Mathematics - 2019-09-25 (page 1 of 3)

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12:18 AM

Hello, I was doing some multivariable limits and my teacher suggested converting to polar using x=rcos(θ) and y=rsin(θ) to check if the limit exists (as it measures the limit from all directions). I was working on a problem in which I know the limit doesn't exist, and can't figure out how to parametize to polar.

user image

Akiva Weinberger

Do you even need to do that? Try approaching on the line y=0 and on the line y=x @AfronPie

Remember that $(\sin x)/x\to1$

Ah, ok that makes sense. for the y=0 line I got the limit to be 1, while for y=x it is 2. I was just tempted to parametrize as the denominator was $x^2+y^2$ (it seemed like a good idea).

Thanks @AkivaWeinberger

Akiva Weinberger

12:45 AM

@AfronPie Yeah that was my first thought too but I realized the numerator would be $(\sin(r\cos\theta)+\sin(r\sin\theta))^2$

which isn't a hundred percent useful

Yup same

Thanks again

2 hours later…

2:22 AM

@AkivaWeinberger there is a valid identity there, actually. See the real-valued formulae here: Jacobi-Anger expansion

(That said, valid does not mean productive)

Akiva Weinberger

That seems incredibly unproductive yeah

but yeah that's cool thanks

I thought it was gonna be something using the sum-of-trig identities

which would give 2sin[(rcos+rsin)/2]cos[(rcos-rsin)/2] I think?

squared

And then $\cos(\theta)\pm\sin(\theta)$ is something involving $\cos(\theta\pm\frac\pi4)$ or something similar

Doesn't seem like a useful road

@rapasite yes

Akiva Weinberger

What's the hardest problem whose solution was known to the Greeks?

Maybe superlatives don't make good questions

What are some hard problems whose solutions were known to the Greeks?

(I haven't actually gone through Elements so I'm not so familiar with what's in there)

I know they knew the surface area and volume of a sphere

The fact that they could do stuff with spherical triangles is pretty cool in my book

(I mainly have Menelaus in mind for that )

Akiva Weinberger

Oh for some reason I thought he was German, 'cause of the name

Mene"lay-us", not Mene"louse"

OK

(or me-NE-la-os/osh in Modern Greek)

(but Euclid is efKLEEthees(h) in Modern Greek so whatever)

/efˈkli.ðis̠/

2:49 AM

@Semiclassical a n g e r y

@rapasite yes

Akiva Weinberger

3:07 AM

The 21st century, much more than any other, is the age of hyperbole.

αιλ σήγλω βαιηνντηούνω, μούτσω μας και κουαλκηαίρ ώτρω, αις λα αίρα δαι λα υπέρβολε

@AkivaWeinberger

Akiva Weinberger

Wow

Why did you do this

:P

Akiva Weinberger

I see what you did, I just wanna know why

because they say Greek sounds like Spanish

Akiva Weinberger

3:16 AM

The phonologies are remarkably similar

or are you asking why I chose the long vowels

Akiva Weinberger

Same vowels and /ɣ/, /ð/

I had to think for a few seconds for "ch"

I suppose τσ will do

Akiva Weinberger

That's how they do loanwords

oh nice

also some people pronounce "ch" like "ts" sometimes

Akiva Weinberger

3:17 AM

The /s/ in Greek is retracted so it almost sounds like /ʃ/ anyway

/s̠/

'Bar' in Greek is ΜΠΑΒ

μπαρ

/bar/

No other way to write /b/

(since β is /v/)

(and not /β/ because that would make too much sense)

how would they transliterate "ei" as in "veintiuno"?

