Pi and the Mandelbrot Set - Numberphile

This video features Dr Holly Krieger. More videos with Holly Krieger: bit.ly/HollyKrieger
More links & stuff in full description below ↓↓↓
Extra footage from this interview: • Pi and Mandelbrot (ext...
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Since this was filmed, Holly has become a mathematics Lecturer at the University of Cambridge and the Corfield Fellow at Murray Edwards College.
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[@jacoboking36]

Pi is like that one uncle who just shows up out of nowhere in every scenario

[@DirtBikes_MathGarden]

nah that's e

[@robertakerman3570]

I like Pie-must B someone's Unc.

[@kasturylbs5905]

Lur minta 4d buat malam ini. Hkong

[@leeyenherngcalebpeircess6121]

No it's more like he's nnever invite becaise he always turns up anyways

[@brad1ey50]

e is the other uncle

[@HansenSWE]

"This video features Dr Holly Krieger." Viral.

[@HansenSWE]

So wait. Math geniuses, gather around, gather around. So if the number is within the mandelbrot set... if we do that calculation she was doing, will we never get a number larger than two? Regardless of how many times to do it? We just roll around inside that weird picture? Is that the whole point of the mandelbrot set? That we can do an infinite number of calculations without going higher than two? That sounds incredibly useful, but I just can't see how right now. It's like a safety mechanism for something. If I'm somewhat correct, how it this applied? It's very interesting

[@bodhifyer]

+Herr Hansen im no math genius at all, quite the opposite really. If I had to guess at the usefulness of the mandelbrot set, it would be to get a better understanding of the fundamental math of the shape of an abstract object. If it can be mathed then such an object could be a physical object somewhere in the universe which we could study from here on earth

[@HansenSWE]

Terrence Zimmermann Wow... I feel like less of a man than I did 5 minutes ago. Please tell me this was university physics and it's ok to not understand it perfectly.

[@HansenSWE]

***** Your comment was marked as spam and I had to restore it, brother.

[@HansenSWE]

Terrence Zimmermann Wait, you're talking about when magnetic fields change? Like from ferro to para? I've seen some stuff about that. Wish I paid more attention.

[@panda4247]

Holly: "...maybe the least efficient way possible to approximate pi" Matt Parker: "challenge accepted"

[@aaronleperspicace1704]

😂😂😂😂😂

[@Thomas_Bergel]

panda4247 Matt would probably get a method that is almost as complex... a classic parker square

[@gormster]

3blue1brown: hold my sliding square

[@Triantalex]

??

[@Nilguiri]

It's amazing how pi can just pop out when you are least expecting it, innit?

[@aaaab384]

It would be if it did. But since it has a simple definition, of course it pops up everywhere. I always expect it to pop up, and I am not amazed when it happens.

[@Gray-beard]

Lol. Try 'cross training your brain' with Gematria, Pi is everywhere...

[@FreymanArt2024]

It's like the Spanish Inquisition!

[@soso3792]

Are you surPised? ...sorry my puns are horrible *leaves*

[@N3bu14Gr4y]

Just like Garfield on a Monday!

[@KonJamo]

You english speakers might not realize this: Mandelbrot actually means Almond bread...to use Almond bread to approximate Pi is a very funny approach to baking :D a fan from Berlin, GErmany

[@PoweDiePie]

Wow.

[@thesentientneuron6550]

I do know a bit of german, but it did not occur to me

[@Pterry23real]

But Mandelbrot is just a surname. But another relative unusally name for the mandelbrot set is Apfelmännchen (little apple man), so you could bake pi with little apples ;)

[@earomc]

sockington1 what do you mean

[@mackan2277]

maybe make some pie out of almond bread :D

[@kevnar]

I created a game using vectors when I was young. I made a point in the center of a sheet of graph paper and called that "The Sun". It was basically just a source of gravity. Then I chose some other random point on the page and called that a "meteor". Then I assigned some arbitrary motion vector to the meteor. -2x,1y or whatever. Then I'd move the meteor by that much. But every round, you have to adjust the motion vector, + 1 if you're x is negative, or -1 if your x is positive, and likewise with Y. The point was to simulate gravitation toward the sun and get the meteor into an orbit. I tried all day to figure out numbers that would make the meteor go into a perfect circle, or at least an ellipse, but all I ever got were these strange, repeating wiggling patterns. Years later, I saw a video on Lissajou curves. Here I thought those curves in my oribits were a failure to create a circle, but all along they were something people were trying to find in and of themselves. It's amazing what curiosity and boredom produces.

