin

Oh, Andrew is back.

robjohn

@AndrewMicallef there is a solution on the site for that.

I won't link to it because that would give it away

Andrew Micallef

Ahh!

Ted Shifrin

Most questions have a solution on the site.

Andrew Micallef

;)

Hiya, yeahI actually have a dumb question

robjohn

2:43 AM

@TedShifrin I wrote one of the solutions to the problem to which he alludes

Ted Shifrin

A different one? :)

Andrew Micallef

How do i write the equation of a line given two known points :)

Ted Shifrin

That's how you knew so quickly, @robjohn.

robjohn

yeah

Ted Shifrin

Seriously?

Andrew Micallef

2:43 AM

(Im trying to get a geometric solution)

Ted Shifrin

What does equation mean, @Andrew?

Andrew Micallef

Yeah I forgot. I know y=mx + b

Ted Shifrin

Where do your points live?

Andrew Micallef

But when i have x1 x2 y1 y2 there was a way if writting that ad a line equation

They live on a circle

Ted Shifrin

In the plane?

Andrew Micallef

2:45 AM

Yeah

Ted Shifrin

Hint: SLOPE.

Andrew Micallef

Was it deltax = deltay?

Ted Shifrin

Draw a picture.

leslie townes

don't do that, it's a trap

robjohn

@TedShifrin just a joke

Ted Shifrin

2:47 AM

lets leslie deal with it

I didn't get it, robjohn.

robjohn

it was a non politically correct joke

Ted Shifrin

Of. Course.

robjohn

. o O ( slope )

leslie townes

i don't think i get it either.

Andrew Micallef

Geez i feel like my brain has already eaten itself

This shouldnt be hard

I drew a picture, have dy / dx which is slope

robjohn

2:50 AM

do you know the formula for the slope?

yes

Ted Shifrin

slaps robjohn

copper.hat

@robjohn the linearity on mine was a joke, so that was a failed experiment.

Andrew Micallef

I work with a vietnam vet who gets 0 respect

Ohhh

I plug in coordinates from one point to get b

robjohn

slope $=\frac{y-y_0}{x-x_0}$

Andrew Micallef

Yep my brain did eat itself

leslie townes

2:53 AM

if my bank makes me create new login credentials to 'sync' my savings with the investment brokerage they sold their investment arm group to one more time in the next month i'm going to go nonlinear.

robjohn

@leslietownes that is the same as a password you used before. try a new one

leslie townes

it's 'too close' to the old ones. tell me the rules. this is like being in algebra 1 again.

tell me what characters have to go in it before you ask me to choose a new one.

Andrew Micallef

Thanks for humouring my senility. Gotta get back to work

robjohn

@AndrewMicallef: so you have the equation?

copper.hat

@leslietownes i found a solution to that

leslie townes

2:56 AM

i have a heuristic to generate new passwords from old. but it isn't working with Schwab. talk to chuck, my a$\star \star$.

copper.hat

so far i have been happy with schwab. modulo a few margin calls.

and they used to waive my $25 wire fee, now they always charge it

leslie townes

i think it's my bank. they wanted to get out of investment but not lose customers, so they kludged away to 'sync' portions of schwab into my banking.

Ted Shifrin

So I am being told by mods that deletion by OP once an answer is arrived at in comments (as opposed to a posted answer) is perfectly legitimate. I'm fed up with these blokes.

leslie townes

and it just keeps breaking.

Ted Shifrin

I just will quit commenting and answering.

copper.hat

2:58 AM

i am with you there

robjohn

@TedShifrin I don't like that, but we don't always agree

leslie townes

things have gotten really weird lately when we all agree.

copper.hat

i don't like feeling i have to fit into the borg

Ted Shifrin

Well, I'm about to quit, seriously.

Andrew Micallef

@robjohn yeah yeah I do; my strategy, when I get more time is to try getting all the intersections of each edge, and construct faces from there somehow (havent fully thought it out yet)

copper.hat

2:59 AM

if there was some consistency it might be a little easier to deal with.

leslie townes

like schwab. just tell me the rules. i bet it's all connected.

copper.hat

today was a bad day. no vaccine to blame. son has #2 tmr

you are trying to link an outside institution with schwab?

its been a long time since i did that.

i have a customer, and whenever i need them to restart our software, they need to effectively login to 32 or 64 machines at startup (not exactly, but sortof). anyway, as part of their security setup, each new login needs a fingerprint confirmation. so they need to sit there, wait & tap 32 or 64 times at strtup.

progress.