Akiva Weinberger

εϊ @LeakyNun

Diaeresis

oh, that exists

Akiva Weinberger

Ex: πρωτεΐνη

(Diaeresis and accent on the same letter)

hmm, they transliterate uno dos tres as ουνο ντος τρες

Akiva Weinberger

3:25 AM

What about seis

word-initial "d" doesn't undergo lenition in Spanish I suppose

Akiva Weinberger

Utterance-initial, at least

@AkivaWeinberger the source only has ουνο ντος τρες :P

comicsmania.gr/pages/18a_mikrosserifis.htm

item 146

ooh I found another sauce

Akiva Weinberger

castellano.gr/articles/158

I was about to

Akiva Weinberger

3:28 AM

They use an accent mark - if it's on the first letter in a digraph then it separates

(and if it's on the second letter then it remains a digraph)

Accent marks in Greek, unlike Spanish, are required for any word with more than one syllable

Note: διέθ, not διέσ

ούνο δος τρες κουάτρο θίνκο σέις σιέτε ότσο νουέβε διέθ όνθε δόθε τρέθε κατόρθε κίνθε διεθισέις διεθισιέτε διεθιότσο διεθινουέβε βέιντε

βέιντε is a little suspicious

Akiva Weinberger

Why

because ντ = d?

Akiva Weinberger

Veinte

@LeakyNun Word-medially I think it can also be /nd/?

Might be free variation

I don't know about that

Akiva Weinberger

3:30 AM

In any case I don't think you can enforce /nt/

hey we just did a Rosetta stone thing

@AkivaWeinberger and θίνκο not σίνκο

Akiva Weinberger

Santorini = /s̠a(n)doɾ̠ini/?

@LeakyNun Right 'cause ceceo

za ce ci zo zu = /θ/

sa se si so su = /s/

> Santorini (Greek: Σαντορίνη, pronounced [sandoˈrini])

Akiva Weinberger

> It may be an alveolar approximant [ɹ] intervocalically

Ooh

A little American-sounding

(Full quote:

> The only Greek rhotic /r/ is prototypically an alveolar tap [ɾ], often retracted ([ɾ̠]). It may be an alveolar approximant [ɹ] intervocalically, and is usually a trill [r] in clusters, with two or three short cycles.[8]

)

hmm

Akiva Weinberger

3:35 AM

> [o voˈɾʝas ˈco̯iʎoz ˈmalonan | ʝa to ˈpços aptuz ˈðʝo ˈineo̯ ðinaˈtoteɾos | ˈota ˈnetiçe napeˈɾasi apo broˈstatus | ˈenas taksiˈðʝotis pu̥ foˈɾuse ˈkapa]

From Wikipedia's example sentence

Hmm

> broˈstatus

what's the [ç] corresponding to?

Akiva Weinberger

Nonsyllabic /i/ after a voiceless consonant I think

what is [ˈnetiçe]?

Akiva Weinberger

Ο βοριάς κι ο ήλιος μάλωναν για το ποιος απ’ τους δυο είναι ο δυνατότερος, όταν έτυχε να περάσει από μπροστά τους ένας ταξιδιώτης που φορούσε κάπα.

@LeakyNun ν έτυχε

"όταν έτυχε" = [ˈota ˈnetiçe] interesting

so they do they German thing with the ch

fronting before front vowels

Akiva Weinberger

3:38 AM

Oh so I was wrong about nonsyllabic /i/

That's just /j/ as usual

ha, they converged!

Akiva Weinberger

/ɣ/ fronts to /ʝ/ though

a merger in plain sight!

Akiva Weinberger

Ooh:

> This also accounts for Greeks having trouble disambiguating voiced stops, nasalised voiced stops, and nasalised voiceless stops in borrowings and names from foreign languages; for example, d, nd, and nt, which are all written ντ in Greek.

ooh

wait why is "ταξιδιώτης" pronounced [taksiˈðʝotis]?

ʝ instead of ç?

Akiva Weinberger

3:42 AM

δ is voiced

oh right

Akiva Weinberger

Notice all the sandhi as well

I dunno what acoustically the spaces in the IPA transcription refer to

but they're clearly not the same as the word boundaries in the orthography

Like French?

Except without the dropping consonants

Hm

Conlang idea: the only consonants are /h̪͆/, /s/, /sʲ/, /ʂ/, /ɕ/, /ʃ/, /ɬ/, and /ɧ/

@AkivaWeinberger are you jewish?

Akiva Weinberger

Yes

like religiously?