[@qwertyTRiG]

That's impressive!

[@ColHogan-bu2xq]

Problem is : do we really know what the very "nature" of gravitation is ?

[@rivaldobox]

I really like how energetic she is when explaining things, it makes the video even better.

[@yttrv8430]

Dr Krieger: Well, I've already used "c", whatever, I'm gonna call this point "c", too. Gottsta love people like this.

[@uritibon17]

"This is something that's like, totally natural to be interested in" - Dr Krieger

[@SOCAKRKA]

but what is the connection? i would really love this to be expalined in depth. othervise very nice video

[@numberphile]

+SOCAKRKA there's a link to more on Numberphile2 - see the video description

[@SOCAKRKA]

+Numberphile thanks, i guess i have to pay more attention before posting comments :)

[@axiezimmah]

+Numberphile that link isn't working though

[@axiezimmah]

***** virus?

[@ccosborn2000]

I don't think anyone knows why the iteration counter will stop (ounce the function produces an answer greater than 2), and then recording that counter, why those digits are PI. Incredible. When you find out why let me know.

[@Christian-Rankin]

*insert filthy comment about obvious pretty women*

[@Peteminator]

+Christian Rankin The maths is not the only beautiful thing in the video.

[@Holexification]

+Christian Rankin I though pretty smart women were imaginary, turns out they are real! Gosh this is complex...

[@asdmla8777]

+Christian Rankin 3:20

[@gumenski]

+Christian Rankin Annnd you were the only one to be that guy. Nice one

[@TheLuxma]

+Aelianus Lucius Decimus +1

[@totallyunmotivated]

That woma- I mean that handwriting is perfect. Very legible.

[@zh84]

Well, that is extraordinary. I never would have guessed. Will you post a link to an explanation of why this works?

[@numberphile]

+zh84 see link to extra footage in the video description

[@Epoch11]

+zh84 I totally agree with you. This seems like witchcraft to me and I would love to see a detailed explanation about why this happens as it does.

[@strangelyjamesly4078]

+zh84 It happens because it does. The lower the value of epsilon the closer to the value of Pi the number of steps required to "bust". Coincidence.

[@NotaWalrus1]

+Strangely Jamesly Do you seriously believe that's a coincidence?

[@moonanddarkness]

+Strangely Jamesly Damn dude, your teachers must have been really bad. "It happens because it does." It's the most mediocre answer one can get." And I'm surprised you actually believe this is a coincidence, specially when there is a link in the description to a video that explains that this number is actually Pi, not just a coincidence, that if you keep iterating to infinity you will get Pi and it also explains why.

[@pokestep]

Wow these videos about the Mandelbrot set were great, I finally know what it actually is! Holly Krieger is also great, I loved to see her explain it all, thanks for that. I sure hope to see more in the future!

[@theulair]

This is the first simple explanation of the Mandelbrot Set and how it is drawn that i have seen in my life. Thank you!

[@BigBoatDeluxe]

Amy Adams discussing mathematics. I could get used to this.

[@ESL1984]

*getting arrival flashbacks*

[@carlos77121]

Underrated comment here.

[@Triantalex]

??

[@BigBoatDeluxe]

@@Triantalex Dr. Krieger kinda favors actress Amy Adams. Kinda sorta.

[@eddotron1224]

Numberphile's 314th video is about Pi, coincidence? I think not...

[@tilmohnen6521]

Watch 3Blue1Brown's Video of aproximating pi with block collisions. There are also powers of 100 involved... I wonder whether its actually the same method in disguise...

[@anraiduine1483]

Waaaa!! So cool!! What I find especially cool too is the fact that all functions (theoretically) eventually burst out, but the smaller the values, the longer they can be iterated and still remain within 2... It's an asymptote! With infinitely small values being infinitely iterable within the mandelbrot set! And to bring it back to the video, the greater the number of iterations, the closer they come to approximating pi! Insane!!!!

[@TheArunster]

6:14 "That's Cool!" Love the enthusiasm!

