Complex Fibonacci Numbers?

[@3blue1brown]

Whoa.

[@ObjectsInMotion]

Wau.

[@unspeakableanomony5340]

Wow.

[@cemgecgel4284]

Wow.

[@ayushdeep7900]

Wow

[@dipi71]

You mathematicians with your ultra-technical terminology, haha!

[@robspiess]

Fun fact: Phi (1.618) is really close to the ratio between miles and kilometers (1.609) which means you can use adjacent Fibonacci numbers to quickly mentally convert back and forth between them. For instance: 89 miles is nearly 144 km (it's actually 143.2), or 21 kilometers is roughly 13 miles (13.05). You can even shift orders of magnitude to do longer distances! e.g., 210 miles is around 340 km (multiplying 21 and 34 by 10) which is close to the actual answer of 337.96 km.

[@witerabid]

🤯

[@nuklearboysymbiote]

OH MY GOD

[@Kebabrulle4869]

I use this trick all the time, it’s so useful

[@jonathanfeller]

I find it easier to just do x+(x/2)+(x/10)

[@XMarkxyz]

Finally a good way to do it quickly, but I still think the imperial units are hideous, just a little less than what I thought before

[@Dodgerific]

I like the Fibonacci series where you start with 0, 0. Its easy to remember

[@jetison333]

I can even calculate any item in that sequence in my head ;)

[@NStripleseven]

The formula for a term in the sequence is the simplest I’ve ever seen

[@darthgeeek]

Math truly is amazing

[@cmyk8964]

Ah yes, the sequence that correctly predicts the exciting things that happens in my life

[@minerforstone4136]

I love how you choose the simplest possible sequence and it's "golden ratio" is undefined

[@punpcklbw]

The moments when his amazed face perfectly merges with himself are really trippy. Nice touch =P

[@niccy266]

It's called the buttercup challenge, he links it at the bottom of the description

[@ryla22]

It made me look up the song Really good song

[@tomppeli0]

@@niccy266 1

[@mrss649]

Lol yes

[@flisboac]

Kinda freaks me out, tho

[@BH-2023]

Well, I mean... the Fibonacci sequence was discovered thinking about the ideal procreation of rabbits, and it's pretty hard to have a negative rabbit mate with a positive rabbit

[@themathaces8370]

That's what mathematicians do... Extending simple ideas to random dimensions...

[@jayfredrickson8632]

You can but then they mutually annihilate and you get a huge explosion

[@reubenmanzo2054]

They say opposites attract, don't they?

[@fders938]

Don't even get started with imaginary and 4D rabbits

[@mr_waffles_the_dog]

All you have to do is swing the rabbits around your head at a moderate fraction of the speed of light, and you get a handy anti-rabbit

[@NyscanRohid]

φ : Let's see what's at the end of this infinite sum... φ : π!? π : Hey. φ : What are you doing in complex space? π : I work here. It's my job to be here at all times.

[@weckar]

π : I was here long before you got here, and will be here long after you leave.

[@mr.champion7304]

*rational numbers in geometric sequences intensifies*

[@FourthDerivative]

"Wait, it's all pi?" "Always has been"

[@sharpfang]

C'mon. You've messed with complex numbers. How are you *not* expecting a π there? Also, mandatory e, this time wearing the disguise of 'ln'

[@lucyf6516]

sharpfang dude most people don’t intuitively know that pi has something to do with the complex plane lol. I’m sure you’re very smart. Here’s a gold star ⭐️.

[@haydenhoes]

this aint no sit down maths. we standin up now

[@Vaaaaadim]

Rise up gamers

[@volodyadykun6490]

I think Matt isn't stand-up comedian, he's sitting all the video, he's more of sit-down comedian

[@thecuriousgorilla6005]

Calm down Nolan

[@standupmaths]

Ha. I’m filming in a small room at home during the lock-down.

[@demonking86420]

He's doing the Parker Square equivalent for standing up (dead meme I know)

[@HugoBDesigner]

The synchronized "Matt Parker's Maths Puzzles" cards were... _chef kiss_

[@standupmaths]

thankyouverymuch

[@caster-]

I hadn't even noticed! Very nicely done!

[@Danilego]

6:49 this is so oddly satisfying

[@Nate9273]

MammamiaDasAhSpicyMeatball

[@hecko-yes]

a channel after my own heart

[@brian554xx]

I've always preferred the 0, 1 start. With these numbers often found in nature, adding a moment of creation feels profound.

