Simply Beautiful Art's realm of calculus and analysis

55 Cancri

01:05

Woah

hardcore parkour over here!

Simply Beautiful Art

@55Cancri You are welcome here any time. Mostly do calculus, and some other funky maths. Feel free to read the transcript as well.

55 Cancri

I most certainly will Simply. Thank you for inviting me!

Simply Beautiful Art

No problem, feel free to enter any time... I shall put this room on my profile page... why isn't it already there? >.<

Typhon

@55Cancri Also please check out my room as well. it needs a populous.

$Mathematics$ The Universe of Quack

Where you are free to do proofwriting and generally hang out.

Simply Beautiful Art

lol

@Typhon Think it needs to attract a more specific audience to get a good running

Typhon

01:09

meh

:p

Simply Beautiful Art

Essence of calculus, chapter 1

►

@55Cancri

55 Cancri

Ooooh thank you so much SBA, I am so excited to watch this and all of his other videos

I will become as knowledeable as Typhon!!!

Simply Beautiful Art

@55Cancri I suppose I specialize in some combination of calculus, large finite numbers, and large non-finite numbers.

So if you have any questions in those, I can probably answer them

Typhon

@55Cancri One day you might. Machine learning is definitely an ambitious thing to learn. :-)

55 Cancri

Well, I still have to learn what much of that is first before I can even ask question about them!

Typhon

01:12

perhaps one day you will put into being what I once wrote stories about

Simply Beautiful Art

Mmm... actually, one of my teachers is interested in machine learning if you'd like to talk to him a bit @55Cancri

Typhon

granted, if that were to happen I'd probably be quite upset. That would be... evil.

Simply Beautiful Art

Not sure if he knows much

55 Cancri

And what did you write about?

Yes Simply, is he someone on here?

And I don't know any machine learning let

I still need to better my program skills

I am still wrapping my head around the logic

I have html and css down

Simply Beautiful Art

@55Cancri No, he's on Discord though

55 Cancri

01:14

I am currently trying to learn javascript, sql, and node.js

Simply Beautiful Art

I do Ruby

Horrible Ruby x.x

55 Cancri

That is for data analysis right?

or is that R

Simply Beautiful Art

Nah, I use it for writing programs that would theoretically output extremely large finite numbers

Typhon

@55Cancri oh just a really twisted story way back when. It was basically a bunch of people on a forum making a story making fun of the forum and me inadvertently calling a forum game about spamming the word duck a "cult" instead of 'spam'. The ultimate plot twist was that it was all a computer simulation on someone's desktop computer. Everyone was just a program.

55 Cancri

haha that is awesome Simply!

Like how many exponents worth?

Simply Beautiful Art

01:15

Also for golfing my code down to the point that its just unreadable jibberish, but short code

55 Cancri

lmao Typon!

Simply Beautiful Art

@55Cancri Oh dear... nowhere near anything I can explain using just exponents I'm afraid...

Typhon

@55Cancri recursive formulae that literally brute force using a googleplex tower of exponents of googleplexes fails to get past f(5,5)

55 Cancri

graham's number?

Simply Beautiful Art

@55Cancri even bigger

55 Cancri

01:17

Oh, the tower thing!

Typhon

that is smaller than f(1,1) of his function

Simply Beautiful Art

I can exceed Graham's number in 71 bytes

55 Cancri

I once read about that on xdxc article

you like keeping stacking these towers

Simply Beautiful Art

Ah yes, xkcd...

xD

Typhon

@55Cancri we're basically making numbers that exceed the collective ego of all presidents and kings in the DC multiverse.

55 Cancri

01:17

and it go so large that just "writing" the tower would extend far beyond the sun

Typhon

XD

Simply Beautiful Art

@55Cancri Its not quite towers... its a bit worse than exponential towers to say the least

55 Cancri

lol

Typhon

our towers would be so large if expanded that they would wrap the entire universe and beyond.

55 Cancri

their egos are quite large so that is an incredible feat

Simply Beautiful Art

01:18

@55Cancri Aren't you supposed to be taking a leave?

