Fibonacci Numbers hidden in the Mandelbrot Set - Numberphile

With Dr Holly Krieger from Murray Edwards College, University of Cambridge.
Have a look at Brilliant (and get 20% off) here: brilliant.org/Numberphile
More Numberphile videos with Dr Krieger: bit.ly/HollyKrieger
Her Twitter: / hollykrieger
Extra detail via Holly: Here's a link to an article (meant for a somewhat general audience) by Bob Devaney explaining the numbers of components, which are called periods: plus.maths.org/content/os/iss...
And more from Bob: math.bu.edu/DYSYS/FRACGEOM2/FR...
If checking out brilliant.org/Numberphile ---
Try the complex algebra course at bit.ly/Brilliant_Complex
Editing and animation by Pete McPartlan
Farey Sums: • Funny Fractions and Fo...
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[@kevnar]

This video about Fibonacci numbers was as good as the last two combined!

[@knvids2812]

not going to like because likes are at a fibonacci number

[@benloud8740]

Underrated comment

[@raspi1983]

'B.' in Benoit B. Mandelbrot's name stands for Benoit B. Mandelbrot

[@Paine137]

raspi1983 Old joke.

[@jony4real]

Wait, so what does the second 'B.' stand for? :-)

[@martinmartinmartin2996]

the second "B" stands for Benoit B. Mandelbrot observerms

[@Pacvalham]

kzbin.info/www/bejne/opKrnWCXe7-IgKs The EDM in EDM Detection Mode stands for EDM Detection Mode.

[@rcredidio]

I saw what you did there :)

[@prime2920]

It is so interesting how literally everything in math is connected and intertwined. This is really cool because if you don’t quite understand a certain topic or problem you can look at some things you do understand and connect it to what you are having trouble with.

[@waynedarronwalls6468]

That is the essence of what is known as the Langlands Program, named for Robert Langlands, who essentially created the whole schema...it relates to what are two entirely separate fields in mathematics, harmonic analysis and number theory, and the bridge that links them together.

[@EquaTechnologies]

EXACTLY! I also find fascinating how this figure is encoded in math and anywhere you go in the universe, the figure is still the same!

[@adavewiley]

Kudos to the animator. The scuttling mandelbug was a delight.

[@trucid2]

David Wiley The sound and the animation cracked me up.

[@Lucaazade]

No it was in fact the very opposite of a delight .

[@qwertyasdf66]

Yusss. I came down here to find the comments about it. That made me so happy. 4:24

[@requemao]

It's a Miyazaki Mandelbug!

[@snbeast9545]

It's a Scuttlebug jamboree.

[@simoncarlile5190]

Two of my favorite concepts in one video. Today is looking like a good day.

[@busTedOaS]

Women and Paper?

[@jacobshirley3457]

Audio and Visual Stimulation

[@me_hanics]

fibonacci and grills

[@HiItsSalty]

markers and brown paper?

[@rmm2000]

Fibonacci Numbers make it 3!

[@tgwnn]

She's so cool!

[@fantasick8880]

And cute!

[@UnimatrixOne]

@@fantasick8880 😍

[@megamillionfreak]

Angelic.

[@linyenchin6773]

and preeeeeety!!

[@Kowzorz]

Everything about this video was great! The visuals were tuned perfectly. The explanation was thorough but succinct. And the enthusiasm of the presenter really brings it all together. Great work!

[@OlafDoschke]

I love the little slot machine illustrating the iteration and the ping sound it makes. That's the way Mandelbrot sets should be computed.

[@CrucialMuzic]

Wow mind *BLOWN,* this is amazing Who else just wanted it to keep on zooming in until infinity? [8:41]

[@sebastianelytron8450]

You mean an infinitely long KZbin video? No thanks.

[@ZipplyZane]

Yeah. I wish it would've just kept doing numbers and faded out, to create the impression it could go on forever. Stopping makes it look like it fails at that number.

[@sebastianelytron8450]

Nobody with half a brain thought it "failed" at that number

[@SapphicRain]

Here's one of the deepest zooms fellow Mandelbrot enthusiast kzbin.info/www/bejne/ZpuqkpylbZyoaaM

[@littleboylost1o1]

+

[@meanmreen]

Every time I’m feeling particularly sharp or intelligent, I click on one of these videos and it instantly puts me back in my place 😅 Still, for my limited understanding in advanced math, it was quite interesting.