[@rarebeeph1783]

it also feels even simpler than a 1, 1 start, like if you had to enumerate all the possible starts, you'd start something like "(0, 0); (0, 1)"

[@KarstenJohansson]

I think both are found in nature. In some spiraled plants, there is a gap in the center which is effectively 0. Others have something in the center which is effectively 1. Until your comment, I'd never considered that. I'm pretty sure that the vast majority of fib-nth() functions consider the 1st nth to be 0.

[@Kinaibhlan]

0 1 is the official way to start it, and the fact it shows up so much in nature and has that creation aspect tells me there's a creator :)

[@antoniofigueroa887]

No 1 0

[@dragoncurveenthusiast]

I have to say, I'm a tiny bit disappointed that his amazed face didn't follow the graph. It even pointed at his face! 6:45

[@trickytreyperfected1482]

Love the name

[@brandonfrancey5592]

I was waiting for the line, "And so I contacted Ben yet again and for some reason he blocked me and stopped responding to my e-mails."

[@brianwestley6985]

The Fibonacci convention was huge this year -- it was as large as the previous two put together. ThankyouladiesandgermsI'llbehereallweektrythechicken

[@zaraak323i]

Tipyourwaiters!

[@quesoestbonne]

@Idiot Online Wondering Aloud 👏, 👏, 👏👏,👏👏👏,👏👏👏👏👏,👏👏👏👏👏👏👏👏,👏👏👏👏👏👏👏👏👏👏👏👏👏.....

[@stevemattero1471]

Wait there's a fibonacci CONVENTION??? When and where!?

[@demonking86420]

Oh get the heck out, I just got that

[@Lawrencelot89]

@@stevemattero1471 Location: just add the coordinates of the locations of the last two conventions. Time: just add the dates of the previous two conventions to get the new date.

[@asailijhijr]

Missed opportunity: you could have had your amazed face trace the path of the graph shown on the screen at the time.

[@jamesthomas3871]

I think he tried. It was awfully close, wasn't it?!

[@arranmcgown2386]

I thought he was going to

[@pronounjow]

He sure Parker Squared that one!

[@sxygrandpa8061]

The Binet formula for the Lucas sequence is actually simpler than the Fibonacci sequence: (ϕ)^n + (-1/ϕ)^n = nth Lucas number

[@yogi30051972]

That's amazing.

[@derrickmelton5844]

This reminds me of an experiment I did with Conway's life. I started wondering what would happen using the standard life rules with a bounded game, but set a cutoff for how many steps the game would iterate. I then took the union of each iteration of the previous game to create a seed for a new game, and continued to repeat the process. I mainly was doing this to see if you could use GOL to generate interesting height maps when I found an interesting property. For some reason if my iteration value was 2 meaning 2 distinct steps after the initial state to create a new input, the mean value of my bounded inputs approached pi. When they surpassed pi they would eventually trend back down to pi. I have no idea why pi arose because I am not that skilled at math, but I still wonder why that generation of inputs for a board state would trend towards it. The most I discovered was that method of generation retained symmetry if it existed in the initial board state meaning a blob in the very center would create symmetry along the diagonal, horizontal, and vertical axes.

[@av3stube480]

Okay I lack the mental capacity to imagine what you did but I'm really interested in why would Pi appear there...

[@adityaekbote8498]

Yo idk what you are saying but that looks exciting let us know if you find anything

[@creativenametxt2960]

Interesting. Would you provide more details?

[@pranavkondapalli9306]

you did an experiment with 'Conway's life' lmao what PS: ik what GoL is

[@trickytreyperfected1482]

@@pranavkondapalli9306 wow, I didn't even notice that first read through. That's an unfortunate typo for OP to make.

[@Rubrickety]

To explore this further would clearly require a large investment of time and effort. I suggest you apply for a Grant. Sanderson, ideally.

[@pamdemonia]

I see what you did there and I approve!

[@anirudhranjan7002]

For those who don't know, Grant Sanderson is the host of 3Blue1Brown

[@anuwamy969]

@@anirudhranjan7002 he already comment

[@davidgrupp7662]

Can we just take a moment to appreciate the editing involved for the amaze face

[@Woe_YT]

Its actually pretty simple, you just cut a still from a frame of the video and then move it to the time and place in the video in reverse. It is a cool effect though.

