## e to the pi i for dummies

Recently we did a video on the most

mysterious and beautiful identity in mathematics e to the pi i is equal to minus one.

Comes up three times in the Simpsons which, of course, makes it even more

important. Now, afterwards a few people challenged me to

come up with an explanation that even Homer can understand and I’ve actually been agonizing over this ever since. And today I want to do just that, I want

to explain e to the pi is equal to minus 1 to

someone like Homer. Ok, someone like Homer who can only do addition, subtraction,

multiplication, division. So, we have to remind him or tell him

two things. The first one is that i is this strange complex number square root of -1, i squared is -1. Second thing is just

kind of a reminder. If you’ve got a semicircle of radius 1 then the length of

the semicircle is pi. Ok, so keep those two things in mind, we

have to use them later on. The first thing I have to explain to

Homer is what is e. So to do that I give him a dollar and tell him “Go to the

bank”. Now I’ve arranged with the manager here to give him 100% interest

over a year, ok. So what happens to this one dollar

when Homer puts it in? Well after one year he has 1+1=2 dollars. Now this is actually not the best you can do with a 100% interest, you can do better if

you find a better bank. And we found a better bank, the Second Bank of

Springfield. At the Second Bank of Springfield they calculate and credit

interest twice a year. So after six months what happens? You get fifty percent on what you’ve got

there. So that’s 50 cents which gives 1.5 dollars. Now another six months pass. Half of 1.5 is 0.75, so you have to add that to 1.5 and that gives you 2.25. That’s what you’ve got at the end of the year if you calculate

and credit twice. Now, at the Third Bank of Springfield they do it three times. So what do you get? After four months you

get that, after eight months you get that and at the end

of the year you’ve got that, even more. And it’s actually quite easy to figure

out the general formula for this. Well, maybe Homer cannot do it, … , but I can do it. So

it’s this one here and you can probably do it, too, if you’re watching this video. So, it’s (1+1/n^n. So, if you credit n times throughout the year that’s how much money you have at the

end of the year. Let’s just check this for the simplest cases 2 and 2.25. So for n=1, we’ve got 1+1=2 hmm is 2 – okay. 1+1/2=1.5 squared is 2.25, ok,

works. Ok, works in general. Now, this is really

good news but maybe what you think now, Homer definitely thinks this is:

Well I divide more and I get more money. So, if I just divide enough maybe I get a

trillion dollars at the end of the year. Sadly that doesn’t work. So, for example, if you divide in 125

parts you get that much money at the end of the year, or have that much money

at the end of the year. Now, if you crank up the n what happens

is, well, that number goes up but it goes up very slowly and actually settles down to

a number. So, if we push the whole thing to infinity, take the limit of this, we get this

number here: 2.718… dollars and that’s the absolute maximum,

that’s “continuously compounding interest”, that’s what it is, so we can’t do any

better than this, that’s e. And that’s also where e comes up for the first time historically, exactly this sort of

consideration, ok. Cool, so now we’ve got e. We’re ready to move on.

e^(pi i). Well, not so fast. Let’s just go to e^pi first which is

actually almost as mysterious as e to the pi i. Why is that? Well it’s got a

special name it’s called Gelfond’s constant and eventually

I’ll definitely make a video about this one, but just for today just ponder it a little bit. What

does this actually say. Well it says weird number to the power of another weird

number and you supposed to calculate this. How

do you calculate something like this? I give it to you on a piece of paper and

you don’t have a calculator. That’s … strange. I think nobody will be able to do this. Well, it would be doable if pi was equal to 3 because then we know we just

have to multiply, you know, maybe chopped off bits here (at the dots),

three times together and we get a rough approximation to what we are

looking for. But, no, we have this one here. So we really want to calculate this, we want to really know for some strange

reason how much money Homer has after pi

years if you’re compounding interest continuously. That’s what I want to know, I can’t go

to sleep tonight if I don’t know. Ok, now the trick here is, we have got this

bit here, which gets us closer close to e the more we crank up the n. Ok, and so if I put that one up here and

put a large number in here we get the right thing, or approximately the right

thing, or as close to the right as we want. Alright now that looks still pretty

awful, okay, and, well, let’s muck around a little bit with it. So, the first thing we do is we multiply by pi here and there, and

see when you do this on the top and the bottom obviously nothing changes and it actually looks a bit uglier than before. But what’s nice is that these two bits are the same and,

you know, what you have to do now to get this is to just crank up the bit in

the box. So n=1 we have this, n=2 we

have that, and then that, and …. that’s still pretty awful ! Except what’s really important here, and

that’s a really really nice trick, is, what’s really essential here is, that

we’re going up. It doesn’t matter how we’re going up, as

long as we’re going up towards infinity, we can go up via nice numbers: 1, 2, 3, … That will also get us there, and that’s

actually what we do and this here is exactly what we’re looking for. So here

it’s like really awful to the power of awful, but now we’ve just got addition,

division, multiplication that’s all we have to do … just a lot.