Euler 2

Seems like I love old things

Lisp is attracting me

copper.hat

i like lisp, esp. scheme. it is consistent

but other than a few places, it is not much used any more. not in fashion

robjohn

(I (used (to (know (lisp)))))

leslie townes

3:10 AM

copper, i achieved success, but only after some unprintables.

my daughter was running in and out the whole time and grabbing things off my desk. mades it really easy to type in all the 2FA codes and work out the rules of the password.

copper.hat

i have found most systems, after a fashion, allow passwords of the form blah<n>otherblah where the <n> isreplaced by 1,2,...

maybe the sequence needs to be a littl more creative.

python is my poor man's lisp replacement.

at least you can do some form of closures there.

it had functional programming before it was fashionable.

leslie townes

yeah that's awfully close to what i use.

that password schema, not python.

copper.hat

passwords are a weak link in security but hard to avoid.

leslie townes

anyway, schwab and i are back in business. i just put $3000 on the ponies. i think that's what i logged into, anyway.

copper.hat

hmm, unfortunately my schwab action today could have supported a herd of ponies.

i'm too old for this stuff.

i don't understand why forex transfers for small amounts are so painful. it typically takes about 10mins on the phone, numerous confirmations and a lot of recorded agreeing to.

leslie townes

3:21 AM

copper i have an interesting cryptocurrency opportunity. it's called LeslieCoin. and saying it's an opportunity is really an understatement.

this is not an issue with LeslieCoin.

transactions are very quick. seamless, even.

copper.hat

the best of finagletech?

since i'm in a stinky mood, and i'm here, here is what i like least about mse

the demphasis of polite human interaction.

Euler 2

The best password is NIhSdslIw164WEg6hD/yJw==

copper.hat

there. that's it. not a single person who keeps hounding me.

Euler 2

Decode it :)

copper.hat

its too short.

Euler 2

3:25 AM

Yes i know that

copper.hat

too much entropy

Ted Shifrin

Copper, are you saying I'm impolite or inhuman?

copper.hat

the complement

which is a compliment, i suppose

Ted Shifrin

Yes. Interestingly, out of nowhere I got a meta thanks from a Brazilian geometer I've helped over the years — on a 2013 meta post about this same shit.

copper.hat

That is nice.

Ted Shifrin

3:29 AM

I wonder why.

copper.hat

The closer you look the uglier things get.

Ted Shifrin

My post to the mods surely isn't known.

Sorta like US lying politics.

Actually, I had noticed him posting some answers to things he has learned, so I was sorta proud.

copper.hat

It is good. Teach a person to fish and all that.

leslie townes

then there won't be any jobs for people who need to teach fishing. think it through.

copper.hat

yeah, look what brexit did

leslie townes

3:34 AM

my advisor always said that mathematics is a hoard of jewels that you must guard with your life. then something about never smartening up a chump.

robjohn

@TedShifrin meta post?

copper.hat

wow. not sure i am entirely in agreement with your advisor's sentiment.

leslie townes

i may be blending together a few things that he didn't say.

copper.hat

maybe you heard 'c' instead of another letter?

leslie townes

he said this while crying into his whiskey after i somehow graduated.

no it's definitely chump. this isn't ireland.

copper.hat

3:36 AM

:-) laughing

maybe it was his groening moment

that really does not sound good when you say it

Ted Shifrin

@robjohn No, in order to flag again I had to dispute a message from mods on why they denied my last flag request.

robjohn

ah

Ted Shifrin

Who knows. I'm pretty fed up.

copper.hat

Regardless, you are of great help to lots of folks here.

Ted Shifrin

And then they delete.

Sodam is one of em. Although I only help his whining self here.

Covid is partly at fault, but the culture had negative derivative long before.

copper.hat

3:43 AM

Pronouncing that username is about as close as my dad ever came to cursing.

Ted Shifrin

Think middle east.

copper.hat

I know. I am not very pc.