[@bacon.cheesecake]

I've always been a fan of the 0, 1 start, glad to see it got some recognition

[@andymcl92]

I too like that start, although the 1,1 makes most sense with the origin story (breeding rabbits).

[@OscarCunningham]

@@andymcl92 In the 1st generation you have one pair of young rabbits and no mature rabbits. So in the 0th generation you must have one pair of mature rabbits and one pair of young antirabbits. Then the mature rabbits give birth to the young pair we see in the 1st generation, but there are no mature rabbits left in the 1st generation because the antirabbits grow up and annihilate them.

[@Chris_Cross]

But to me, it doesn't seem like it should work. The reason there is the two "1"s is because there's nothing before it. So if you start at 0, there's nothing before it, so you put another 0. "0, 0". But then, if you try to make the sequence by adding the two previous numbers to get the next, it just becomes and infinite string of "0"s.

[@peglor]

I like to start with two zeros - makes the maths much simpler...

[@crustyoldfart]

Why not make it completely general and start with the integers A,B ? So the series progresses A,B,A+B,A+2*B,2*A+3*B,3*A+5*B.... And we see that adjacent Fibonacci numbers occur in the coefficients. We can legitimately make A,B anything we chose including +ve and -ve values chosen at random.

[@theddrman]

I love that you've made a living of messing around with interesting numbers and sharing it with us. I used to do things like this on my TI-86 graphing calculator, but never got far enough to make these kinds of incredible graphs (it was far beyond my mathematical understanding). Thanks for sharing your passion!

[@clockworkkirlia7475]

This is just such a cool maths revelation, with an amazing payoff and one of the absolute best editing jokes I've ever seen. That's pi outta pi from me, even if I apparently can't read 3d plots very well.

[@hexeddecimals]

Now I want to see a 3b1b style animation of the 2d inputs moving around to their 2d outputs

[@fibbooo1123]

One of the “results” of the 3b1b videos is exponentiation moves complex numbers around in circles- so presumably like that? But maybe not since there are 2 exponentiations being added

[@volodyadykun6490]

Also just all the colours ordered as inputs mapping to their outputs

[@violetsteele350]

He commented on this vid. You could comment on his commeng

[@Chlorate299]

Find someone who looks at you with the same excitement that Matt gets around numbers.

[@georgplaz]

with detached heads floating in space? no thanks!

[@duskyrc1373]

Heh, that random pi at the end. That's something I love about maths, if you're ever hungry you never have to go far to find a delicious pi.

[@vblaas246]

I want to know why though. Is it because every periodic system has a (circumpherence/2r)*dt relation? What about an 'oval', it can always be projected back to a circle right? Giving you a pi in every periodic system somewhere?

[@TheBasikShow]

V Blaas I’m not sure exactly why this particular pi shows up, but complex analysis is absolutely riddled with pi so it isn’t that surprising. In particular, this function is made of exponentials, and complex exponentials are inextricably linked with pi.

[@ottolehikoinen6193]

2/5ths make it sound he could've used τau and get rid of the pesky 2.

[@wilddogspam]

@@ottolehikoinen6193 2/5 * 1/π =4/5 * 2/τ though.

[@criskity]

@@TheBasikShow I remember when I took Complex Analysis in college, the answer to the exercises we did was almost always pi. If not, it was zero, 2pi, or pi/2.

[@claytrav2153]

The line looks like my Doctors Signiature

[@georgplaz]

underappreciated comment!

[@AlanTheBeast100]

Looks like my doc's prescription for ... well anything and everything.

[@DarqIce]

I'm actually thinking of trying to align my signature to this plot just for my internal giggles :D Would also make a nice company logo.

[@mosconirmg]

When I learned Fibonacci sequence in 99 (i was 15), I tried to extended backwards, but I lacked the math to understand this whole "bi infinite" sequence. Watching this video was a real time travel to the past. Nice work!

[@HipNerd]

The ‘face’ bits were great. Nice effect.

[@mangusto666]

You could utilize time representing one variable. An animated 3D graphic may be used to visualize a 4D equation.

[@himanbam]

You can kinda already do that with his program by sliding the complex input value.

[@sherlock_norris]

Or you can colorcode the complex plane and then color it according to the complex output.

[@elvis_mello]

That seems like something a physicist would do

[@buddyclem7328]

Yes! I want to see this!