But that’s basically all we have to do, so we are getting there. And, of course, the pi here stands for

really any number whatsoever. So what we’ve done is actually

we’ve figured out how to calculate the exponential function with basically

nothing, with just this. That’s what we’ve just figure out, that’s a pretty pretty good effort. Alright, so let’s graph this (e^x) and a

couple of those guys (the functions on the right) and see what happens. So I’ve graphed the exponential function

and I graphed the first one of these guys, well the second one really where we take m=2. Not a terribly good fit but if you crank

up the m you can really see how good this gets.

And, actually, when you press, you know, the button on the calculator that’s what

your calculator does at some level. It just adds and multiplies and divides

and these sorts of things. That’s all you can do. Anything, anything complicated in

mathematics, you know, when you do it numerically has to be reduced to just basic

arithmetic otherwise it doesn’t work. Okay here we

go. Almost there now. Just chuck in your pi i, that’s what we’re interested in, and go for it. And actually we could go for it at

this stage. It’s actually not very hard to multiply things like this. Well, this is basically a complex number in

here, so we’ve got a nice number plus a nice number times i. It’s actually not that hard to multiply a couple of those things together I could

teach you in a second actually i’m going to teach you in a second. Let’s just do it on Mathematica and see

what Mathematica spits out. So for m=1 we get this number,

it’s also a complex number. Doesn’t matter what you put up there, doesn’t matter what m is, the result is

always going to be a complex number. So, let’s crank it up now. Crank it up, crank it up, crank it up, all

the way to … what did I do, a hundred. And you can see that this first bit here

gets closer and closer to -1 and the second bit here, that nice

number in front of the i goes to 0. So basically the ugly part goes away and

were left with the -1 if you kind of go to infinity. We could stop here,

but actually I’ve got this really, really nice way of multiplying complex numbers

which we can apply to this, multiplying complex numbers with triangles. Let me just show you. Ok so here we go. Now complex numbers you can draw. Real numbers you can draw on the number line, complex numbers you can draw in the xy-plane. Actually Homer stands right on

top of the xy-plane so we might as well use it and he can really relate to it at

this point in time. So here we’ve got the real number line

there is 0, there is 1, there is 2, and so on. And, well, we’ve extended this real line by the complex

plane. It’s just this whole thing. Every complex number corresponds to a

point in here. For example, 1.5+i is just the point where you go 1.5 over here along

the x-axis and then one up in the direction of the y-axis. And then this guy here,

for example, 1+2i. Well, 1 over here and then 2 units up. Ok, now multiply those two things

together. So what do we do? Well, we do 1 x 1.5 is

1.5 then 1 x i is i. 2i x 1.5 is 3i and then the last one that’s where we have to remember

that i squared is equal to -1, so this is -1 x 2 is -2. Now we just combine things together in the

obvious way, so there and there and that’s the product. And, of course, that

corresponds to a point, that guy out there. Well how do you get from here to there? Not

obvious, right, we can do this, but you can actually see at a glance, you can see at

a glance that these two guys get you up there. How? With triangles! Okay, so to every point, to every complex

number we associated a triangle and the corners are 0, 1 and that point here. So that’s the first triangle. Let’s just

save it. Second triangle 0, 1, point. Now we align them like that, stretch this one, the red one,

so that these two sides are the same and there is your product. Brilliant isn’t it. So you just kind of

aline and stretch these triangles and you know what happens. And actually if

you know the triangles it’s pretty easy to predict where the

product is going to be. Let’s do another example. Let’s do this one here

squared. So what you do for squared is this triangle twice. Ok stretch it, that’s the square. Now, cubing and we’re going to have higher

powers so we need to see what happens here, so just make another triangle,

stretch it, that’s the cube of this number here. Alright now higher powers. That’s the

complex plane. For the higher powers this circle here, the unit circle, the circle

of radius 1, around 0 plays a very very special role. Why is that? Well here is a complex number

