Epic Circles - Numberphile 

Ain't nobody who touches you tangently like him

Both thing have 1 thing in common, they kiss

boi, u haven’t watched his question 6 video have u😂

@@baras9700 I have. He said he literally cried after solving that question because he was too happy.

I keep looking at girls like that, and they freak out.

I know it's called "circle inversion" but I too do not have the foggiest idea what circle inversion actually does.

When it's finished..what do you do with it?

Imagine you're inside a circular mirror and you draw a shape on the floor, the inversion of this shape through the circle is how you'd see it in the mirror

Welcome to the world of math!

try searching for something called hypobolic geometry,hopefully it helps

It's a long way to go just to get the radius of a circle, but it's kinda worth it for the look of pure, distilled insanity at 21:50.

He's like the Bob Ross of mathematics

@@YtseFrobozz 😂😂😂

The one thing I got from this video is that there are many many circles, and this guy is having a great time.

Ytse Frobozz 🤣🤣🤣🤣

i feel bad for my inversion now

@@wknw1442 i pulled a few muscles trying to kiss my inversion IRL.

@@nrm224 I presume that was before you figured out you just go right up to the circle of inversion, and kiss the perimeter.

"Kiss" is a terrible way to describe two lines being tangent to each other. In real life, kissing still keeps the two people or things separate: just very close together. Tangents actually share a point, so the kissing point is necessarily the same in both the real and inverted world. I've watched this video a few times and this question always bugged me.

Im thinking Simon originally thought I'd be an oval

13:52

And don't get me started on the kissing circles...

Say no Mohr...!!

@@SriRamDasariChandra - nudge nudge

Ever been kicked in the tangetiallys?

Chausies thanks I might need this eventually

Fluffy Boi pretty simple it turns out that you have to add 8*n at the denominator each time

Finally, some actual information! Thanks.

sat thru the entire thing, even tho it is absolutely beyond me. however, despite not understanding the circles per say, the pattern jumped out within seconds, but i could only think of it iteratively. well done finding the general. i wish he had related the geometry to that closed form explicitly. anyway, learning about the inversions was cool.

@@ayaipeeoiiu8151 exactly my thought indeed. I was satisfying hearing the 1/95 at the end, because it was a predictable sequence from the beginning.

I see this as a absolute win

Omg. And not only with the same enthusiasm but practically the same cadence!

​​@@TeganCantEven and body language... well it was similar

Francisco Ibarrola glad you enjoyed it

+Francisco Ibarrola Purple crayon much

+Numberphile awesome video guys. brilliant

I wish I could watch that when I was younger really!

me too!!

Phew!!! This is epic!

I was thinking the exact same thing! In honor of his enthusiasm, I might actually really try and really understand what's going on. :)

Frankly, he's doing that which is the sole function of GOVERNMENT... Taking something **SIMPLE** and **needlessly complicating it** ... ;) (Yes, that's a reference to Burt Gummer from Tremors...)

Yes, Adam! I agree!

He's just high

I did the same thing! It was very satisfying to see that I worked it out correctly! I definitely would need to study it a bit more to understand how the circles all worked though... 😂

@@matthewziemba7526 - thank you for reminding me of that video. I watched it again :-)

kind of seems like a waste of time really when basic sequence math gives you the result in 5 seconds not 26 minutes. I bet the analytics on this video show no watch time between 3:37 and 26:24

@@B.M.0. - I see the meaning of that video in introducing the Inversion At A Circle and giving an example. While I find the inversion elegant and appealing, there seem to not many examples, unfortunately: Ptolemy's axiom, Pappias circles like here, geometrically constructing circles that touch other circles. I would like to know more examples. But I found the construction here marvelous from 14:00 on. Have other watched it, too? If not, I consider it their loss. Watching videos is often "wasting time", but it can be also very entertaining. . . . About basic math giving you the results instead: you would still have to prove it. Also here, some steps are cut short, like the radius of the inverted circles being R/16. Yes, I can prove it myself, but they should have added it to the video, I think.

Same here 😅.