[@davidpalomino9138]

While this does work in theory, it's not going to be like what most may imagine. Since the full plot is a 2D manifold embedded in 4 spacial dimensions, a 3D cross section would just be a 1D manifold embedded in 3 spacial dimensions.

[@SeptimusBlyth]

19:44 Here comes Matt’s π day calculation 2021.

[@twojuiceman]

I thought the same thing. Use that absurd formula for area under the curve to calculate pi

[@BrodieEaton]

Matt just looks so happy, and it makes me happy. This is actually a really cool find! Well done!

[@StrangeIndeed]

1:34 I just love Matt's humor, where he randomly does stuff, never addresses it, etc. Plz never change

[@sacielo]

The "other" thing I loved about this was the "how we got there" story. A great example of the mindset to approach problems scientifically and what to look for :)

[@OH-pc5jx]

Yeah I really liked this video start to finish - but I *really* wish he’d done a domain colouring/colour wheel plot!! I find them so much more intuitive

[@scanerang]

3Blue1Brown has a nice way to represent 4D graphs. What he does is draw the transformed gridlines of the input space. It's like what you did with the graph with the real number line as input.

[@olmostgudinaf8100]

Did no one think of using colour for the 4th dimension?

[@MPSpecial]

He also used colour gradients in another video (about finding the zeros of a complex function I believe)

[@f.p.5410]

Those graphs don't always look good, and they can even be more confusing for non-injective functions. Watch 3b1b video on Riemann's zeta, the map looks cool but it doesn't tell you anything about the function. You can't really recognize slopes and shapes, it's a mess. Unfortunately, this function looks like the kind of function which would be too messy to represent as a grid mapping.

[@f.p.5410]

@@olmostgudinaf8100 Yes, and it's really useful from a topological perspective. For example, a Klein bottle is quite intuitive if you colour the overlapping part because you can see the neck part moving in the "colour dimension". It's less useful for complex functions because you can't really see slopes. It's hard to tell if a colour is shifting at a parabolic/exponential rate. It's still used a lot through a plotting technique called domain colouring, but it's still not a perfect way to plot complex functions. There isn't a perfect way unfortunately, you'll always have some drawbacks.

[@teamupleft7097]

What about two overlapping 3d surfaces attached to the 2D complex plane? Like the thumbnail for this video but with one real surface and one imaginary surface.

[@TheSpacecraftX]

It hit me near the end how good of a job you've done of editing this. The virtual plot that you're actually pointing to points on like a weatherman. Also I suspect you just learned how to do the face thing and it's really cool.

[@HYPERPLASMATIER]

The positive only values look like a growing spiral from the side, while the negatives create a spiral we serve head-on. If you used them as different POV, you could maybe plot out the tips of leaves or the sharp bits of a pinecone. It's really neat..

[@DavidDyte1969]

That was really sweet. I saw the title and started trying to imagine an equation describing a curve like that, with zeroes where the Fibonacci numbers are. Didn't realize that such an elegant parameterized version already existed.

[@mr.neverspeaks7884]

I “enjoy” math and this is WAY out of my understanding of math ,but I just love the content. Thank you!

[@Ragnarok540]

I was going to say this is not complex at all but yeah, is a bit complex. Get it? Is easy, thought, except for the 4D visualization part.

[@sebastianjost]

@@Ragnarok540 4d visualization can be done a lot better when using colors. I've explained that in this comment section before so I won't do it here again. But if you search, you will find how it's done.

[@carrotfacts]

@@Ragnarok540 For someone who watches math youtube videos for fun, it's quite difficult. Glad you get it so easily, though

[@tirocska]

@@carrotfacts Could you explain which part(s) you find difficult? Just curious.

[@andrewosborne1447]

I mean, all he did was say “here’s a solution to a recursion. It’s continuous on C”.

[@fozzzyyy]

90% of the budget for that amazed face effect at 6:47 Edit: I stand corrected 7:28 btw for plotting complex functions, I've been trying for a while to make a program the plots the path of f(x + ti) in 3D where t is just the time. This could be a 4D plot

[@louismichaud9379]

What is the song called

[@leomadero562]

Buttercup

[@qwertyuiopzxcfgh]

"What a stupid idea! Who wants a video about Fibonacci numbers at 3 in the morning!?" Matt Parker: "Oh boy, 3 AM!"

[@soupisfornoobs4081]

Now me at GMT+2, knowing sleep is a social construct

[@alexandermcclure6185]

Literally anyone awake at 0300 just wants something to do.