[@mamoonblue]

I lost her at 1+1 is 2.

[@teovinokur9362]

Nathan Thames That's kind of Numberphile's comment section in a nutshell

[@JK-ff8xf]

"PewDiePie's personal account" kek

[@EvanRustMakes]

Muhammad Mamoon 2+2 is 4, -1 is 3, quick maths

[@deschain1910]

Well, you have the basics down.

[@KnakuanaRka]

Who thinks that’s funny?!!

[@Tehom1]

7:20 Sometimes called the "naive sum" as well. It's also how you construct the Stern-Brocot tree, which enumerates all the positive rational numbers without repeating any.

[@PlayTheMind]

Riemann: My zeta function hides primes Mandelbrot: My set hides Fibonacci Ramanujan: -1/12 __

[@SparHD]

PlayTheMind riemann is way above mandlebort and ramanujan

[@AryanTheMentalist]

Sharklops Haha.. nice one

[@livedandletdie]

What do you mean wrong, the limit of the nonconvergent sum of 1+2+3+4+5...+n where n=Alephnull-1 Does end up as -1*(1/12)

[@DABATTLESUIT]

jawad mansoor YOU CANT JUST MAKE THAT CLAIM AND LEAVE

[@SparHD]

+jawad mansoor Riemann has done much more besides his hypothesis eg: introduced the term manifolds, riemann integrals, was one of the pioneers of non euclidian geometry (with gauss and some other russian guy), also physics and probably tons of things that im not aware of, he was one of the best mathematicians to ever live

[@albertb8999]

It's pretty cool, that these two things have such a connection

[@Nukestarmaster]

But not terribly surprising, Fibonacci numbers pop up just about anywhere.

[@albertb8999]

Nukestarmaster I do not think so_about these_numbers! Fibonacci_numbers_are definitely_not_anywhere,_you_idiot

[@jpphoton]

it leads me to speculate that *everything* is, in fact, encoded in the Mandelbrot Set.

[@MarsLonsen]

fukin druggos

[@HeartAndMind34]

@@albertb8999 I see what you did there, incorporating the sequence into your sentences. Well played, Albert B, well played.

[@gabrielthompson9800]

Test question: In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, what would be the next number? Answer: So you take a point called c on the complex plane...

[@philpayton8965]

So is not a valid word with which to start a sentence except in very rare circumstances, for example explaining the purpose of doing something.

[@nodezsh]

@@philpayton8965 It is valid for a joke though. The joke was that he's explaining a very easy concept in the most complex way imaginable using the most casual language possible.

[@philpayton8965]

@@nodezsh sorry man it was a bit pedantic of me it just used to be a pet hate of mine, probably fuelled by the fact I had a horrible micro-managing supervisor who started every sentence with "So...". it was a me problem, not a you problem, just ignore me. was a long time ago now anyway.

[@aardvarkman6919]

The beauty of this math overwhelms me with emotion. Perhaps that seems strange, but the beauty of how all this works out makes me want to cry.

[@justaphotographer]

I thought I was the only one who feels this way! I completely agree! There is just so much order and beauty in all the world I don’t know how to take it all in emotionally.

[@AntonioKowatsch]

Holly's laugh touches my cardioid :P

[@zixuan1630]

oof you better motice (this is not a typo)

[@dxrpz1669]

Simp

[@waynewalls5033]

@@dxrpz1669 incel

[@anthonymarcelino8460]

Incel

[@maggiehernandezreni]

I used to hate Mathematics. Long story short I developed Arithmophobia since an early age. Until tonight I watched a video about Fibonacci Sequence that introduces me a total new prospective of Math into my life. And for the first time in a long time 33 years more or less (I'm actually 37) I understood Mathematics 😱🤯😍 And after that I found this video is like a double 🤯🤯 sorry I had to is literally mindblowing. I think I can start saying I HAD Arithmophobia. Thank you!

[@nodezsh]

I would 'guess'? it usually happens because of how cumbersome it is to get used to it from such a young age and to basically drill math into your skull by brute force. Maybe, for whatever reason you had a knack for math but never developed the skill to use it because of some bad experience learning it growing up, at a very critical time. But this here, this makes no sense. It's like finding a glitch in the matrix. That's why it's so fun.