I dislike non positive electronic interactions. If it is going in that direction I much rather do it in person. So much can get out of hand electronically.

leslie townes

you wouldn't say that to my face.

copper.hat

surely behind your back is in front of your face?

leslie townes

it's all relative.

copper.hat

3:51 AM

unrepeatable joke popping into my head.

took 4 attempts to get google to give me directions to the correct entrance for vaccination

i mean freeway exit.

leslie townes

oh i had this experience on a hospital campus. they sent me on an eulerian path.

they were all trying to help, the directions were just useless, and no signs for a covid injection clinic. why would you want people to find it on the campus of a hospital is beyond me.

copper.hat

that is why i want to go to kaliningrad (after broom bridge, of course)

my kids (& wife) used to get mad at me when asking directions because i would basically ask everyone i encountered (slight exaggeration) as i reckoned people's desire to help often outstrips their knowledge

i'm surprised Euler didn't jump in at the mention of kaliningrad .

leslie townes

i once gave someone directions in berkeley and reversed a key direction down shattuck avenue. my wife was there and said nothing, and then when the woman left my wife was like "isn't it the other way?" great timing.

not on purpose i just mix up my left/right

copper.hat

my brother was getting directions to somewhere in kerry, and the gist of it was, "so, you know place XYZ?", "yes...", "Ok, if you get there you have gone too far".

So the idea was to overshoot and then backtrack, but they had a funny way of putting it.

leslie townes

"you can't get there from here" is a perennial favorite

copper.hat

3:59 AM

dang, that was the story i was just typing

but someone did tell us that :-)

i asked how do you know it exists, but got a quick kick in the back of the seat

Ted Shifrin

Maine

leslie townes

seems like every country has its Maine

Akiva Weinberger

4:27 AM

If you pick two points at random from an n-dimensional unit cube, for large n, the distance will be almost surely almost exactly $\sqrt n/6$

leslie townes

truly a result that we are all basically alone in an indifferent universe.

Akiva Weinberger

Therefore, three points at random from a high-dimensional cube almost certainly determine a near-perfect equilateral triangle

Three points in a square determine an obtuse triangle 72.5% of the time. Three points in a cube determine an obtuse triangle 54% of the time. As the dimension goes up, the obtuse triangles go down, as everything slowly becomes equilateral

leslie townes

which succulent plants or fungi do you recommend i consume so that i can see this in action.

Akiva Weinberger

Three points in a 4-cube determine an obtuse triangle 41% of the time

@leslietownes fractal ones

leslie townes

i like the simple result that most of the volume of the solid n sphere, n large, is basically right at the boundary. i think it's just a thing with the gamma function.

Akiva Weinberger

4:31 AM

Not even, it's just a thing with the power function

In a radius 1 sphere of dimension n, 1/2^n of its points are located a distance 1/2 or less from the center

leslie townes

i'm impressed by polynomials in r, i guess i just admitted to one and all.

Akiva Weinberger

and r^n of its points are located a distance of r or less from the center

leslie townes

are these proved with the aid of some integrals and the gamma function? or is it just like the pigeonhole principle or something. throw me a bone here.

copper.hat

it is a consequence of the $L^2$ weak law.

leslie townes

it sounds cool, even if it's somehow not

Akiva Weinberger

4:33 AM

Think about 2D

When you scale any shape up by a factor of 2, its area goes up by a factor of 4

copper.hat

${X_1^2+\cdots+X_n^2 \over n} \to {1 \over 3}$ in probability as $n \to \infty$.

Akiva Weinberger

In 3D, when you scale up a shape by a factor of 2, its volume goes up by a factor of 8

leslie townes

or so you would have me believe. it's starting to make more sense.

Akiva Weinberger

So the formula for the volume of a radius R sphere in n dimensions is [some constant]*r^n

copper.hat

$X_k$ iid uniform on $(-1,1)$

Akiva Weinberger

4:35 AM

and I just care about Vol(r)/Vol(1) to find what proportion of a sphere's volume is distance r or less from the center

(Vol(r) meaning volume of radius-r n-sphere)

copper.hat

i'm still on the first statement.

Akiva Weinberger

Which statement

leslie townes

In the beginning,

Akiva Weinberger

9 mins ago, by Akiva Weinberger

If you pick two points at random from an n-dimensional unit cube, for large n, the distance will be almost surely almost exactly $\sqrt n/6$

This one?

Proof sketch: $\int_0^1\int_0^1(x-y)^2\ dx\ dy=\frac16$, go from there

copper.hat

yes. most points in an $n$ dimensional cube are on the boundary of a ball.