+charles paradise *groan*

+charles paradise I see what you did there

+fluffy I'm an app developer and I'm absolutely *APP*-auled by your statement - I'm afraid I'm going to give you a *PUN*-ishment. *bad trombone sounds*

*dies from pun

+charles paradise It's a shame this video doesn't circulate online more. It deserves a round of applause.

It is also 1/95th as efficient as just measuring the tiny circle. Though a video where they just do that would only be about 1/95th as interesting to watch.

@@-42-47 Measuring the tiny circle isn't efficient because error gets amplified at small distances

Haha yes. I wish he was clearer

@@alexalt2630 Which part needs to be clearer?

@@alexalt2630 What he shows should make you able to do unlimited iterations of this if you have enough paper; or pixels.

It is epic

trequor Hah..yup. Shalom

Me too!

think it is a series look for a common difference in the denominators. I mean you can even write the equation for f(n) of it

@@jobliar937 it must be a series since as you see towards the end what changes for each of those small little blue circles is just one addition of purple circle that is between two parallel lines

Ahmed Uygun Still trying time figure out how to describe the formula, but am I off in thinking the next in the series should be 1/135?

@@PercyPortland You are correct. As others have pointed out, the general formula for 1/radius of nth circle is: (2n+1)^2 + 14 where the first blue circle has n=0.

@@jimvj5897 I know it might not be prettier to everyone in an "a*n^2+bn+c" form, but I did it anyway. 4n^2+4n+15. I hope. Been a long time since I touched algebra. I like it in a form that hides the magic.

For sure!

Same idea...

Here you have another one 15 23 15+23+1=39 23+39+1=63 39+63+1=103. Ooooopsy.

Me 32 seconds in: "1/95" 26'35" pass Me: "Yup."

Standard

lol, if I had this problem, then i wouldn't do it either

Kitty Forest fires

why though..? that's just disrespectful.. just remember how much effort he puts into teaching you things. that's valuable time...

It's my time. I paid for it.

Read this exactly when he says it.

So the whole video pretty much

I like the part when the pen was on the paper

the part where h

So did i

I liek potamto

You might say he's a numberphile

@@FairyNuffMuffin2 badum tssss

Rusty Shackleford : )

The explanation took forever after editing, it's nearly a half an hour long.

MathHacker42 You'd have to be a rather impatient type of person to consider a half hour as "forever"

Bluemilk92 Maybe in normal circumstances, but this is a RU-vid video. They tend to be "long" at just /ten/ minutes. (Mind you, I don't necessarily disagree with you. It's just worth considering.) I, for one, loved this video :)

Bluemilk92 Yeah, I may have been a bit hyperbolic, I just meant that it was much longer than a typical youtube video.

MathHacker42 I guess it just matter what type of content you watch. Since I often watch video-game related videos, I rarely watch anything under 20 minuets

Perhaps if you were a watchmaker :)

Functional

Stephen Parker True point! Didn’t think about how gears are kissing each other in a similar manner

LittleMikey this is related with hyperbolic geometry.

LittleMikey Regrets?

derci ferreira Euclidean geometry is not the same as hyperbolic geometry

I don’t understand....

I recommend a computer.

DarkArachnid, that should've been rather obvious, no? LOL

He didn't use it for a reason could you guess the reason......?

Doesn't change the way your brain computes. Besides, you still need a decent algorithm from your brain for the computer to compute.. That is, if you're the one actually giving it a thought. Personaly I would recommend a brain before getting the computer. Which surprisingly gets all back to the original point : where is Numberphile going today ??

He actually already has one, he just doesnt use it ^^' :D Painful to watch ^^

At first I thought it was mm but I got super confused when it looked more like 3cm and not 6

I suppose the thing about maths teachers is that they're not just teaching you for maths, they're teaching you for physics, engineering, etc., where having specific units of distance is important. In pure maths, it's enough to say "60 arbitrary units of length" or even just "60".

@@jared8515 6:24 dude it's clearly 6cm.

@@PaulPower4 In engineering you often find that parameters are "non-dimensionalized" to obtain a general solution, and then you slap on some scaling factors to get the result that fits your needs.

Not the higher ones. IIRC mine stopped caring beyond Algebra I (or whatever equivalent), unless the problem calls for units (How much thread? or How long of a fence? when your numbers have units).