[@RicardoRoblesJr]

Thanks for the incredibly fascinating video. The more I learn about the Mandelbrot set, the more I like it. Dr. Krieger is excellent as always.

[@Wayne_Robinson]

Despite being quite familiar with both the Fibonacci sequence and the Mandelbrot set, my mind was indeed blown. It's even more amazing how "number games" like this can relate to the physical world (at least the one we can perceive).

[@trailtrs1]

This is one of the very best layouts of this fractal relationship with the Fibonacci sequence

[@mathsmoica]

*The reason I love mathematics*

[@microbuilder]

"Is the universe a fractal that can be calculated in equation? Is it Fibonaccis perfect golden spiral or is it just my imagination?"

[@kennethwalker3939]

From what I can tell, the world is defined by mathematics and patterns naturally. Math is the translation for the patterns that took the chaos or the earliest known parts of the universe up till as far as we can see. When Mathematics fails is the day I'm lost lol. @@microbuilder

[@aarongoodwin4845]

Beautiful Ladies teaching us?🤫

[@realeyesrealisereallies97]

Peng broads?

[@baptistebauer99]

Amazing.... please do more videos about the Mandelbrot set. It is the most interesiting mathematical object I know of, in my opinion... Loving your videos!

[@RupertBruce]

I wrote a program to generate the Mandelbrot set many years ago and the interesting part was outside the iconic shape - the colours are formed as visual representations of the number of iterations (like a contour map) with the iconic shape merely the set of values that kept on iterating. They were the boring bit! Thank you for showing me what I was missing. I'll have to revisit that code with these extra features to explore!

[@CountKoski]

These inserted animations make all the difference - great thinking mr Haran!

[@numberphile]

Glad you liked them - they were done by Pete McPartlan

[@Cyberspine]

I would tell a joke about Fibonacci. But it's as bad as the two previous jokes you heard combined.

[@wierdalien1]

Cyberspine #groan

[@raphielohnef4678]

You shouldn't start with two zeros... :D

[@Grizzlywer]

0 + 0 = 0

[@bradleylomas7525]

Cyberspine and where does your joke end? If you are going to be funny, at least have an educated punch line to go with it. Those are hilarious

[@josuke6869]

@@bradleylomas7525 HAHAHAHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHHAHAHAHAHAHAHAHAHA AHBAHSBHABSH HAHAHAHAHAHAH

[@stevefrandsen]

Very interesting and well explained Holly as always. Thank you!

[@Manusmusic]

Glad to see her again!

[@winmine0327]

Next: how to cut a cake via prime numbers, Graham's number created by Conway's game of life, and the fractal dimensions inside Parker squares.

[@erik-ic3tp]

Do you want existential crisises? Anyway, cool subjects!

[@matttondr9282]

…while doing a dice trick represented by playing cards printed on the surface of a Klein bottle.

[@shubhamshinde3593]

2:12 even non-mathematicians love this for different reasons xD

[@Myrslokstok]

Shubham Shinde Yes it is fun. They kind of look down on us as children.

[@pizeblu]

When you really uncover it, it is for the same reasons, it is a way to describe or show the nature of the universe and consciousness. Just mathematicians see it in numbers and other people see it more spiritual, but it dissolves into the same sensations one has.

[@raffaelepiccini3405]

@@pizeblu well.. except that the way mathematicians see it actually makes sense, the way you see it doesn't.. it really have nothing to do with consciousness, the nature of the universe... It's just math As a non matematician myself I love it because it shows how something so complex and weird can come up from such a simple rule.. also because fractals are just weird, counterintuitive and fascinating.. but nobody who understand this even a tiny bit would say that it's connected with things like consciousness or the nature of the universe.. get your feet on the ground mate

[@ManlyBog6448]

@@raffaelepiccini3405 I just don't understand how so many people who had never communicated before are able to "figure out" fractals and how they relate to consciousness on their own.

[@declinescore]

2:13 made me smile

[@Schnitz13]

This channel never ceases to amaze me.

[@CybranM]

I can see that nail and gear flag in the background :)

[@paulkingtiger]

and a Reunion swamp hen!