[@connorconnor2421]

5:34 this excites me uncontrollably it's impossible not to smile

[@tony_yt5967]

“Ofc you’re dividing it by the sq root of 5, big fan!”😂😂made me happy made me smile nice 👍🏽

[@chrisxd146]

It's insane how often pi shows up in any level of math. Funnily enough it's the first example I given when helping students to better understand infinite series and what they're useful for (alongside Euler's identity). Very cool video that I wish I wouldn't have waited so long to watch.

[@nutronstar45]

always with a 2.

[@TheNefari]

i just realised that the curve that goes through 1 twice is actually a spiral/cone looked at from the side :D

[@Microtherion]

I think it could even be described as an 'epicycle'. (Oy, Ptolemy: no! I respect your attempt to maintain the geometric integrity of our planet's immediate locality, but if you was to come round here, and start arranging *my furniture* into a highly idiosyncratic theological exegesis, I would say - 'Ptolemy, nooo! Outside now! You are not in the bustling multicultural milieu of ancient Alexandria. This is Lambeth. Now get your pharaonic physog out of my impromptu courtyard knees-up, you stripy antediluvian muppet!' Etc). :)

[@DerKlappspaten]

6:44 actually looks like an inwards spiral beeing (exponentially) accelerated to the right

[@PiercingSight]

That was my thought too. Could help explain why pi shows up a few times. The Fibonacci numbers may just be a 1d slice of a 2d projection of a 3d spiral.

[@mryellow4633]

Wow interesting observation. Meanwhile the negative numbers in the Binet formula formed an actual spiral 7:09. If the positive inputs can be described as an "inward spiral" then the negative numbers would be an outward spiral.

[@kylecow1930]

Yeah it is!!, the (-phi)^-n term acts as a spiral exponentially decresing in radius and the phi^n acts to push the centre of the spiral to the right exponentially

[@Turcian]

Matt: Uses Python for computing the values Also Matt: Uses Excel to plot the values computed using Python We need to talk about Matplotlib. Or should I call it Mattplotlib?

[@silverzero9524]

Mattdoesntplotlib

[@MaxDiscere]

Matplotlib sucks, excel is far better if you want to be fast

[@trueriver1950]

@@MaxDiscere Agreed Excel is great for a fast and dirty first look. But it's no good at all if you want to be able to zoom, change point of view, etc

[@iabervon]

Mattplotlib will give you graphs that are interesting and look good, but if you happen to look at them diagonally one of the ways, they don't quite add up. Also, some of your numbers appear in two places for some reason.

[@pascalschetelat7198]

Wow, lots of people here need to learn to use matplotlib which is arguably both faster and and more powerful than excel. Plus it is interactive and give nice looking graphs

[@lovingfoe1444]

This guy went insane. Really maths “y”. Imaginative. I love how he opens he mouth to show his excitement.

[@ChuckHenebry]

Love the various faces. Nice video editing! The goodbye face kept me watching all the way through the Jane St. promotional-a first for me. Nice audience hook, Matt!

[@KarstenJohansson]

6:29 this is the shadow of a spiral (3D onto 2D plane). Then the next part of the video shows a spiral, which is still a shadow of the spiral, but seen from a fairly easily guessed angle in 3d space.

[@timh.6872]

That's a good catch! It does look like a projection of a decaying helix.

[@atomictraveller]

i saw nonlinear damped mass spring (have a vid on using quadrature osc to appx sine and cosine) s0 = 1.f; s1 = 0.f; // init s0 -= w * s1; s1 += w * s0; // loop .. where w = angular frequency 2 * pi * hz / samplerate

[@gordonwiley2006]

The way Matt's goodbye face's hand was animated was wigging me out for some reason. Does not tarnish the good maths though.

[@MCAndyT]

Really kept my attention while he did the sponsored portion. Very clever, that one...

[@fennecfoxfanatic]

That animation kept me on my toes! More intense than the bouncing dvd logo

[@gordonrichardson2972]

Gordon Wiley Tom Scott did a video on green-screen perspective errors being creepy.

[@kuromurasakizero9515]

it seemed like it was both timed to the beat of the music and slightly off at the same time. Methinks he learned a new editing trick and enjoyed it a lot :)

[@partywumpus5267]

@@kuromurasakizero9515 definitely looks like he's having a fun time with it

[@Yazoocxi]

Matt! You are already in python. Take a look at the library "matplotlib" it can do zoomable/movable 3D plots directly from python.