Typhon

by this i mean the collective current size it has expanded to

the universe has infinite spacial volume to expand into

Simply Beautiful Art

@55Cancri Actually, it wouldn't fit inside the observable universe

55 Cancri

lol, you guys absorbed me back into conversation

Simply Beautiful Art

xD

55 Cancri

I was just going to watch attack on titan

Simply Beautiful Art

01:19

Would you like to learn about my function?

55 Cancri

but as long as you guys have interesting things to say, I am all ears!

Simply Beautiful Art

A function @Typhon could not exceed xD

55 Cancri

certainly!

Typhon

imagine a titan where every molecule is a googleplex tower of googleplexes

55 Cancri

lmao!!!!!

that is quite a large number

incomprehensible.

Typhon

01:20

@SimplyBeautifulArt not even typhon is that big and yet he is the largest of all giants.

granted, he's also the only giant and most of him is just a rebuilt talos...

Simply Beautiful Art

@55Cancri Nothing is so large

Typhon

then again, I never finished that so I cannot technically cite it's size. XD

I also screwed up and thought talos was the colossus of rhodes

Simply Beautiful Art

def f(a,b,c=a)a.times{a*=c>0?f(a,b,c-1):b<0?a:f(a,b-1)}&&a end

Typhon

the latter is just a famous statue. XD

the former is a giant robot

Simply Beautiful Art

That's my function

62 bytes

55 Cancri

01:22

And I have no idea what it is telling me.

Three variables?

Simply Beautiful Art

f(64,0,1) is larger than Graham's number

Yup, and its written in Ruby

Typhon

It is basically googleplex level recursion

i have yet to beat it

truly beat it

Simply Beautiful Art

@55Cancri You aren't supposed to read it, its golfed to fit in as few bytes as possible

@Typhon lol, what's that supposed to mean?

Typhon

i think forcing it to take itself as input will beat it ultimately

55 Cancri

oh I see, like when websites compress code to make the site load faster

Simply Beautiful Art

01:23

@Typhon Nah, nesting itself is already built into itself

Typhon

@SimplyBeautifulArt a really large amount of recusion

55 Cancri

haha

Simply Beautiful Art

@55Cancri Well I do a lot of weird things...

55 Cancri

Simply that is awesome

Typhon

@SimplyBeautifulArt i was thinking of nesting the final input back into itself

for another round

maybe do it about 1000 times

Simply Beautiful Art

01:24

@Typhon Yeah... I already got that covered

Typhon

but...

Simply Beautiful Art

Nope

Typhon

you can next itself within itself again

double recursion?

55 Cancri

so do you mean the output of that is only 62 bytes?

Simply Beautiful Art

Nope, already built into the function

Typhon

01:25

@SimplyBeautifulArt dude I can write f(f(f(f(f(x)))))!

and that will be larger!

Simply Beautiful Art

@55Cancri No, the program size is 62 bytes. As there are no special characters, it is 62 characters including spaces

@Typhon In the amount of characters it took for you to write that, I can go much bigger

Bigger than you trying to nest if a Graham's number amount of times

Anyways, would you like to see how my function approximately behaves? @55Cancri

Typhon

i can use string_execute

55 Cancri

you two are like my role models.

Simply Beautiful Art

lol...

55 Cancri

definitely simply

Typhon

01:26

and just use itself to determine the amount of recursion

Simply Beautiful Art

Anyways, so it starts off rather simple.

f(a,-1,0)=a^2^a

Typhon

and then use that to make another level of recursion.

Simply Beautiful Art

For clarity, exponents are right to left, so a^2^a=a^(2^a)

@Typhon I told you! I already got that covered!

You could do a Graham's number amount of levels of recursion, but you won't beat me @Typhon

So, for example...

f(3,-1,0)=3^2^3=3^8=6561

Typhon

What if I did a grahams number of a grahams number recursions?

Simply Beautiful Art

Its kind of big, but no so big...

@Typhon Nope, dude, I got you way past covered

55 Cancri

01:29

What do the -1 and 0 do?

Typhon

what about f(grahams number) number of recursions?

Simply Beautiful Art

Nothing. They are the initial values @55Cancri

@Typhon Any sorts of recursion is almost completely covered, trust me

And any sorts of recursions over the amount of recursions is already covered

And recursions over the amount of recursions that are over the amount of recursions!