[@TheMrMerudin]

4:30 theory of T H I C C N E S S

[@jugglars]

This is amazing. I'm sure that somebody can show that this connection is absolutely natural. But it definitely is not obvious and that makes it so beautiful and funny. Thumbs up!

[@beekarola]

Great explanation, thanks. By this construction the numerators are also the Fibonacci sequence, two terms behind the denominators. Since the ratio of subsequent terms in the Fibonacci sequences approaches the Golden Ratio as n --> infinity, this means that the ratios that you are considering approach the Reciprocal of the Golden Ratio, Squared. [I think this is right - and surely pretty well known. I just realized it from your presentation.]

[@xanokothe]

Love you Dr Holly!

[@billburr1347]

Intelligence makes people more beautiful

[@NwoDispatcher]

Racial purity makes humanity beautiful

[@NwoDispatcher]

@AccuracyIsGone I agree... come go after the heretic of the g4y empire

[@arthurmee]

@@NwoDispatcher the exact opposite is true. Racial purity leads to an amplification of genetic defects over time. The largest gene pool is the healthiest.

[@andrew7taylor]

@@NwoDispatcher If you truly believe that, don't ever have a DNA testing. You'll find out that you're anything but. Most of your ancestors had more IQ than you and had this idea that screwing around is more fun than raging about a concept that doesn't exist.

[@FrankACai]

@@NwoDispatcher come on, evolution needs tension. How about you leave it be when it's so minor

[@vizart2045]

Numberphile really nails it by explaining math in an entertaining and lighthearted way.

[@miksurankaviita]

Wow, this was the best video from you for a while

[@wntu4]

A new video of Dr. Holly, aka she who commands my heart, mind and soul. This is a great week indeed.

[@ibmicroapple9142]

oh yes, Holly Krieger

[@dlee645]

IBMicroapple There needs to be more Dr. Krieger videos.

[@fantasick8880]

I think I have a new crush.

[@EVILVIKTOR]

@TheronQRamacharaka I'm guessing it's a perfect match. But something tells me the carpet is gone.

[@takotaw8453]

IBMicroapple simp

[@waynewalls5033]

@@takotaw8453 still a virgin

[@cobaltbluesky2276]

i was obsessed with the fibonacci sequence when i was little and i'm obsessed with the mandelbrot set now, seems like a perfect video for me lol

[@Quadflash]

Elegant, beautiful illustration of how math describes our universe. And, how most everything is connected. Thanks for the deep sense of awe I'm feeling right no.

[@marekbanaszkiewicz6338]

Mandelbrot set is amazing. It's incredible how quite simple definition leads to infinitively complex structure.

[@ouss]

she is BACK

[@sumitmandal3901]

my mind is blown off right now. I am amazed and mesmerised almost ecstatic to find out the relation between Julia, Mendelbrot and Fibonacci Thanks a lot

[@djcarter85]

Not only is the maths in this really cool, but I also loved the cheeky Nail and Gear hiding in the background :)

[@Mobin92]

So I guess it's time to fall in love again...

[@takotaw8453]

Mobin92 simp

[@dxrpz1669]

Simp

[@cyberhexreal]

Simp

[@masonhunter2748]

Hi

[@wrexshunt]

Ha ha ha - he may be a math geek and be on about that !

[@Purpial]

Person: "what the heck happened to your mind?!" Me: "oh dont worry it was just blown"

[@MikeReggie95]

What I've learnt is no matter how many Mandelbrot videos I watch, I still have no idea how it's made. Only that it looks amazing on a projector!

[@fishandchips8813]

seriously- you guys and this channel have had an actual, noticeable effect on my life. i am filled with awe more often than i was before folowing you. i have begub studying math in my spare time and i f*ing love it! i cannot thank everyone associated with this enough.

[@Traumtheater0]

The Mandelbrot Set will never be as beautiful as Dr. Holly.

[@Darker7]

I disagree :Ü™

[@SoumilSahu]

Math is amazing. Who even discovers this stuff?!