[@vibacourtney-battista6918]

What software could I code an interactable fractal zoom using python?

[@RubenMoor]

Seems crazy to me to rely on excel when you have matplotlib - or at least I wouldn't admit it 😬

[@andrewosborne1447]

Matplotlib, pandas, numpy.

[@ThaiNguyen-bl1pi]

Ruben Moor you underestimate the obsession of Matt with Excel

[@jacobwolf5640]

Very interesting, that plot of the Binet sequence appears to spell out 'Jeremy Bearimy'...

[@pedrosso0]

Jeremy Bearimy you say?

[@nymalous3428]

Coincidentally, I just taught my class graphing complex numbers on the complex plane yesterday... and today I get this recommendation.

[@nicolaom]

I’m wondering what are the properties of the loop that the two 1’s form... I don't know why, but it was the part that I found the coolest

[@theot1692]

I wonder what the area of the loop is

[@richardpike8748]

@@theot1692 I was about to say this lol. I also wonder the area of the loop. And if you wanted to go deeper I guess you could also do analysis of curvature, length, etc... never know what you might find.

[@gajbooks]

The real question is does the loop shrink in the complex plane, and if so where does it reach zero size?

[@richardpike8748]

@@gajbooks volume of the loop? o_o

[@hwliebenberg5487]

I dont think the line crosses the x axis at all, I believe that from the point of perspective where you looking from the X/Y axis vantage point it looks like it crosses the x-axis, but it doesn't, it loops around it, just like a inverse spiral if looking from the vantage point of Z/Y axis. (I dont know, it just looks like it)

[@eldabys]

playing buttercup while he does the amazed face... LMAO

[@zozzy4630]

Started reading through the comments hoping someone had already ID'd the song for me. Thank you

[@ALZlper]

@@zozzy4630 You mean Darude Sandstorm?

[@Rabbit-the-One]

@@ALZlper I think he does indeed mean Sandstorm by Darude.

[@colinsanders3124]

@@ALZlper The song is Buttercup by Jack Stauber

[@colinsanders3124]

@@zozzy4630 kzbin.info/www/bejne/m2PUeGirrKl2jZY

[@DemoniteBL]

I love how excited Matt is about everything.

[@tekkorb778]

I never expected that the graph of the negative positions of the fibonacci seqeuence would give a fibonacci spiral, amazing!

[@eekee6034]

That first graph made me the most excited I've been about math, *ever!* :D

[@surrealdynamics4077]

I love your videos! I don't really understand the complex maths involved, and I don't think I ever will get to. But maths really spark an interest and curiosity in me, I love to learn more and take a peek into this otherworldly stuff!

[@romajimamulo]

What I would do is the way 3blue1brown did the display of the Zetta function: start with a grid in the complex plane, and animate distorting it

[@mistec34]

I love your enthusiasm, my dude. Keep learning, growing and challenging yourself and others! :)

[@dianedong1062]

Beautiful! I never thought about using anything other than positive real numbers in the Fibonnaci sequence until today.

[@plaustrarius]

Was able to graph the 2D slice with real inputs, working on the complex input/complex output graphs excellent project thank you Matt!!

[@ImDoubleDelight]

"A third" incorrectly stated as 0.333, yet time stamped at 03:33 is some fine trolling... 🧐

[@i_booba]

That's some fine detective work also, dang.

[@GunganWorks]

The amazed face absolutely cracked me up!!!

[@Poldx]

6:50 - next level videoediting - I love it

[@lmitz]

I like how at 1:35 it is perfectly synced between the text in the previous video and the current video

[@yukelalexandre8885]

Hell yes! Been thinking about this for two years but couldn’t visualize it without the tools!

[@YuVW]

I can't believe you did all of this teasing and then didn't show the plot across the line containing the zeroes

[@Gusquoise]

If the surface have not a name yet It could be named "the Parker's Blanket"

[@jamesbra4410]

Very informative, lots of effort put in. Some of the best math content I've seen.

[@scottytremaineplays9461]

This should have been in my complex analysis module. Also, the limit of the integral of the Binet function - mind blown 🤯

[@deldarel]

That goodbye face floating around caused a form of anxiety I've not yet felt before. I loved the video. I kinda lost it at complex inputs, but those 2D graphs were super satisfying

[@helloworld-ru9tz]

This video just tells us how amazing Matt Parker's editing and programming skills are. 6:45 and 7:24

[@Lyssinator]

You’ve likely already heard of it, but you could also look at the 5-adic interpolation of the Fibonacci numbers; this yields a 5-adic continuous function in fact! Really cool stuff. Unfortunately, I think you‘d run into the same difficulty (or more) getting a visualization of the result.