55 Cancri

So now it is just a matter of how large you make a

Simply Beautiful Art

Believe me, I am good at what I do @Typhon

@55Cancri Oh no, that's not the goal. Making the second and third arguments bigger makes things go crazy

$\text{You got the math rendering?}$

Typhon

what if I let it run for 24 hours continually doing more recusion at the level of f(graham's number)^f(grahams number) number of recursions per nanosecond and then make it close so that it prints out whatever number it was at that point?

55 Cancri

01:31

yep

Typhon

fair enough

I'll stick to the geometry

@55Cancri I actually know how super mario galaxy works in theory

Simply Beautiful Art

$$f(a,-1,1)\approx \underbrace{f(f(f(\dots f(}_aa,-1,0),\dots) ,-1,0),-1,0),-1,0)$$

For example...

Typhon

<=== that is an image of a grid one could theoretically walk on as an approximation of its geodesics. the math is all there. Just haven't made the walker.

Simply Beautiful Art

@Typhon and yes, I've got you well beat

Typhon

@SimplyBeautifulArt then I will use induction to find the larger number.

I will take your largest possible output and I shall.... add 1

Simply Beautiful Art

01:34

$$f(2,-1,1)\approx f(f(2,-1,0),-1,0) = f(2^{2^2},-1,0) = f(16,-1,0)=16^{2^{16}}$$

Typhon

wait!

I just realized

we are going about it all wrong

we need to find really really really small numbers

and plug them into 1/x

Simply Beautiful Art

$$f(3,-1,1)\approx f(f(f(3,-1,0),-1,0) ,-1,0)$$

@55Cancri So you can see how this goes?

Typhon

@55Cancri here. Read it for yourself. the-cult-of-duck.wikia.com/wiki/The_Cult_Of_Duck_Wiki

Simply Beautiful Art

@Typhon The smallest non-zero real numbers are equivalent to 1/large numbers, where said large numbers are found separately, so that's not a useful idea.

55 Cancri

damn, I thought I would figure it out on my own, but unfortunately I still do not see what you mean by initial values or the purpose of -1, 0

Typhon

01:36

but finding small numbers will be easier...

55 Cancri

The first number is the number of times you plug the function into itself right?

Simply Beautiful Art

@Typhon So you say

@55Cancri Yup

Typhon

meh

i dont like large numbers anyways

i like geometry

and number theory as in proofs and stuff

Simply Beautiful Art

@55Cancri Approximately

55 Cancri

you like geometry but not trigonometry?

Typhon

01:37

large numbers is... boring to me

Simply Beautiful Art

@55Cancri IKR? xD

Typhon

@55Cancri what do you mean?

55 Cancri

or were you the person who failed trigonometry?

Typhon

oh yeah

i forgot i mentioned that

i was

55 Cancri

haha

Simply Beautiful Art

01:39

Oh, do say, think you can make a larger number than me without being restrained to programs? @Typhon

@55Cancri Its not nice to laugh at people

55 Cancri

I think that is awesome considering how intelligent you are

Typhon

he wasn't

@55Cancri yeah...

55 Cancri

I made the highest grade on my trig exam, but during lecture, I didn't understand a single thing! yet this guy who made lower than me followed along perfectly!

Simply Beautiful Art

Is @55Cancri a 'he'?

If I may ask

55 Cancri

Yes I am

Simply Beautiful Art

01:40

k, just checking

55 Cancri

But that just goes to show, grades and test taking don't matter

because I think that guy will walk away with a much better understanding of the subject than I will

Typhon

@55Cancri i wouldn't know about that, but whatever. Everyone else did fine in my class.

Simply Beautiful Art

Well, anyways, f(a,-1,1) gets very large, and it gets large much faster than f(a,-1,0) does

55 Cancri

and I was totally envious. The grade felt undeserved and like a slap in the face

I just crammed the night before

Typhon

I probably just didn't pay enough attention

55 Cancri

01:41

okay thats it Simply!

so what does the third number represent?

how does it make the function increase faster?