Akiva Weinberger

4:38 AM

If you pick a random point in an n-dimensional unit cube its distance from the origin should be $\sqrt n/3$ I think

Oh crap

I meant $\sqrt{n/6}$ in the first instance

and $\sqrt{n/3}$ in the second

You see the difference there: the first is picking two points in the cube and finding the distance to each other; the second is picking a point and finding its distance to the center

copper.hat

this is an exercise in durrett's probablity: theory & examples

Akiva Weinberger

(the angle AOB is gonna be around 120 degrees)

(A=first point, O=origin, B=second point)

er is that right? hold on

no that doesn't make any sense

uhhh don't quote me on this but I think the angle is gonna be $\arccos(3/4)\approx41.41^\circ$

copper.hat

in $n$ dimensions???

Akiva Weinberger

yeah

i could be wrong

leslie townes

that real number does happen to be gamma of something.

Akiva Weinberger

4:45 AM

I feel like AOB should be obtuse half the time

copper.hat

i am not sure why it would not be $90^\circ$ on average?

Akiva Weinberger

yeah i must have made a mistake somewhere

OHH

I see where I went wrong!!

The distance to the corner (any) will be $\sqrt{n/3}$ on average, not to the center!!

And so it's really ACB that I was measuring, where C is a corner

leslie townes

math.stackexchange.com/questions/4126392/… love to see the effort really go in.

if anybody in here did that i owe them a bottle of wine.

Akiva Weinberger

And the distance to the origin will be $\sqrt{n/12}=\frac12\sqrt{n/3}$, assuming this is $[0,1]^n$ and not $[-1,1]^n$

copper.hat

i'm tempted to provoke my provoker

Akiva Weinberger

4:51 AM

and finally: the angle AOB will be almost always almost exactly 90 degrees

copper.hat

$n$ not $2$, presumably

leslie townes

it's one of those things where the more you look at it you begin to notice more stuff.

Akiva Weinberger

which actually makes sense

And, like, Pythagoras checks us out

leslie townes

it's a phone screenshot of an exam due next week, they didn't even crop it, and there's random numbers and List Item at the bottom. i'm selling NFTs of this.

Akiva Weinberger

because $\sqrt{n/12}^2+\sqrt{n/12}^2=\sqrt{n/6}^2$

By the way

copper.hat

4:53 AM

well, to be fair, there is really no hint when you ask a question that such things are not appreciated

i know basic common sense and all that

Akiva Weinberger

There is a dimension n where a cube of sidelength 1cm and sphere of radius 1000km have the same diagonal

leslie townes

it's efficient. i bet this person wouldn't overbill by the hour.

Akiva Weinberger

Obvious when you think about it (the former is 1cm*sqrt(n) and the latter is constant 2000km so of course they eventually cross)

copper.hat

i need to learn from her

Akiva Weinberger

Once the diagonal of the cube is around $\sqrt{2\pi e}\approx 4.132$ times the radius of the sphere, they start to have the same volume

(proof involves Stirling's formula)

Therefore, there exists a dimension n where a cube of sidelength 1cm has the same volume as a sphere of radius 1000km (mind the units!)

leslie townes

4:56 AM

stirling's formula, we all know what that's related to.

Akiva Weinberger

This is in dimension $1.7\cdot 10^{23}$

Oh it's a fun one

leslie townes

avogadro would almost be pleased.

Akiva Weinberger

Stirling says $n!\approx \sqrt{2\pi n}(\frac ne)^n$

leslie townes

i know n! by another name. i'm sorry, this bit is tired, i'll give it up.

copper.hat

i like avogadro on toast

Akiva Weinberger

4:58 AM

If you toss a fair coin 2n times, the odds that it shows up exactly half heads and half tails is asymptotically $1/\sqrt{\pi n}$

Proof: use Stirling's approximation, you're basically asking for $\dbinom{2n}n/2^{2n}$

So turns out pi shows up randomly in probability!

(heh, "randomly")

Rover

@leslietownes Hi

leslie townes

hello.

Rover

What's the name of your YouTube channel?

leslie townes

i haven't got it yet. still working on how to brand it. might got tiktok instead.

Rover

Oh ok

Akiva Weinberger

5:03 AM

Take a stick and break it into 3 pieces

What's the probability that the pieces can form a triangle?