[@Quantiad]

Holly Krieger + Hannah Fry = Heaven

[@littlemikey46]

+iSquared My two favourite female mathematicians :D

[@Quantiad]

+Jeremy Watts Haha! Mate, that's harsh but hilarious. Both geniuses and both very cute in my opinion.

[@adymode]

+iSquared So special women, suz Dancso ahhhh

[@Quantiad]

+Andrew Input Great shout. Another beautiful genius.

[@whiterottenrabbit]

+iSquared Doctor, Doctor, gimme the news, I got a bad case of lovin' you!

[@hey8174]

Dr Holly Krieger is increasing my intelligence.

[@pacinpm2]

I see what you did here.

[@Epoch11]

+Tucense I'm sure it is expanding.

[@hey8174]

+Mark G ᕦ( ͡° ͜ʖ ͡°)ᕤ

[@iDEaXANA]

+Tucense i don't see it. mind explaining?

[@iDEaXANA]

+Tucense there must be something wrong with me.

[@sandeepsreehari9188]

How very interesting to see the link between Pi and the Mandelbrot set. It's really about taking particular values for C and Z. Thanks for the informative video.

[@stevoofd]

The mandelbrot set is like the gift that keeps on giving

[@weirdshamanwizzard3156]

"This is a totally natural thing to be interested in" this statement just... i don't have any words for it (but i don't disagree)

[@Dazbog373]

Least efficient but one of the most interesting ways of approximating pi

[@BobStein]

Brady makes a good talk great with his graphics and just the right touch of comments and questions.

[@margarett.newman7574]

Glad to have been shown that. Thank you!

[@ComicBookGuy82]

0:43 and that's why you should pronounce Z as Zed

[@geuwglesuxballz6074]

+LaVelle I am open to zed being the preferable pronunciation. How was that demonstrated here?

[@ZPSBestProfileName]

+GeuwgleSuxBallz It sounded a lot like the c, which, wasn't too confusing, but I could see it irking some students at the back of a lecture hall.

[@unvergebeneid]

+LaVelle As a German, I prefer zee British pronunciation as vell.

[@ZPSBestProfileName]

Penny Lane Nice.

[@ZPSBestProfileName]

Penny Lane Yeah I presumed you were joking about being German.I thought going with French would be better as they literally say "ze", but appreciated the wordplay all the same.

[@TheRocketMaster2013]

Numberphile! Congratulations on 314 videos!

[@virondata]

This actually blew my mind. WOW thank you for this video.

[@KC9MDO]

The number of steps (N) to get over became closer to representing PI the larger N became. I love this kind of number theory, thanks.

[@michaelbauers8800]

A friend and I were delaying going back to programming, so during lunch break, we did a bit of python to show this works in base 2. I will paste the program here. Note that you might see a discrepency in digits due to rounding. Also note I used an alternate starting value.

[@michaelbauers8800]

import cmath import math def f( c, z): return z*z + c def doit(c, e): i = 0 temp=f(c,0) while abs(temp)

[@michaelbauers8800]

You can modify the value of 20 used in range, and see more digits. For those not familiar with Python, bin gives the binary representation of the number.

[@SunriseFireberry]

Can you use the Mandelbrot Set to approximate e? Or other transcendental numbers?

[@hyperbole5726]

+TimeAndChance nah man, but try series and sequences

[@hearueszueke6206]

+TimeAndChance maybe...

[@simoncarlile5190]

+TimeAndChance Try to do it.

[@allanfloyd8103]

+TimeAndChance Doubtful... If you watch the extended explanation, it boils down to the fact that this function approximates the Tangent function, and that is basically made of PI. ;)

[@canerkorkmaz9428]

You can approximate ( and find ) using its Maclaurin series ( which is Taylor series at 0). e^x = 1+X/1!+x^2/2!+x^3/3!+... e = 1 + 1/1! + 1/2! + 1/3! ...

[@meredithhargrave1178]

I'm in algebra 1. This is all way over my head, but I still love watching these videos.

[@alihijazi4451]

I giggled a lot at the end of the video, from pure joy. That's AWESOME. I love math.

[@megamillionfreak]

She is so intelligent. And a great voice and presentation. I could listen to her for hours, days.

[@jestemqiqi7647]

"Mandelbrot" is german and means "Almond bread"

[@jestemqiqi7647]

gabriel schilhan No.

[@shorterneilisbored7078]

Dammit, i'm allergic to numbers.