Simply Beautiful Art

lol

Well, its quite natural actually...

$$f(a,-1,b+1)\approx \underbrace{f(f(f(\dots f(}_aa,-1,b),\dots) ,-1,b),-1,b),-1,b)$$

55 Cancri

see that loses me again

Simply Beautiful Art

So $$f(3,-1,2)\approx f(f(f(3,-1,1) ,-1,1),-1,1)$$

55 Cancri

I see two, so I am looking for two of something

but I dont see the 2 corresponding to anything on the right side

or the -1 for that matter

Simply Beautiful Art

Well, it sort of works like this

55 Cancri

01:45

okay change it to 4 and show me what it looks like then

Simply Beautiful Art

-1,2 corresponds to -1,1

-1,1 corresponds to -1,0

-1,0 corresponds to a^2^a

55 Cancri

you're blowing my mind Simply!

Simply Beautiful Art

And on each step, you do 'a' amounts of whatever it corresponds to

So while f(a,-1,1) grows very fast, f(a,-1,2) grows much faster

And then f(a,-1,0) << f(a,-1,1) << f(a,-1,2) << f(a,-1,3) << ...

55 Cancri

So you are subtracting -1 from the third number

sorry for not understanding this

it is probably something I just haven't come across yet in my math classes

we just do f(x) for now

Simply Beautiful Art

@55Cancri Yup, but don't forget, we are also doing a lot of the functions

55 Cancri

01:48

or f(g) for composition

Simply Beautiful Art

Note, for example, the following:

Typhon

what do you think?

Simply Beautiful Art

f(3,-1,1) is very big, right @55Cancri

55 Cancri

I know its big because the function will go into itself 3 times

Simply Beautiful Art

Then f(3,-1,2) ≈ f(f(f(3,-1,1),-1,1),-1,1)

55 Cancri

01:50

I also know the 1 will be subtracted from the third 1

Simply Beautiful Art

The inside f(3,-1,1) is really big.

Let's call x = f(3,-1,1)

So we have... f(3,-1,2) ≈ f(f(x,-1,1),-1,1)

55 Cancri

typhon, how did you make that?

what did you use?

Simply Beautiful Art

We know x is gigantic, and that's how many times we have to make f(...,-1,0)

So while x is big...

f(x,-1,1) is insane

55 Cancri

I see, this is a massive recursion

Simply Beautiful Art

as its equal to f(f(f(f(........f(x,-1,0),....),-1,0),-1,0),-1,0),-1,0)

emphasis on how much stuff is inside the .... portion

And then we have to do f(f(x,-1,1),-1,1)

Which is crazy larger

And so on and so forth to higher numbers

55 Cancri

01:53

my powers of abstraction fail me

Simply Beautiful Art

And then f(a,0,0) corresponds to f(a,-1,a)

55 Cancri

I feel as though you have this very precise and incredible understanding of what is happening here,

but I am unfortunately just not there

Simply Beautiful Art

xD Well, it happens to be my night now

So I think I'm gonna head to bed now

55 Cancri

lol I am so sorry Simply!

You are just too smart for me

Simply Beautiful Art

No problem xD

55 Cancri

01:54

I was really trying to understand

but I just couldn't get my head around those last two numbers

you subtract the second from the third,

but what did that mean?

Simply Beautiful Art

As I like to say, it helps to try and make a larger number than me to start comprehending what is going on

@55Cancri Try making a large number yourself and see where it gets you :)

g'night

55 Cancri

lol alright Simply

enjoy your night

and I hope you top your record in the future

with as few bytes as possible :)

Typhon

@55Cancri I made it by making function objects and then building a 3D model off of the outputs of the functions.

Brevan Ellefsen

05:35

@SimplyBeautifulArt Looks like you have managed to fill your Calculus chat with large numbers XD is there an Integral and Series chat?