(You need to make sure no piece is longer than the other two combined)

leslie townes

there's some good constant appearances in shuffling card problems too

Akiva Weinberger

Spoiler: its 1/4

Neat calculation I did: there's a $3\ln(2)-2\approx0.07944$ chance that it's not just any triangle, but an acute triangle

wait

copper.hat

all triangles are equally pretty

leslie townes

yeah, sheesh. i see there's no site rule against triangleism.

copper.hat

i'm justifying it by age

leslie townes

5:08 AM

i'll propose one through appropriate channels

Akiva Weinberger

I think that means that if I break a stick into three pieces $x,y,z$, there's a $\ln 2$ chance that $x>\sqrt{y^2+z^2}$

copper.hat

i think it is simpler than that

Akiva Weinberger

meaning I've labeled one of the pieces, and either it can form the longest side of an obtuse triangle, or its so long that it straight up can't form a triangle

$\ln 2$ is a suspiciously simple result, maybe there's a simpler way to see why that is

Ted Shifrin

@copper.hat Hiss.

Akiva Weinberger

oh no hold on sorry I have it backwards

copper.hat

5:10 AM

you need $P[l_3 \le l_1+l_2]$.

Akiva Weinberger

$\ln 2$ chance that $x<\sqrt{y^2+z^2}$, should be

copper.hat

for a triangle

leslie townes

the draw a picture portion of my brain is winking to life again. this really is obnoxious.

copper.hat

i had my order wrong

Ted Shifrin

@leslie Just have those cells surgically excised.

Akiva Weinberger

5:11 AM

so either it's an acute triangle, or my labeled piece is too short to form a triangle

copper.hat

sorry, i'm not getting into cute or not

Akiva Weinberger

@copper.hat Yeah

Well, plus the other two permutations

Ted Shifrin

Damn, @copper is full of hiss-able things tonight.

copper.hat

no, that is sufficient

Akiva Weinberger

Given that $l_1+l_2+l_3$ is constant (it's the length of the original stick)

copper.hat

5:12 AM

language is such a delicate thing

@leslietownes the assignment question got closed.

leslie townes

i printed out a PDF for my NFT. closed but not forgotten. forever pointed to by a blockchain.

copper.hat

i really think the question asking place could have a sentence or two that would head a lot off at the pass

Akiva Weinberger

alright i need to sleep, night

Ted Shifrin

question asking place?

Night, DogAteMy.

leslie townes

i prefer a list of links to something resembling a legal document, ideally several of them.

copper.hat

5:16 AM

brain is not functioning. Where the OP goes to ask a question.

leslie townes

how about a list of links posted as comments immediately after the question is asked.

copper.hat

Should have a hint that pure homework is not ok.

:-)

leslie townes

you can click through to that. i think article 2 section 5 subsection (c). it's in there.

copper.hat

that is tantamount to what happens, albeit a bit randomly

it feels a bit 'clique-y' to me. (excuse diction)

or some little power grabby thing going on

leslie townes

i never went to a posh boarding school in england in the mid 20th century but there may be parallels

copper.hat

5:23 AM

i suspect my schooling was marginally different to British public schools...

i have an inherent lack of respect for authority

leslie townes

the british public schools are indoors and fewer sheep attend

copper.hat

5:54 AM

it is always hard to separate history from a certain photoshopping born of pride, but there used to be hedge schools held outside. this were illegal, but catholics were not allowed to attend the anglican schools (neither were presbyterians).

well, outside might be an exaggeration

leslie townes

i walked right into that one

copper.hat

ding ding

i've got rid of my mse angst

for today

i need to learn to accept a certain level of subliminal bullying

leslie townes

i adopt the view that none of this is real and even my day job, which ceased being live in 3/2020, is just my CPU come to life

Euler 2

i don't like how se is changing its design

copper.hat

did you let se know?

Euler 2

6:07 AM

@copper.hat they have almost stopped listening to feedbacks

copper.hat

maybe the are listening but doing nothing.

Good morning, folks

good night :-)

haha

Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. == Life and work == Brook Taylor was born in Edmonton (former Middlesex). Taylor was the son of John Taylor, MP of Patrixbourne, Kent and Olivia Tempest, the daughter of Sir Nicholas Tempest, Baronet of Durham.He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees in LL.B. in 1709 and LL.D. in 1714. Taylor studied mathematics under John Machin and John Kei...

copper.hat

6:12 AM

My alma mater was where Boole taught.

Transcript for

$Mathematics$ Mathematics