Akiva Weinberger

3:53 AM

and the only vowels are /ə/ and /ə̥/

@LeakyNun Yes

can I ask you a religious question?

Akiva Weinberger

Sure

is Hebrew the original language?

Akiva Weinberger

No

and do you know Jews who insist that?

Akiva Weinberger

3:55 AM

Yeah

I haven't discussed it with them though

Everyone knows the true original language is Sanskrit though

(joking)

(it's actually Tamil)

and have you heard of people trying to connect the meanings of triliteral roots to the individual letters?

Akiva Weinberger

I probably heard that somewhere

like how A means strong and L means something I forgot so A-L is god

Akiva Weinberger

I know Kabalah attaches a lot of meaning to the letters making up the words for things

I'm really not the person to ask about Kabalah though

hmm

Akiva Weinberger

3:59 AM

You could probably find people who do a similar sort of thing with hanzi to be honest

(Would probably be easier, too)

so 90% of the Chinese characters are formed from a radical denoting meaning and a word denoting pronunciation

Akiva Weinberger

Fun fact: the letter 'chet' used to refer to two separate phonemes that merged

Phonosemantic compounds, yeah

@AkivaWeinberger well a lot of mergers happened in Hebrew

Akiva Weinberger

This one happened in early Biblical Hebrew

@AkivaWeinberger so I don't know if you were referring to that

or maybe you're talking about things like fengshui

and I don't really know about that

Akiva Weinberger

4:01 AM

Don't know what that is

@AkivaWeinberger Biblical Hebrew also. just look at https://en.wikipedia.org/wiki/Semitic_languages#Consonants

leaky original how?

the Jews believe the world has been created with the word

Akiva Weinberger

> Biblical Hebrew as of the 3rd century BCE apparently still distinguished the phonemes ġ /ʁ/ and ḫ /χ/, based on transcriptions in the Septuagint. As in the case of /ɬ/, no letters were available to represent these sounds, and existing letters did double duty: ח /χ/ /ħ/ and ע /ʁ/ /ʕ/. In both of these cases, however, the two sounds represented by the same letter eventually merged, leaving no evidence (other than early transcriptions) of the former distinctions.

guess what language that word would have been in

Akiva Weinberger

Interesting

4:03 AM

"... and G-d said, let there be light"

Akiva Weinberger

Different people will give different levels of literalness to that

but I don't really see a problem in believing that Proto-Semitic existed and evolved into Hebrew and friends

sure

Akiva Weinberger

Re: Sanskrit (or Tamil) being the world's first language, though — which are surprisingly common claims — it turns out that with the power of nationalism you can argue anything

There's a lot of people who believe "I speak a language therefore I'm an expert in linguistics"

(See also: "I have a brain so I'm an expert in psychology" and "I have a body so I'm an expert in medicine")

(Maybe that last one is not as common)

@AkivaWeinberger stares at essential oils

Akiva Weinberger

Fun fact: They're called that because they contain the "essence" of the plant

It's basically a bad pun

4:08 AM

well the problem is that Hebrew is a Canaanite language (closed related to Phoenician) but the Hebrews are supposed to have originated from outside Canaan

Akiva Weinberger

Hm never thought about that

but Abraham (and sons) are able to converse with the locals pretty well

if im not mistaken, i think it was the lingua franca of the time in the region

but triple check me on that

so Hebrew can't be the original language!

Akiva Weinberger

When Joseph spoke to his brothers and they thought he was an Egyptian official what language did that happen in

why not?

Akiva Weinberger

4:10 AM

I imagine Egyptian... Did Joseph have an accent?

here's another fun thing to ponder -- while the New Testament is written in Hebrew, the Koran is written in Arabic. The Muslims accept the Hebrew Bible and the New Testament as the word of G-d. Why did G-d cross over to Arabic in the Koran?