[@WilliametcCook]

I was surprised at how easy it is to graph in Desmos: \frac{\left(\phi^{t},0 ight)-\frac{1}{\phi^{t}}\left(\cos\left(t\pi ight),\sin\left(t\pi ight) ight)}{\sqrt{5}} Set your preferred boundaries for _t_ Or, if you want animation, restrict _t_ to [0,1] and replace every instance of _t_ with _at_ for some variable _a_

[@nbee4981]

I was taking a course regarding the Laplace Transform , lo, about 40+ years ago, and, as a part of it, the prof introduced the notion of "difference equations" (cf. "differential equations") and the difference equation analog of the LT called the "Z Transform". As he went into the idea, I realized that you could use the Z transform to redefine a Fibonacci sequence as a function of the two initial values (this was a variable Fib sequence, not just the uniform standard one) and the "n-th" value you wanted -- that is, rather than have to calculate all the intermediate numbers, you could get the n-th term by simply plugging in N, F-sub-0 and F-sub-1. And, in fact, this was the subject for the next day's class/lesson. I always love it when I see where the class is going ahead of time. Not sure if that can be turned into a segment, but you might enjoy looking over it either way. Transforms are pretty cool things. And the LT is actually pretty primitive, being one of the first tools invented to manipulate, analyze, and understand the concepts of differentials.

[@belladoralastname6096]

This is one of my favorite channels specifically because I think this is the only person I've ever seen excited as I get for math

[@MaximusMuleti]

This was one of the coolest Fibonacci maths I've ever seen!!!

[@rtleitao78]

Trippy. You kinda lost me when you started plotting 4D, but I stayed an it was cool. Way over my head, but cool. I loved the limits at the end. Gorgeous. No wonder some folks believe there is something magical about these number. It is pretty.

[@bentbliley]

In regards to where Fibonacci starts, I’d always been taught it starts 0 1.

[@ComradeTiki]

Zero indexing... nice.

[@kyrilcouda]

6:45 Liked and subscribed just for that meme. Good job, Matt :D

[@DevashishGuptaOfficial]

The complex plot for the positive analytic continuation of fibonacci sequence, kinda looks like another spiral (spiralling inwards) but viewed from the side! Very interesting!

[@DjImpossibility]

Gosh darn it, now I want to look at Fibonacci quarternions!

[@jeremy.N]

Hey Matt, looks great. However, you should try taking the logarithm of the absolute value, when plotting, since the fibonacci series is an exponential series and thus diverges quite fast. That would also help showing the zeroes and the "waves" you can see in the function.

[@joshcork2201]

This works when the output is large, but for small values, log is a very large negative number. Furthermore log is undefined at 0.

[@aliasalias2131]

Nice new point of view, thank you :). Also, by the way, in the log abs plot, you can see the two binet terms as two planes, which I find constructive. Remark: Personally I like to plot the abs and use colors for the output phase, to keep it 3d. It distracts a little bit from the phase, but often you don't really need it, and e.g. with the log abs you can see the zeros and poles quite well.

[@MajikkanCat]

That sounds awesome! 😸 It will both show the angle and make it 🌈rainbow, which automatically boosts the awesomeness of a mathematical plot by about omega! Sadly, no version of Geogebra I've ever tried can make multicoloured outputs 😿 so it's gonna have to be a new file, not an updated version of this particular interactive.

[@switch1e]

This is so cool. Thank you for making this video

[@asherdp]

I saw the thumbnail and derived everything in this video and now I finally saw it and was like "why are you so surprised?", I was already trying to go in between the fibbonacci numbers and then I saw "Complex Fibonacci" and immediately thought binet's formula

[@parodysam]

First time I’ve seen e, pi, and phi all together like that

[@jasonpatterson8091]

You said "deposit" in your presentation of that puzzle. That implied positive numbers only. I don't know how many other people considered negatives and discarded the idea as outside the rules as presented, but I did. I'm still bitter about the bonus points.