Simply Beautiful Art

12:19

@BrevanEllefsen xD Idk man!

$$C(\alpha)_0=\{0,1\}\\C(\alpha)_{n+1} =C(\alpha)_n\cup\left\{\gamma+\delta,\gamma \delta,\gamma^\delta,\omega^{\mathrm{CK}}_\gamma, \omega_\gamma,\chi_\gamma(\delta),\psi_\gamma( \eta):(\gamma,\delta,\eta\in C(\alpha)_n) \land(\eta<\alpha)\right\}\\\psi_\beta(\alpha) =\sup\left\{\max\{\gamma_n:( \gamma_n\in C(\alpha)_n)\land (\gamma_n<\omega^{\mathrm{CK}}_{\beta+1})\} :(n\in\mathbb N)\right\}$$

$$D(\alpha)_0=\{0,1\}\\ D(\alpha)_{n+1}=D(\alpha)_n \cup\{\gamma+\delta,\gamma\delta, \gamma^\delta,\omega_\gamma^{CK}, \omega_\gamma, \psi_\gamma(\delta),\chi_\gamma(\eta):(\gamma,\delta,\eta,\pi\in D(\alpha)_n)\land(\eta<\alpha)\} \\\chi_\beta(\alpha)=\sup\{\max\{\gamma_n: (\gamma_n\in D(\alpha)_n)\land (\gamma_n<\omega_{\beta+1})\}:(n\in\mathbb N) \}$$

@BrevanEllefsen Mind you, large infinite ordinals are also of interest

Simply Beautiful Art

13:26

Good morning @Nilknarf

Nilknarf

@SimplyBeautifulArt Hi

Simply Beautiful Art

Well...

what you wanna do?

Nilknarf

Can you help with this?
https://math.stackexchange.com/questions/2326194/proving-the-relationship-between-the-definite-integral-and-area

I posted it as a question for anyone in case you aren't up to it

Simply Beautiful Art

xD

@Nilknarf the proof involves telescoping sums and the mean value theorem

Nilknarf

...

Simply Beautiful Art

13:29

Can't find a good duplicate immediately, so...

So, we start with the mean value theorem...

$$f(x)\in C^2\implies \exists c\in[a,b]\implies f'(c)=\frac{f(b)-f(a)}{b-a}$$

This is the mean value theorem

Nilknarf

Yes I know

Simply Beautiful Art

Particularly, I would like you to consider $c\in[a+k\Delta x,a+(k+1)\Delta x]$

Applying the mean value theorem, we end up with...
$$f'(c)=\frac{f(a+(k+1)\Delta x)-f(a+k\Delta x)}{\Delta x}$$

Nilknarf

Mhmm...

Simply Beautiful Art

Note there exists a $\Delta x$ in front of the sum which then cancels with the denominator here

Oh wait, I've done it wrong. Gosh darn it...

$$f(c)=\frac{F(a+( k+1)\Delta x)-F(a+k\Delta x)}{\Delta x}$$

Where $F$ is the anti-derivative of $f$.

Nilknarf

Ok

Simply Beautiful Art

13:35

Mmm... I've still done it not quite right....

Anyways... multiply this by $\Delta x$

And then sum it up

and you more or less get a telescoping series

Whereupon you are left with something like $$A=\lim_{n\to\infty}F(a+(n+1)\Delta x)-F(a+\Delta x)$$

And by letting $n\to\infty$, we get $a+(n+1)\Delta x\to b$ and $a+\Delta x\to a$

Hence, we end up with $$A=F(b)-F(a)$$

As claimed

@ÉtienneBézout Hello and welcome to my realm!

Nilknarf

Ahh, okay. I hadn't tried using the mean value theorem to reduce it.

Simply Beautiful Art

:) It reduces quite beautifully I do say...

Nilknarf

Yes, it does.

ooh, I was also wondering about some other things

regarding hyperoperations

Simply Beautiful Art

Haha, imagine calculus without it... we'd all be ripping our hairs out trying to do them integrals

@Nilknarf yes?

Nilknarf

Ha, yeah... that would be frustrating expressing every single area as a messy limit of a sum...

Okay

So how is a fractional hyper operation defined? If it even is defined?

Simply Beautiful Art

13:41

Yeah... we're still working on making tetration extended to fractional heights.

The worst part is that hyper-operations are, in general, non-analytic

Nilknarf

...

Well.

Simply Beautiful Art

Meaning we can't invoke the uniqueness theorem

Nilknarf

I was trying to differentiate tetration

Simply Beautiful Art

Yeah no

Nilknarf

I had to define it recursively

Simply Beautiful Art

13:43

Of course

Nilknarf

I started with $f_a(x)=^ax$

Simply Beautiful Art

32

Q: Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers?