on that unit circle. The triangle that corresponds to it has two equal sides, there and there. So, when you align two such triangles what happens? Well you don’t have to stretch, right. Let’s see what happens when I kind of

raise this to the power of 8 … 2nd power, 3rd power, 4th, 5th, 6th, 7th, 8th. That’s the 8th

power of that guy here. So this power spiral, or whatever you want to call it, is just kind of wrapping around the unit circle. So it doesn’t matter how high a power

you choose, it’s just going to end up somewhere here on that circle. And what happens when you move

that guy here off the unit circle? Let’s just move it inside. So we move it

inside, what do we get? Well, we get this nice spiral here, kind of spiraling inside. And,

actually, to go higher and higher it goes closer and closer to 0. If we move this guy outside, well it’s

always going to be a spiral, but the spiral kind of spirals outside. Main

lesson to take away from this is that the closer you start at the unit circle

the closer the spiral, this power spiral will wrap around the unit circle. So now

let’s go for the real thing, the one we’re really interested in. Ok, this guy here. So there’s the complex

number, here in the middle. What is that? Well it’s 1 over here and

then you have to go up pi /m so that’s kind of going

up there. Let’s go for m=3. Ok, let’s just draw this. There we go 1 over here, pi/3 up

there, and then we have to do cubes, right. So 3 times same triangle, scaling, and so

on, what we’ve just done. And it gets us over there. Ok, right. Now what’s going to happen when

I make this m bigger? Well, the 1 stays the same. I make the m bigger, that means that this number here gets ….

smaller, right? That means that it’s going to wander down

here, that it’s going to get closer and closer to this point, and actually I can make it

as close to this point as I want, as close to 1 as I want by making m bigger and bigger. It’s just going to wander down here, down here, down here. This means that the spiral is going to

wrap close to the unit circle. Well let’s do it. So, crank up to

4, four triangles. Now crank up to 5, five triangles now. It’s wrapping closer, right. 5, 6 now let’s just let it go and see how

that guy here gets closer and closer to -1.

It’s real magic about to happen. Ready to go for the magic? There we go, cranking it up all the

way. Well not all the way, up to a hundred 🙂 We can see it’s really getting closer

and closer to -1 and it’s pretty obvious why, right? I mean the bit that’s obvious so far is that

because that guy here wanders down and down, it gets closer and closer to the

unit circle we should get a closer and closer wrap

around the unit circle but what’s not clear at the moment, maybe, is why we

don’t wrap further or closer. Why do we only go halfways around. And for that you have

to remember what I said at the very beginning this reminder about the length

of this semicircle. What’s the length of this semicircle again? It’s pi, okay it’s pi. And what is this? This is

the mth part of pi. So, basically we’re starting out with the mth part pi here and then we’re doing this m times so pi/m times m

is pi. So we’re going to eventually wrap around

halfways, smack on, and we’re going to get e^(pi i)=-1 and I think this is the way to explain it. Hopefully, well I don’t know about Homer but

you know hopefully you who are masters of plus, minus, multipl,y and so on got something out of it.

## ALiNa DoNG

Jul 7, 2019, 11:29 amSee ya numberphile

## pepe6666

Jul 7, 2019, 5:33 amBAHAHA holy shit where he shows you how to multiply complex numbers by drawing the triangles – what a fucking hero. you sir are a champion of the human race.

## Witold Domeyko

Jul 7, 2019, 9:11 amIs it correct to deduce that e to any complex number = -1 ?

## Watching The Earth

Jul 7, 2019, 5:11 pmI guess this video named as : "

pi over ei for dummies" but youtube auto-translate name to different languages as "pi over ei for stupids" . So maybe manual translation required.## FuckYouWhosNext

Jul 7, 2019, 3:50 pmyou should wear tight black turtle necks and dance around like Gunter on Sprockets!

## Naz Doga

Jul 7, 2019, 9:18 pm8.32 / How ( 1 + … ) suddenly becomes ( -1 + … ) ?? That's the key point I needed but missed and so couldn't follow the rest.

## tsiggy

Jul 7, 2019, 11:00 pmMathologer, you are a teacher.

## Александр Захаров

Jul 7, 2019, 5:03 pmХуйня какая-то

## Tony Detroit

Jul 7, 2019, 2:59 amI'm confused by the proof strategy. Can someone explain to me why e^pi was explored as part of the proof. The pi seemed to play no role. He just as easily could have considered e^5. Or could have begun with e^x, no?