Akiva Weinberger

It's such a shame Escher never saw aperiodic tilings (like the Penrose tiling)

'cause doubtless he would've used them in his art

@JoeShmo 'Cause the Hebrew versions are corruptions or something like that?

to them

Re: Escher, it's neat how his circle limit stuff came about

'cause he wanted some way to portray the infinite in a finite space

and experimented with some ideas and couldn't quite get something he was satisfied with

and then randomly saw an illustration by Coxeter (I think) in some math talk

of a hyperbolic tiling

and immediately realized that that was the solution to his problem

idk how much of it is modern reinterpretations.. still. Quite difficult to ignore the fact and the deed -- the language of choice of both the Hebrew bible is.. Hebrew.

Akiva Weinberger

and he asked Coxeter how he drew it, Coxeter IIRC gave some incomprehensible answer that assumed Escher had more mathematical background than he did, and eventually Escher figured out how to do it on his own

(He wanted a straightedge-and-compass construction)

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The infinite in a finite space

pretty

not sure what im looking at though

Akiva Weinberger

4:20 AM

Right so anyway I dunno what he would've done with an aperiodic tiling but I'm sure he would've had fun with it

i see an elephant in distress

Akiva Weinberger

@JoeShmo Are you familiar with Escher's "Circle Limit" series?

Or his tiling stuff in general

At home I have a coffee table book full of Escher prints

Great artist

yeah vaguely

impossible staircase constructions etc.?

Akiva Weinberger

user image

He did that too @JoeShmo

^Circle Limit IV

Angels and demons tesselating

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Birds

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These are neat

especially the fish turning into the birds

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alright. gotta hit the bed. good night everyone!

Akiva Weinberger

4:25 AM

Here's a previous attempt at an infinite tiling

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He didn't feel satisfied with it because it had an artificial outer boundary

he is a wonderful artist

Akiva Weinberger

so it still felt "partial" (you could complete it infinitely outwards)

whereas with the Circle Limit stuff he eventually did there's no artificial boundary

well the irony is that he is constrained to finitely many recursive drawings

Akiva Weinberger

user image

He's a fan of tilings changing as you move along them, I guess

^

Escher's Metamorphosis II

like so^

@AkivaWeinberger איל גﬞודיאו־איספאנײול איס לה לינואה פﬞאבﬞלאדה די לוס גﬞודיוס ספﬞרדים ארונגﬞאדוס די לה איספאנײה איניל 1492. איס אונה לינגואה דיריבﬞאדה דיל איספנײול אי פﬞאבﬞלאדה די 150,000 פירסונאס אין קומוניטאס אין ישראל, לה טורקײה, אנטיקה יוגוסלאבﬞײה, לה גריסײה, איל מארואיקוס מאיורקה, לאס אמיריקאס, אינטרי מונגﬞוס אוטרוס לוגאריס

Akiva Weinberger

4:29 AM

user image

^the same but not a video

"Favlada"? Sounds Portuguese, interesting @LeakyNun

I'm gonna learn a song in Ladino for my choir at some point

Dunno the phonology. I think they still have a /v/ and /b/ (unlike Spanish where they're merged)

Jewish Sephardic - Sien Drahmas Al Dia - Ladino | Yamma Trio

^Ladino

It's a shame Judeo-English never developed. It would be interesting to see how the orthography would work

(Is that song in 9/8?)

seems like it

35 mins ago, by Akiva Weinberger

Re: Sanskrit (or Tamil) being the world's first language, though — which are surprisingly common claims — it turns out that with the power of nationalism you can argue anything

@AkivaWeinberger we get the same dose in HK

Akiva Weinberger

@AkivaWeinberger Not often you see someone compose music to a painting

where people say that Cantonese has a history of 1000 years while Mandarin only a few hundred years

so Cantonese is more authentic

Akiva Weinberger

There's a demand/Mandarin pun somewhere

but I forget what it is

@AkivaWeinberger so f>h happened after 1492!