[@Cellkist]

Soumil Sahu mathematicians

[@SoumilSahu]

+Cellkist obviously, but who goes out of their way to say, "today im gonna pull out the Fibonacci sequence out of a weird shape"

[@axemenace6637]

Soumil Sahu They don't. They explore a weird shape and say "Wow! Fibonacci sequence relates to it!" How do they find the weird shape to being with? Well, mathematicians make random problems up and hope they lead to something interesting. The Mandelbrot set was a lucky discovery!

[@Pete-Logos]

Soumil Sahu sometimes it's a Mathematician, sometimes a "non- Mathematician" notices a pattern and wants to know: "does this 'thing' ever stop or does it go on forever?" They may get bored of it, or keep studying it, or even become obsessed with it (especially if their pattern appears to present itself everywhere; it's a constant reminder.)

[@JM-us3fr]

People who think it's amazing

[@JeffSelf]

Mind blown. Again. These videos are so amazing.

[@feetnincheees]

Gorgeous illuminating presentation :) thanks

[@essimcaldin8294]

Hey, thanks as always

[@Deathranger999]

The picture of the freshman sum they showed was wrong. Is it a Parker freshman sum? 🤔

[@Morstius]

I was searching for the comment pointing that out, wonder who misunderstood the freshman sum joke

[@mikeo759]

For some reason they showed the multiplicative

[@FrankBatistaElJibaro]

Everytime Holly laughs a furry baby kitten is born.

[@davidwilkie9551]

Fabulous presentation, thank you for the intellectual depth charge -> factual recognition time bomb of realization.

[@scottjampa6374]

You had me at Fibona..wow those eyes...

[@robleytrades]

ik i dont understand what they are saying on numberphile but i still like to watch the videos

[@bemerald6079]

I just saw this live at the Cambridge maths open day. So cool!

[@urbaniv]

Am I allowed to say that you look amazing and I just love to watch and hear you explaining very intelligent things I don't understand :-)

[@dmytronadtochyi9116]

She has one of the cutest laughs

[@StopItGarrison]

Your a creepy dude

[@dmytronadtochyi9116]

Michael Eaves what? Why?

[@anthonymarcelino8460]

@@dmytronadtochyi9116 haha incel

[@ElektrykFlaaj]

I'm watching this on high, sounds awesome

[@danielrodgers3002]

Fractal sets and the Fibonacci sequence seem to be a base geography of our world. In this video you seem to show that the Fibonacci sequence auto-magically flows out of the Mandelbrot set. Extremely fascinating, thanks.

[@RationalDisconnect]

I love how Brady has the Nail & Gear in the background.

[@BlacksterVFX]

This was very interesting and well presented 👍🏻

[@skroot7975]

Math never ceases to amaze me.

[@MathAndComputers]

I was wondering if Farey sums would come up! I loved that other video on them, too. :)

[@hamsterpoop]

That was really well explained... thanks!

[@russhendrix9674]

SHE BLINDED ME WITH SCIENCE!!

[@GDQuaza]

This is math, you’re even blinded by vocabulary.

[@67PhilR]

Thomas Dolby......luv science

[@unbreakablefootage]

when shits stormy outside but a new numberphile video is up

[@Dyslexic-Artist-Theory-on-Time]

We also have the start of the Fibonacci numbers 0,1,1... in the complex plane. The zero in the centre can represent t=0 the moment of now in an individual reference frame. We also have negative 1 and positive 1 with a rotation 2π that is a constant represented by ħ=h/2π. Therefore we even have the start of the Fibonacci numbers 0,1,1,2,3,5,8,13,21... forming spiral on all levels of creation!

[@donerickson1954]

I could listen/watch Dr Holly Krieger all day.

[@sebastianelytron8450]

listen? yeah right

[@prophecycat5058]

Gotta love that 9:59

[@alejandronq645]

Dr. Haran really makes great questions

[@WiseGuy508]

He is not a doctor.

[@alejandronq645]

Wise Guy he is indeed

[@navinmeshram3226]

Such an amazingly explained. liked it

[@WiggysanWiggysan]

*NEW RECORD* - Congratulations Numberphille. Only 12 seconds to completely confuse every atom in my head. I await your next video.

[@NicosMind]

*Brady always asks some great questions.* Correct me if im wrong but his qualifications are in engineering and not math(s)? He obviously has some math skills or at the very least a good math instinct, but how good are his actual math skills? Has he done any papers and if he has has he been cited much?