## name2

Jul 7, 2019, 3:36 pme^pi*i is equal to -1. Which, coincidentally, is i squared. So that means e^pi*i = i^2. Then with this, you can figure out that:

e^pi = i^(2/i)

e^i = i^(2/pi)

And finally…

e = i^(2/pi*i).

And when I checked to see if this is legit, it is! (e = 2.72…)

So not only e is transcental, it's also made from an imaginary number!

## Bryan Carter

Jul 7, 2019, 12:06 amGreat video, except he completely lost me in the last 30 seconds, he just re-stating the formula. There's so many cool concepts here that I do understand, but none of it helped me understand i as an exponent. I learned a lot from this video, but I feel like I'm no closer to understanding e to the pi i.

## Rajesh Sharma

Jul 7, 2019, 1:51 amYou are a genius sir

## Mark Durham

Jul 7, 2019, 10:20 pmSeems relevant somehow that "pie-eyed" means "very drunk." 🙂

## TheJ4RyD

Jul 7, 2019, 11:27 pm"i squared is 1". He means negative 1, right?

## HBH ZTH

Jul 7, 2019, 5:01 amAt first I wondered where I had heard the intro tune first. Then I remembered Kate Bush.

## priyank2626

Jul 7, 2019, 2:16 pmIn the final case we substituted pi i with m . So a imaginary number with real number.

## That Guy

Jul 7, 2019, 7:08 pmI'm shocked how dumb I am

## Falling Oranges

Jul 7, 2019, 12:11 amThe intro chords remind me of the intro to Babushka by Kate Bush

## 도꼬마리

Jul 7, 2019, 12:06 pmwell now i know!

## Chuck Desylva

Jul 7, 2019, 7:32 amOk, please do in 3 dimensions.

## Angel IA

Jul 7, 2019, 2:40 pm3.17157

## Rajpal singh Dawar

Jul 7, 2019, 1:46 pmI really want to know what type of calculator you used for the calculation of infinite sums of the pattern and plus one by N to the power n

## Rajpal singh Dawar

Jul 7, 2019, 1:47 pmAt the time 3.43 seconds what type of software or calculator you used please let me know this is very important

## JeevS GOne

Jul 7, 2019, 4:09 pme^{i x} = cos x + i sin x

## Khang Nguyễn Trường

Jul 7, 2019, 3:21 amstonk

## WireDog

Jul 7, 2019, 4:10 amMy IQ is 8 points short of genius level. Literally. I can’t understand this.

## Nilo Alvarado

Jul 7, 2019, 5:01 amSe agradecen los subtítulos, renace nuevamente el deseo de perderme en el universo matemático

## Cocoabine

Jul 7, 2019, 1:19 pmSooooo.. because triangles?

## Andreas Andreotti

Jul 7, 2019, 6:08 amThank you sir, it could not be more pedagogical presented. I like mathematics because it has always given me peace of mind. I often try to I solve problems on algebra, geometry and trigonometry in order to get away of everyday concerns. The fascination of it is to reduce a complex problem into the four main calculation forms as you wisely pointed out. You call them tricks, I call it maneuvering, or favorable manipulation. Being a Greek I always have the Aristotelian logical categories in mind that help a lot in mathematical thinking.

## vLinh dh

Jul 7, 2019, 10:12 amho ho ho

## Ryan Fuxa

Jul 7, 2019, 7:59 amThat awkward moment when you realise Homer Simpson is smarter than you….He works at a nuclear facility….?

## Antivlog

Jul 7, 2019, 8:15 pmTEH UNIT CIRCLLLE!,!!!

## Wang Wei

Aug 8, 2019, 3:34 pmWhat is the curve if you connect those points from m=1 to m=∞?

## rajesh sarkar

Aug 8, 2019, 4:08 amOUTSTANDING ……

## Leon Ravenclaw

Aug 8, 2019, 3:39 pm4:44 Engineers entered the chat

## jstarks21

Aug 8, 2019, 2:45 amOh, that's what Elon Musk put on his Twitter

## Red Rock

Aug 8, 2019, 9:15 amInstructions unclear.

Divided everything by 0.

## Brett Ess

Aug 8, 2019, 5:54 amBrilliant! I wish my mathematics lecturer was this good. 😎

## gore buster

Aug 8, 2019, 6:41 amSoo if I deposit one dollar with a banker at an imaginary rate of interest compounded continuously, at the end of 3 years, 1 month, 21 days, 16 hours, 21 minutes, 5 seconds, 923.60771908 milliseconds I owe him a dollar?