[@theleftuprightatsoldierfield]

Knewity Mandelbrot is also close to an anagram to almond bread. Mandelbrot can be rearranged to almond bre. It is missing an a and a d

[@shiva_hardly_sarcastic]

First of all its a name and doesn't has to be translated

[@amawalpe]

It’s almost an anagram because words have the same root : mandel -> almond ; brot -> bread :)

[@brazenzebra]

A beautiful video gift from a math angel of heaven.

[@nintender6333]

Thats absolutly awesome! Im 17 and I write about the mandelbrot Set for school (I have about 18 pages which is more than enough), but I think about getting this into my seminar work

[@nousefor1]

3:19 That's what I always tell the ladies about my Mandelbrot Set :\

[@SeanForeman]

Even after watching the additional video this is still baffling amazing to me. I am a software engineer and I find it fascinating that the number of permutations in a particular calculation could result in a known value... but you have to move the decimal place! It feels like there is something else amazing hiding in that idea that could be used to speed up certain calculation by skipping permutations and knowing the overarching value representing iterations.

[@adheesh2secondsago630]

No, this is very inefficient, u can't skip anything since output of previous is used as input to second iteration

[@jonathanford9354]

I think the exact formula is pi = lim e--> 0 of N(e)*sqrt(e). Hence if we divide e by 100, the output "shifts" by one decimal place (My N(e) is defined such that it's equal to N(0.25+e) in the video).

[@MijinLaw]

On 3 blue 1 brown they have several videos of pi showing up in weird places, and he manages most times to show how there is some analogy to a circle in the system. e.g. There's one where a heavy object collides with a much lighter one and a wall, with perfectly elastic collisions. As you make the heavy object increasing powers of 10, the total number of collisions tends towards pi times a power of ten. It's mind-blowing, but he manages to explain how it actually can be mapped to movement around a circle. In the case of the mandelbrot set, multiplication of complex numbers directly involves movement in a circle, the connection is probably simpler.

[@andersonantunes4257]

This is actually really cool!

[@Tertious89]

I love complex numbers. There are just do many interesting and thought provoking applications when they are involved.

[@PhysicsPolice]

Wow! Why/how does this trick work?!

[@numberphile]

+PhysicsPolice seen the extra footage on Numberphile2?

[@DrDress]

+Numberphile Y U NO put link at end of video?

[@colouredmirrorball]

+DrDress Y U skip commercial?

[@whiterunguard7408]

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[@gckbowers411]

+Whiterun Guard The first decimal place is wrong.

[@wii3willRule]

Blew my mind... why is it like this?

[@hearueszueke6206]

+wii3willRule we dont know

[@daedra40]

+Haris Ziko (PiMathCLanguage) The universe refuses to be all so direct with us!

[@allanfloyd8103]

+wii3willRule Follow up explanation in the description and at the end.

[@jameswilson6352]

This is an absolutely amazing discovery!

Omg, a video about my two favorite topics in mathematics!

[@williamlavie8532]

Great video. I would have been interested to know why this method allows us to approximate pi though...

[@numberphile]

+William Lavie extra video on Numberphile2 channel

[@DimitrisAndreou]

They forgot to explain why it is so!

[@dozer1642]

Dr Holly you are amazing.

[@iaalias4013]

That is absolutely amazing

[@BigNickontheDrum]

Dr. Holly Krieger: Mandelbae

[@stv3qbhxjnmmqbw835]

Man of culture

[@Harlequin314159]

Hang on..... but why!?

[@TheOutZZ]

WOHO :D! A Mandelbrot set video on my B-DAY!

[@josejrtuti]

I love this channel!

[@niteexplorer9934]

i'm depressed now

[@WahranRai]

The sexiest teacher in youtube !

[@Quadflash]

Fun! Fascinating! And, completely surprising. Thanks!

[@lockythindromen3974]

+1 for recommending Sage!

[@kemkyrk8029]

I'm disturbed with that... I mean, the number of required steps to get bigger than 2 is some kind of "continuous" (if it takes n steps to a number to blow out, then you can get another number very close for which it will take exactly n+1 steps to blow out). Therefore, for every integer n, there is a real number which blows out of 2 in exactly n steps. Since you can do that for every integer 3, 31, 314, 3141, 31415, you can find a sequence of real number whose number of steps converge to π*10^smth. I think it's interesting that this sequence is just some kind of .25 + 10^-2n but, it's not that magical that it converges to π, since you know that such a sequence exists. You can do that with every real number...