[@bengineer8]

he is a journalist

[@NicosMind]

Bengineer8 He is but im pretty sure ive heard before that he has a qualification in engineering. And most journalists dont know anything about maths. It got me wonderin

[@aarond0623]

Because we keep going between two fractions, does the fraction approach something?

[@Zephei]

trekky0623 I believe it approaches 1 - 1/φ, where φ is the golden ratio (1 + sqrt(5))/2.

[@Myrslokstok]

So when do we get a zoom in at the golden ratio?

[@nasser101]

Approaches 1

[@alekisighl7599]

zoom in infinitely and you will get the golden ratio.

[@rubickon]

Amazing explanation. Thank you.

[@williamsquires3070]

I love mathematics because there’s so much hidden beauty and unexpected patters recur in seemingly unrelated systems. One can keep digging deeper and deeper and still find unexpected patterns that’ll keep you busy for a lifetime... if you let it! :)

[@Pacotille_6529]

Nail and Gear!

[@akikarvonen]

Ah, Dr. Krieger! Must be my lucky day!

[@tomholt1080]

@TheronQRamacharaka jeez chill haha

[@fredflintstone9657]

We should all be so lucky.

[@aloluk]

Wow, my mind is blown!!!!!! I had no idea where mandlebrots even came from and cant believe my favourite subject of complex numbers are hidden in them!

[@jaimeduncan6167]

A very simple and elegant explanation.

[@RickWeberEcon]

But why should the Mandelbrot set have tendrils that coincide with their Fibonacci position?! I feel like I was told I’d get an answer and all I got was an amazing new mystery

[@angelmendez-rivera351]

Rick Weber Isn't that all what answers are?

[@Scurvebeard]

That explanation just made me more confused. The explanation seems like an even crazier way for numbers to function.

[@ColumboJedi]

great hair! ;D

[@navypinkdesign]

The best Mandelbrot tour guide EVER!

[@blazebluebass]

That's what I call a satisfactory video. Simply beautiful.

[@JanSanono]

Exactly 10 minutes... 😍 it's beautiful

[@johnox2226]

Jan Sanono It's actually 9:59

[@O_Chey_]

You are probably a physicist

[@veralapsa]

If you take the frame count of 14975 @ 25 fps it's exactly 9:59, no .something, so John Ox is correct.

[@Oohenry1]

10minutes is the threshhold for more money/view

[@wlan246]

And the video isn't bad either.

[@janablazok8321]

gurl, you made me accutally understand it

[@thetheflyinghawaiian]

I like how all these videos are done in sharpie on cardboard. It's a nice touch. Very satisfying, but I wonder how often they mess up?

[@rivertaig8703]

Mind blown! I think it's so incredibly cool how sites like Numberphile, 3 Blue 1 Brown, and Brilliant are making mathematics fun and inspirational. If you haven't checked out Brilliant's site, I highly recommend it for any curious minds.

[@VoorTrekker88]

Please don't misunderstand me, I am always enthralled by the content of these videos. However, between Dr Holly Krieger, and the absolutely lovely Dr Hannah Fry, I could even watch with the sound off!

[@MasterHigure]

0:45 What's up with the up-arrow-paper appearing in all the videos lately? Oh, and 7:32 should be 1/2 + 1/2 = 2/4.

[@zinnakatt8312]

MasterHigure, That's funny, do more arithmetic.

[@numberphile]

Freshman Sum Freshman Sum!

[@LastRellik]

Baaahahahahahahahahajahahahahahahajajaja

[@MasterHigure]

I mean, you messed up a Freshman sum. That's basically a Parker freshman sum right there.

[@Patrick_Bard]

Yeah, you said that Freshman Sum was in a certain way and showed it differently.

[@pierreabbat6157]

You could draw a straight line from 1/4 to the waist. Since the map is 1/2-sqrt(1/4-z), the straight line from 1/4 is still straight in the circle. And an arc subtends half the angle from a point on the circumference that it does from the center. So the bearing of the point on the circle from 0 is the same as the bearing of the point on the cardioid from 1/4.

[@toma_coffin8013]

I enjoy Dr Holly's videos. Brillance is when you can solve known math, Genius is when you discover/invent NEW math,