(Bcz the money the banker owes me is equal to my principal amount * e^(rate of interest * time) so e^(i*pi) = -1 so the banker owes me $-1 which means I owe him $1.

## Jimin Bang

Aug 8, 2019, 6:36 am"Where's the pie?"

## PRITIPRIYA DASBEHERA

Aug 8, 2019, 4:23 pmit is so neatly "pi"ed

## Avinash SOOKUR

Aug 8, 2019, 7:22 pmYou are the GOD of Mathematics …….

## Floofy shibe

Aug 8, 2019, 9:28 pm3:14

## Muttley

Aug 8, 2019, 3:52 amthat t-shirt is both transcendental and imaginary

## Muttley

Aug 8, 2019, 4:13 amMmmmmmm pie

## Payge

Aug 8, 2019, 9:21 amWhat’s the equation for being dumber than homer ?

## Joan Ferran

Aug 8, 2019, 10:52 pmExcellent explanation tks

## Mike Gordon

Aug 8, 2019, 11:09 pmFascinating, thank you. But one question: By the end, you seemed to have eliminated e from the function. So does it work for any other number in place of e? What is the connection with e as the ‘number of natural growth’?

## John Byrd

Aug 8, 2019, 10:46 pm"for dummies" and then blazes through multiplication of two algebraic functions. Bad teacher. No Twinkie for you.

## Alive Afterall

Aug 8, 2019, 6:55 pme^i*pi = -1;

e^2*i*pi =1;

ln(e^2*i*pi) =ln(1);

2*i*pi =0;

WTF??))

## DIY home tricks automation

Aug 8, 2019, 7:18 pmhe is from NZ or Aus??

## The Sahil

Aug 8, 2019, 8:40 pmWhat software do you use?

## 13IG 0NE THE

Aug 8, 2019, 1:03 pmWait… if we assumed m=n×pi×i, doesn't that make m itself an imaginary number?

## Hjtunfgb

Sep 9, 2019, 12:17 amIf you went a little bit beyond you could easily prove the more general form e^i*theta… genious

## David Ramadeen

Sep 9, 2019, 4:08 pmWhy you say you do not know about Homer? YOU ARE HOMER. 15:34. Just suck the bottom lips in. Thanks for nice job. BUT WHAT HAPPEN WHEN YOUR M=Pi?

## Technical YADAV

Sep 9, 2019, 4:32 amPuzzle lover

Nice! Content

#HUMANTHINGS

## Clive Goodman

Sep 9, 2019, 9:10 amThis identity is actually a special case of exp(ix)=cos(x)+sin(x)i

## Colin Holloway

Sep 9, 2019, 1:16 pmI got some of it. That will do for now. Love this explanation and REALLY like the graphics. So good to add some concrete explanation to the abstractions.

## italianpro 2005

Sep 9, 2019, 12:18 pmso, can we say that (1+e) × (1)^i pi is still equal to 0??

## Aqua rius

Sep 9, 2019, 1:16 pmThank you very much! Now I understand Euler's formula!!!

## Imagine Existance

Sep 9, 2019, 1:40 amAn easier way to understand this is

x^yi=cos(y*log(x))+i*sin(y*log(x))

Plug x to be e and y to be pi

so -1+0

Or cos(pi)+i*sin(pi)

You can cancel out log and e because log(e)=1

## Darki i

Sep 9, 2019, 8:20 pm1:09

## OliBint

Sep 9, 2019, 7:55 amI'm pretty sure Homer would have lost you with the funky shifting triangles on a graph where the y axis represents complex numbers…. at around 11 mins.

## NotValik

Sep 9, 2019, 7:54 pm“This guide can be understood by anyone that knows how to do addition, subtraction, multiplication and division!”

Uses exponents, graphs, imaginary numbers, funcions…## Laureano Luna

Sep 9, 2019, 8:39 amFar from being a proof.

## Jose Lozano

Sep 9, 2019, 1:56 pmI had a hard time decoding the title of this video!

## Marie Meyer

Sep 9, 2019, 4:46 pmI freaking love that video

## omar elbendary

Sep 9, 2019, 7:44 pmThanks junji ito

## дан нестеров

Sep 9, 2019, 8:17 pmThx, you helped a lot

## Sherine Victoria

Sep 9, 2019, 2:11 pmCan believe they made a video just for me lol

## Mark Odern

Sep 9, 2019, 5:24 pmStill not a single word about 2π (tau). Do you want Martians to laugh at us cause we still use a half of the number?

e^2πi = 1

e^ti = 1

And graphically it would make even more sense, since we make 1 revolution, not a half (why we would even do that?)