[@kemkyrk8029]

Oh ok, so then, I can call that wonderful :))))))

[@alainischileno]

I wrote a simple program and it only works up to 0.2500000001, only by adding two zeros, and only when it's followed by a 1 each time. For instance: 0.2501 gives us 312, and 0.250001 gives us 3140, but 0.25001 is 991. And putting 0.250002 instead of 0.250001 gives us 2219. Can someone prove me wrong?

[@RaidChampion]

+Alain Rochette It does work. Formally, it is (epsilon^(1/2))*N(c) which converges to pi. If you use 0.25001 for example, epsilon = 0.00001 and N(c) = 991, then epsilon^(1/2)*N(c) = 3.1338 ...

[@MichaelS-vy1ku]

+Alain Rochette Harmonic oscillation?

[@alainischileno]

RaidChampion Thanks! they didn't explain that in the video, which is why i was confused.

[@alainischileno]

RaidChampion tried it again and this time it works pretty much up to 0.2500000000001, anything smaller than that and the approximation actually gets further away, do you know why this is?

[@linuslee6314]

+Alain Rochette my guess would be that that's the limit of precision of whatever programming language you're using. If you want to get more accurate, you might try using eg MATLAB or octave for mathematical computation.

[@nosuchthing8]

Amazing video

[@Peter-pp6kj]

Please more videos with Holly! :-)

[@aakksshhaayy]

Damn it someone put a ring on that lol

[@misterhat5823]

And here I thought I was the only one who looked.

[@jairochang9505]

already has one 💍 02:01

[@rorycollier7521]

Never been so heartbroken to see a ring on someone's finger. :( On topic though, super interesting video. I love this channel! :)

[@Max_Flashheart]

Mandelbrot has a breakdown and heads for Pi = mindblown

[@TheZenbudda]

Love pi. Love the Mandelbrot set. Thanks for the marriage in concepts!

[@michaelbauers8800]

I read that a student ( at the time of discovery) discovered this. Dave Boll

[@djkafab4boll500]

You are correct, Michael. My husband, Dave Boll, discovered this in 1991 as a grad student in Fort Collins. Aaron Klebanoff did the proof in 2001.

[@michaelbauers8800]

The proof is worth looking at, found it online some time ago. I don't pretend to fully understand it, but I always like trying to understand what I can :)

[@ccosborn2000]

I believe that this wasn't found until a computer was used to do boo-coo calculations and color all those little dots depending on how fast the iteration goes to infinity.

[@random554468]

I understand that the mandelbrot set uses c and z as variables, but it is extremely confusing when talking about it and you are pronouncing both varables as "see" instead of "see" and "zed". And then using another c as the outside variable.

[@geuwglesuxballz6074]

+random554468 That would be confusing, but I have yet to encounter anyone that pronounces z as see. Since this speech impediment is apparently quite rare, this is a non-issue as demonstrated in this video by a person who pronounces them quite distinctly, as is nearly always encountered.

[@unvergebeneid]

+random554468 c is see and z is zee. I didn't have any trouble telling the difference and English is not my native language.

[@omp199]

+Penny Lane But /s/ and /z/ are distinct phonemes in your native language, aren't they? (My understanding is that "reisen" and "reißen" make a minimal pair which establishes the existence of distinct phonemes /s/ and /z/ in German. However, I have seen this disputed, so feel free to disagree.) It might be that they are not distinct phonemes in +random554468's native language. In any case, it is neither polite nor helpful, when someone has difficulty with something, to reply, "Oh, really? Well, _I_ have no difficulty with it whatsoever!

[@Sixsince-dd2eu]

c is see and z is zee. I didn't have any trouble telling the difference and English is my native language.

[@wernerruch6218]

This was fascinating.

[@karlmahlmann]

A very fun to watch video.

[@681726]

The question is "why?"

[@saparchitekt]

JFK

[@bytefu]

+benheideveld Jelly Fish Kebab?

[@DrNPCabd]

Only Mathematics are more beautiful than Holly...

[@ashr2526]

You really are doing the lords work, thanks dude!!

[@Axterrix]

It is very interesting. Thank you!!!

[@mana24]

i am here for the beautiful person in the vid

[@suave319]

I need an explanation for this sorcery!