(ferrum > hierro, filium > hijo, etc)

Akiva Weinberger

4:42 AM

Interesting

from Cantar de mio Cid, 12th century

Ya sennor glorioso, padre que en çielo estas,
Fezist çielo e tierra, el terçero el mar,
Fezist estrelas e luna, e el sol pora escalentar,
Prisist encarnaçion en Sancta Maria Madre,

Fezist = hiciste

Akiva Weinberger

Neat

sennor = señor

nn = ñ

I wonder if that was pronounced as a gemminate or what

> Many words now written with an ⟨h⟩ were written with ⟨f⟩ in Old Spanish, but it was likely pronounced [h] in most positions (but [ɸ] or [f] before /r/, /l/, [w] and possibly [j]). The cognates of the words in Portuguese and most other Romance languages have [f].

Akiva Weinberger

Interesting

Like /hu/=[ɸu] in Japanese

> The palatal nasal /ɲ/ was written ⟨nn⟩ (the geminate nn being one of the sound's Latin origins), but it was often abbreviated to ⟨ñ⟩ following the common scribal shorthand of replacing an ⟨m⟩ or ⟨n⟩ with a tilde above the previous letter. Later, ⟨ñ⟩ was used exclusively, and it came to be considered a letter in its own right by Modern Spanish. Also, as in modern times, the palatal lateral /ʎ/ was indicated with ⟨ll⟩, again reflecting its origin from a Latin geminate.

Modern Spanish's respecto vs respeto distinction is interesting

also calidad vs cualidad

the generalization is doublets

Impeachment Proceedings to be started against Donald Trump

Akiva Weinberger

5:10 AM

user image

@LeakyNun Can you spot the pattern in the design

not really

Akiva Weinberger

Hint: time flows down and to the left

@AkivaWeinberger self-demonstrating statement right there

Akiva Weinberger

@Semiclassical Yes

@AkivaWeinberger it seems that a grid is determined by the three grids on the line above it

000 -> 1
001 -> 1
010 -> 1
011 -> 0
100 -> 0
101 -> 0
110 -> 0
111 -> 1

Akiva Weinberger

5:18 AM

Yeah

Exactly

The seed is a single pixel (offscreen)

Rule 30 is a one-dimensional binary cellular automaton rule introduced by Stephen Wolfram in 1983. Using Wolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour. This rule is of particular interest because it produces complex, seemingly random patterns from simple, well-defined rules. Because of this, Wolfram believes that Rule 30, and cellular automata in general, are the key to understanding how simple rules produce complex structures and behaviour in nature. For instance, a pattern resembling Rule 30 appears on the shell of the widespread cone snail...

ooh nice

Akiva Weinberger

@LeakyNun Reading the right-hand side from the bottom up we get 1000111, which is a unique identifier of the rule

Oh wait I think it's the color-reversed version of Rule 30

atlas.wolfram.com/01/01/135

Rule 135?

https://writings.stephenwolfram.com/2017/06/oh-my-gosh-its-covered-in-rule-30s/

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@AkivaWeinberger guess the rule ;)

Akiva Weinberger

Start from random seed, shift everything by 1?

So I guess somewhere around 256/3

86?

why 256/3?

Akiva Weinberger

5:25 AM

I dunno I want binary 101010 or something

hint: the emoticon

Akiva Weinberger

Rule 170 (which is about 256*2/3 so not too far off)

What's it to do with the emoticon

if you zoom in the image you'll see that it doesn't shift everything by 1

look at the top row

Akiva Weinberger

Oh… adds them?

The top-left and top-right?

Hm no

Ohh

Rule 1

Derp

Hm wait

No, 101->1

Rule 34?

Aight now I get it

:P

Akiva Weinberger

5:31 AM

Fun fact: the station design was meant to honor Conway

It was supposed to be "inspired by Conway's Game of Life"

but they couldn't figure out a good design from that

so they went with the similar looking Wolfram Rules instead

which has very little to do with Conway

(except for both being cellular automata I guess)

(One 1D, the other 2D)

Also Conway wasn't told about any of this until it opened

Rule 110 is known to be Turing complete, fun fact

I think Rule 30 is conjectured to be

what does Turing complete mean?

Akiva Weinberger

You could turn an algorithm into a sequence of pixels, run Rule 110 on it for a bit, halt when it reaches some special state, and convert the output into an answer

I think?