And I meet this misconception in every YouTube video on the topic.

## Mustafa Malik

Sep 9, 2019, 7:26 am0:51 HOMER NO UNDERSTAND

## VronZ

Sep 9, 2019, 12:29 pmI knew it was beautiful but I didn't know it was this beautiful

## Arttu Kettunen

Sep 9, 2019, 2:31 pmDifferent school subjects' levels taught by youtubers:

Anything else: middle school

Math:

University## Floofy shibe

Sep 9, 2019, 10:37 amat 5:07

mathologer: e^pi

Me: you mean 27??

## Jean-Yves BOULAY

Sep 9, 2019, 5:29 pmThe number Pi and the Golden Number (φ) and the inverse of these numbers are made up of a seemingly random digits. This article is about order of the first appearance of the ten figures of decimal system in these fundamental numbers of mathematics. There turns out that the ten digits decimal system (combined here with their respective numbers: figure 1 = number 1, figure 2 = number 2, etc..) do not appear randomly in the digits sequence of Pi and the digits sequence of Golden Number (φ). The same phenomenon is also observed for the inverse of these two numbers (1/Pi et 1/φ).

https://youtu.be/NDj0rJhnO_g

PDF here: http://jyboulaypublications.e-monsite.com/

## Kenneth Payne

Sep 9, 2019, 4:19 pmI like mathologer a lot, but on this topic, I’m afraid that he gets blown away by the 3 blue 1 brown guys, whose 3 minute video is way more understandable.

## Thibo Van België

Oct 10, 2019, 2:19 pmYou lost me at point: 00.00

## wsquix58

Oct 10, 2019, 11:14 amBrilliant in its simplicity in explaining the complex…No pun intended.

## Rednaw

Oct 10, 2019, 3:58 pm3:28 i broke it, if n = 999999999999999 then the value will be 3.03503…

## S T

Oct 10, 2019, 8:48 pmYeaaa … no.

## asmr lover

Oct 10, 2019, 2:19 pmWhat if you listen to something you cant understand but you think that you understands it

## Nimesh Poudel

Oct 10, 2019, 4:45 pmHow (1/2)! = √π/2

## Slawek Zawislak

Oct 10, 2019, 9:59 pmi think we should ask Homer te explane it to us

## Girls Girls Girls

Oct 10, 2019, 8:59 pmawsome, it really is for dummies, but i finally understood it. Thanks

## NUKE

Oct 10, 2019, 6:16 ame^iπ=-1 is so poetic.

All the famous weird numbers in mathematics, 2 transcendental, one imaginary and one negative: e, π, i and -1 all related to each other directly!

## Karl Jo

Oct 10, 2019, 1:50 pmyour explanations are so clear and understandable!

## Sidhant Srivastava

Oct 10, 2019, 4:55 am0:53 i^2 is not 1 lol

## JustAnotherYoutuber

Oct 10, 2019, 9:48 pmSo my question is, why pi? Seems like you are just equating it as a constant * pi = some arbitrary constant. Couldnt this work with any number other than pi?

## Sergio Andres Solis

Oct 10, 2019, 8:10 pmwhy does it get closer to -1?

## Mr. Gang Banger

Oct 10, 2019, 11:43 amP=NP guys! Now Homer is a millionaire!

## TallWaters

Oct 10, 2019, 1:20 pmThis is very very good, thanks a lot.

## Dave Rocket

Oct 10, 2019, 3:40 amLamentably, Homer still wouldn't understand.

## Rojasnetor

Oct 10, 2019, 6:08 pmI really enjoyed this video, I now love maths more than ever!

## Alexander Mizzi

Oct 10, 2019, 7:23 pmWhy underestimate Homer.

## mario mario

Oct 10, 2019, 7:24 pmWhy underestimate Homer.

## mario mario

Oct 10, 2019, 7:26 pmi^2=1 you mean -1

## WeGoTSkiLL

Oct 10, 2019, 10:38 amThis might be a stupid question, but why doesn't this work with any number besides pi? I know it doesn't work, but why? If it's just important to get both sides of the triangle the same, any number divided by a huge m would get that sorted, wouldn't it? What role does the pi play? I seem to be missing a part to fix this 🙂

## Márton Bálint

Oct 10, 2019, 12:46 pm…………………………………………

## S

Oct 10, 2019, 7:52 pmThat "shell"like shape made from triangles awfully looks like the golden ratio spiral shape