[@theadamabrams]

I could have written this exact comment word for word! "Deposit" means a strictly positive number. I even looked up the word at the time because, like others, I thought of the negative answer and then determined that it was not a valid solution (and I did notice Matt's use of the word "integers", but he also explicitly used the word "deposit"). I'm okay with giving equal points for the answer involving negatives, but it seems insulting to give *additional* points for an *_incorrect_* answer.

[@DukeBG]

Same. I was offended by Matt saying that the rest of the people didn't consider the negative numbers. I considered and actually emailed about them, but did not enter it in the answer box because I didn't want to lose points.

[@Huntracony]

@@DukeBG In MPMP: if it goes in the answer box and it works, it is valid. Though, I also didn't try it because I thought positive only was implied.

[@DukeBG]

@@Huntracony Yeah, now we know that and we'll act accordingly in the future. But I'm still going to be cross about "not considered" wording here.

[@Huntracony]

@@DukeBG Rightfully so.

[@jony7779]

You give "domain coloring" a try next time you want to visualize functions of complex numbers.

[@notottomedic]

Ok.... (2/5ln(phi))/pi might be my favorite conclusion of any of Matt's videos. They all showed up. The 3 giants. Pi, e, and phi. That's incredible.

[@ataraxianAscendant]

ok i really like the buttercup challenge thing you had going on, I've been listening to a lot of jack stauber recently and I thought it was really cool to see one of his songs appear in one of your videos!

[@Czxvkq]

That equation at the very end reminds me of Euler's Identity. You could call it Parker's Identity!

[@stephenamy9879]

I liked this so much that I clicked "like" n times, n being an odd number

[@rextheroyalist6389]

Stephen Amy so 2n + 1 ?

[@masonhunter2748]

I clicked like n times, n being an even number

[@revenevan11]

9:00 you could use a 3d plot with the input complex n being the 2d surface of the x-y plane, and the z axis being one output, and then either use some color gradient for the other output instead of a 4th dimension, or animate the 3d plot over time as the 4th dimension so that we could get some idea of how it changes as you slide along that 4d axis. I've done a bit of this sort of sliding through the 4th axis by animating with 3d slices to show simple 4d objects like hyperspheres in wolfram mathematica, but my trial has expired for that and I haven't gotten around to learning python yet to do it myself. Loved this video!!! Edit: you (and Ben) did some stuff like this, I just commented too early lol.

[@revenevan11]

Re-watching this video, I'm realizing how close I was to pushing back into the negative numbers when I was learning about the fibonacci sequence! (Basically, as a kid, I realized that 0, 1, 1.... was more of the fundamental starting point, so when I wrote my fibonacci generator on my TI-84, I started with 0 and 1 😁)

[@criskity]

That loop in the graph is mind-blowing!

[@albertvila5364]

That reaction on 5:34 was exactly what I was expecting and actually made my day

[@nicholasmeara6869]

MATT I had a question? Does a spiral shell follow the spiral graph shown at: 7:15 or does it follow the Fibonacci sequence or is is just random spiral found in nature?

[@prasanttwo281]

Spirals in shells and sunflowers and such follow the spiral you get if you infinitely zoom out from Matt's spiral, because the sequence approaches the goldes ratio towards infinity

[@timh.6872]

It's an approximate golden spiral because the fractional negative powers of the negative inverse of phi rotate and grow like phi, while the negative powers of phi disappear pretty quickly. Is it in nature? Sometimes, but the base usually isn't phi. Humans notice logarithmic patterns all the time, and it's hard to tell different bases apart unless you measure.

[@Owen_loves_Butters]

Plus, not all spirals are based on the golden ratio

[@97ynoT91]

Little known fact that if you substitute "n" in the Binet formula for the amount of AP flour (unbleached, in grams) used in your Binet recipe, you can calculate exactly how much powdered sugar (in micrograms) to apply after frying them...

[@SaberTooth2251]

It seems to me the best way to visualize 4 dimensions would be to have an intractable rotatable 3D structure as you illustrated from Ben, but combined with the idea that you have some slider bar that acts as a perpendicular axis. You can treat that as a temporal axis and watch the time evolution of a 3 dimensional space from any of an infinite number of 2 dimensional views

[@VcSaJen]

What if you make 4th axis a color? I think it's possible, but depending on graph it could be messy/opaque.

[@josephyoung6749]

that curve at the 6 and a half minute mark looks so nice, almost like cursive... well done!

[@F_L_U_X]

I've been binging your videos all night. 6:47 was WAY funnier than it should have been lol