[@manudelmarche]

You just opened the door for conspiracist to tell us that the illuminati have taken off from Planet Nibiru and are headed towards planet earth in an attempt to ensure that 911 was actually a fractal simulation based on alien lasers fired from their bases located on the dark side of the moon and reflected by a giant dyson sphere operated by monsters who escaped the CERN evil portals after scientists have run within the LHC a PI approximation algorithm based on the mandelbrot set drawn by protons shot at each other :-D

[@stevenhernandez8966]

Great videos...

[@LRibeiro97]

I had to comment: THIS IS AWESOME!!!

[@Tobbzn]

But... why?

[@guilemaigre14]

+Tobbzn because we can :)

[@escraftTH]

I HATE THE SOUND OF SHARPIES ON PAPER!!! SOMEONE PLEASE HELP ME. AHHHH!!!!

[@erelde_]

+3.141S92653S29793238462643383279502884I97I69E99 You're on the wrong channel then ^^ (truth is, I dislike it as well)

[@Cr42yguy]

+3.141S92653S29793238462643383279502884I97I69E99 you should really think about that "S2" in your name and maybe change it to "S8" ;)

[@hearueszueke6206]

+3.141S92653S29793238462643383279502884I97I69E99 yes PI, the sound is annoying, but it is a great video, or not?

[@dlwatib]

+3.141S92653S29793238462643383279502884I97I69E99 Mute!

[@KCSutherland]

I agree with Cr42yguy. 3.141 S! 92653 S2!! 979... Try 3.141592653589793238462643383279502884197169399375105820974944 or so.

[@dushyanthabandarapalipana5492]

Thanks a lot!

[@ryanedwards5545]

subscribed. this was great.

[@dentalhams]

Are those lightsaber earrings?

[@OpenMind3000]

nice sache, aber sie sollte aufjedenfall mal Mandelbrot ein bisschen besser aussprechen. :D

[@TheSammy2310]

+OPEN MIND Simon, du hier? :D

[@PurzelPadauZ]

+OPEN MIND Ausserdem könnte sie mal aufhören, Pi wie Kuchen auszusprechen.

[@Serfuzz]

+OPEN MIND Weisst du denn auch, wie er seinen Namen ausgesprochen hat? Er war immerhin Franzose.

[@okaydokey7852]

+Serfuzz ja danke ich nehm einen Hamburger ohne Gürkchen, ne große Pommes und zwei Eimer Popcorn.

[@expertizzlist]

+OPEN MIND Oh Gott und wieder ein deutsches Dummbrot, das unbedingt auf deutsch kommentieren muss - und sich natürlich noch über die Aussprache eines "deutschen" Namens aufregen muss. Komm, geh Pillen flitschen und Traumtagebuch schreiben, aka deine Kernkompetenzen. Dumme Sau.

[@samueldevulder]

Ah Holly... I could spend hours iterating her videos again an again :) I wish there could be more of them on youtube.

[@muthukumaranl]

Holy crap that was awesome!!!

[@Fresse]

How do I conquer this fair maiden?

[@shameekbaranwal]

+idkagoodusernameyet just find the last digit of pi

[@quoabell]

+SKBytes it's 4.

[@shameekbaranwal]

Are you sure? I mean, i need to know how

[@shirankao69]

+SKBytes Alright, here you go. The last digit of pi is 0. Yes, I can prove it.

[@shameekbaranwal]

Prove

[@troll00712]

Guess i have to wait fro the English translation to come out.

[@danielreich2304]

Who has watched this on Pi day and only realizing after it that it is Pi day?

[@HeartAndMind34]

Me, albeit 7 minutes early...

[@djkafab4boll500]

My husband, Dave Boll, discovered Pi & the Mandelbrot Set back in 1991. Aaron Klebanoff did the proof in 2001.

[@idriscarney5299]

Too high for this

[@marcotizzano77]

Dr Holly Krieger, I'm in love with you 😍😁

[@SimonLaudati]

So fascinating

[@yoniziv]

This is amazing!!!

[@ewouthonig371]

What would be someone's motivation to figure out and study something that is of absolutely no value in the real world?

[@chapinward]

by god, she is beautiful..

[@stefanamg63]

This blows my mind!

[@shineisland7447]

“And this is like something that is totally natural to be interested in.” 🤣🤣🤣