This question has been in my mind since high school. We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, then $f(n) = m \times n$. Likewise, we get exponentiation by repeated multiplication. If $g(1)=m$ a...

25

Q: What is the derivative of ${}^xx$

How would one find: $$\frac{\mathrm d}{\mathrm dx}{}^xx?$$ where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$ Work so far The interval that I am working in is $(0, \infty)$. It doesn't make much sense to consider neg...

derivatives tetration power-towers

Nilknarf

And then
$$\ln f_a(x)=^{a-1}x\ln x$$

and I basically ended up with
$$f'_a(x)=^ax\bigg[f'_{a-1}(x)\ln(x)+\frac{^{a-1}x}{x}\bigg]$$

Oh thanks

Simply Beautiful Art

np

25

Q: $n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I was successful, it turned out not to be the most elegant formula but it worked. (Firstly, I inve...

calculus derivatives tetration

Nilknarf

Wow

Interesting reading :)

Simply Beautiful Art

13:50

:)

Nilknarf

I'm curious about something else too (sorry :P)

Simply Beautiful Art

np

Nilknarf

How do people use polar coords to evaluate integrals? What is special about polar coords that makes it easier to evaluate?

Simply Beautiful Art

Sometimes you have a region

and you want to find that area

Sometimes you are given $y=f(x)$, and you can integrate with respect to $x$

Nilknarf

Yes

Simply Beautiful Art

13:53

Sometimes you are given $x=f(y)$ and $y$ is most clearly not writable as a function of $x$, so you integrate with respect to $y$

Sometimes it is most clearly not a function with respect to $x$ or $y$, but it could be seen as a function with respect to $\theta$

Nilknarf

Hmm. Okay.

And converting it makes it somehow easier to antidifferentiate?

Simply Beautiful Art

Well, you need a function to even have a change at integrating (not antidifferentiating)

Nilknarf

Ah, right

Nilknarf

14:17

Okay, another thing

I was playing around with inverses

And I was wondering

If I have
$$h(x)=f(g'(x))$$
then can I conclude that
$$h^{-1}(x)=g^{-1}\bigg(\int f^{-1}(x)dx+C\bigg)$$

For some $C$?

It seems not

Simply Beautiful Art

Take derivatives of both sides

And it should become clear that this is not true

Nilknarf

Yes...

Simply Beautiful Art

However, did you know that if you can find the antiderivative of $f(x)$ and $f^{-1}(x)$ exists, then you can find the antiderivative of $f^{-1}(x)$?

Nilknarf

Yes

Simply Beautiful Art

oh, okay

:P

Nilknarf

14:24

Yeah, there's a formula

Simply Beautiful Art

So... wanna do the Gamma function reflection formula, complex analysis, something else, or nothing at all?

Nilknarf

Perhaps

In the middle of a weird inequality question at the moment though

Simply Beautiful Art

0

Q: What conditions of $x$ such that $e^{x^e} \ge x^{e^x}$?

What conditions of $x$ such that $e^{x^e} \ge x^{e^x}$ ?

inequality functional-inequalities

This one?

Nilknarf

Yeah

@SimplyBeautifulArt Can't figure out how to rigorously prove that the first one is greater until $x=e$, and then it is smaller

Simply Beautiful Art

14:45

I've done it on $0\le x\le1,x=e$

Nilknarf

Bad question, but a fun problem :P

Can you give me a probability problem that uses an integral?

Simply Beautiful Art

I don't do probabilistic integrals.

Nilknarf

:(

Okay then, let's do some analysis

@SimplyBeautifulArt Or perhaps we should do the Gamma function thingy

Simply Beautiful Art

Mmm, its a hard inequality

And I have to go sadly

Nilknarf

Oh okay. Bye!

Simply Beautiful Art

14:58

Cya!

Nilknarf

15:11

@SimplyBeautifulArt Where is TheGreatDuck? I was going to read some of his stuff about step-functions but I can't find his profile (even by searching).

Simply Beautiful Art

18:27

@Nilknarf He changed his name: math.stackexchange.com/users/289789/typhon

$Mathematics$ The Universe of Quack

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$Mathematics$ Simply Beautiful Art's realm of calcu…