No comment on any of the conversions involved being efficient to compute

> Matthew Cook presented his proof of the universality of Rule 110 at a Santa Fe Institute conference, held before the publication of NKS. Wolfram Research claimed that this presentation violated Cook's nondisclosure agreement with his employer, and obtained a court order excluding Cook's paper from the published conference proceedings. The existence of Cook's proof nevertheless became known.

Bit of a dick move on Wolfram's part

> The function of the universal machine in Rule 110 requires a finite number of localized patterns to be embedded within an infinitely repeating background pattern.

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Interactions between "streams"

hmm

Akiva Weinberger

called "spaceships" in the article

possibly by analogy with the similar concept in the Game of Life

@LeakyNun Isn't it interesting how the universe can simulate itself

aka we can build computers (out of pieces obeying the laws of physics) which can simulate physics

can it?

Akiva Weinberger

5:46 AM

This is what video games do

Well

oh yeah isn't there an argument that uses that to argue that we're in a simulation

because we can create many simulations

Akiva Weinberger

We don't know for sure about quantum mechanics being simulatable

and statistically you're more likely to be in a simulation because there's only one "real" universe

Akiva Weinberger

and we don't know what the ultimate theory of everything is

Reminds me of the question: "How do we know that we're awake"

How do we know that we're not asleep right now and that this isn't a dream

Once someone challenged me to prove it to them

and I came up with this really clever proof

and they were convinced that we were in fact awake

…and then I woke up.

2

"Prove to me we're awake right now." "Pfft, I could do that in my sleep! …wait"

6:32 AM

can we write this in the form of sets of R

$x>1\Rightarrow x^2>9$

can we say that is$ ]3,+\infty[$?
thank you

hello

Akiva Weinberger

If I understand correctly, yes

no it's (-infty,1] U (3,infty)

Well, my last night dream took that one step further:

I died without realising I have died

Any body ever herd about God particle

The way it pull that narrative off (almost) is that when I woke up in the middle of the night, the room and vision is so distorted that it looks nothing like my real life room. Before I hit the light switch, I thought I am still dreaming while in fact I am dehydrating

Put it simply, the brain masked the sense of dehydration, which my consciousness misinterpret it as me dying, with a dream state so that even real life is part of the dream

If I did die, the brain will probably go hyperdrive on this and I will not be able to tell the difference between real life and dream as my body functions shut down one by one

Akiva Weinberger

6:47 AM

@Secret You're always five minutes away from death, but the counter resets every time you take a breath

The Oh My God Particle | Space Time

Akiva: Well that's true

Akiva Weinberger

Fun fact: the word "boson" (the type of elementary particle) was named after Indian physicist Satyendra Nath Bose (1894–1974)

@Secret Neat mnemonic: you can last 3 minutes without air, 3 days without water, and 3 weeks without food

(approx)

(and about a week without sleep, but I dunno how to fix that into the mnemonic)

Death will probably feel like dizziness and confusion, to the point that you won't even notice it when life ends

It'll all be fuzz

or static

I dunno

7:04 AM

Yes there are many amazing fact about his theories

8:01 AM

Death in the Microcosmos

Relax and Enjoy the View

hi

@Secret I think this is new

8:42 AM

it is

and it is relaxing

9:13 AM

Hey

is there an algorithmic way to generate all elementary functions

I'm thinking of something along the lines of

Start with a set of $\{ c, \alpha x \}$

The constant and linear function

Apply every unary elementary function to that set and add them to the set

And then apply every binary operation with every pair of elements from that set

Every time with different factors

so

\begin{eqnarray}
\{ \alpha x, c \} &\to& \{ \alpha x, c, (\alpha x)^n, e^{\alpha x}, ... \} \\
&\to& \{ \alpha x, c, (\alpha x)^n, e^{\alpha x}, \alpha x + c, \alpha x + (\beta x)^n, ... \}
\end{eqnarray}

seems like it would work but be fairly inefficient because of getting the same class of functions

ie $x \times x \sim x^2$

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