Just my thoughts

Oh, this isn't *complex* maths, precisely... "i" shudder to think. :p

@@merge3550 Oh, that's okay, I can probably do something silly (yet strangely elegant) with exponents to fix it.

Ernesto Roybal ouch

I don't recommend it.

Ernesto Roybal MY HEAD JUST EXPLODED 100 TIMES MORE! why? well you can see 1 on 15, 1 on 23 (plus 8), 1 on 39 (plus 16), 1 on 63 (plus 24) and 1 on 95 (plus 32). so at the start i thought "is it 1 on 95???" at the end "OMG I FRACKING KNEW IT!!!!!!!".

Lucas Pluijgers i had the same, the question is what has that progression with 8*n to do

Awesome

Yeah, I got to first base last night... I touched Sarah tangentially. No big deal...

Unfortunately, that would have to exclude touching where anything happens to cross the tangent line--

Bro, i had a dream that i touched my crush tangentally (r/outofcontext)

or like the name of an indie rock band

+Kazza FDM genital*

It does sound kinkier than kissing, now that you mentioned it.

You're mixing up tangent with tanga and the string theory

what is this comment chain

something... _sniff_ ...beautiful....

You don't expect to be asked a question so your brain momentarily short circuits even though it's an easy question xD

Not as exciting as a pirate becoming a student union president.

d:^)

LOLOL

best comment so far

AAAAARRRRRRRRRRR

how is that?

The equation to solve for it is 1/4(4+(n-1(n)))-1 when n= the place in the sequence you are solving for

danlmd1 try 1/( ( 2n+1 )^2+14 ) n starts at 0

I dont want to die.

ill be pickled

Drink orange juice

I'm not playing, but I died from alcohol poisoning by proxy

Like Russian-Russian roulette? :D

+Justin Lew (MC Gamer) That are touching the circles you put in your circles.

+Justin Lew (MC Gamer) Reminds me of XKCD 855. Before all the other "great" minds of the web, Zombo.com's designers used the awesome of circles.

+Justin Lew (MC Gamer) You still don't sound nearly as suggestive as Simon saying the circles are kissing... I will never think of tangents the same way again...

Ellipse My Ride!

Even less!

I read this comment at the perfect time.

That's how long the video where he only says circle is

Speeds up by what number?

what if you drink every time? 😂

yyGODyy Ha i was right. Bitches!

yyGODyy yeah i thought the same thing, lol no need for the 20 minute explanation

yyGODyy But what he showed us was the way to prove this using geometry. We could all reasonably assume 95 was the answer (as did I) but using this method it shows the poof of that, very cool.

+yyGODyy Yeah, I spotted the same thing (+8+16+24+32). Geniuses like us don't need to draw any circles :)

+yyGODyy Right-o. Now prove it by induction.

+STaSHZILLA420 Lol, me too.

+STaSHZILLA420 I thought a much younger Bob Ross

you beat me to it!

KILLER KEEMSTAR

***** BUT NOT MINE!

Proof that you've only seen one video on the entire internet

+Kieran Kaempen I might not be a mathematician, but isn't mathematical induction a valid and accepted proof technique? I mean, yes, you could draw a lot of colourful circles for a few hours, but inducing the theory is a lot more practical - especially in real life, when you don't have the time (or even the skills) to elaborate a fancy (and in this case quite unconventional) proof. Nevertheless this was a very interesting approach to the topic and encourages to think out of the box more often.

I think that Simon (the guy from the vid) shouldn't reveal that many numbers in the pattern, since I can also see a pattern among it (nevertheless, a really beautiful and epic proof). If he did, like say reveal 2 numbers, then the general audience wouldn't be so cocky about it having to SuDdEnLy have the next number. Side Note (a REALLY late reply to RYFAMO): The proof Simon presented is a problem WAYYY back when geometry is the algebra of ancient math times.The chain is called the Pappus chain and it was obviously discovered by Pappus of Alexandria on the 3rd AD century. So Pappus actually proved this using the Inversion method, without ANY mathematical induction. So technically speaking, it's a practical proof from ancient times.

RYFAMO noticing a pattern and testing it repeatedly is not induction. induction is testing base cases and then showing that because the base cases worked, the next case will also work. Just saying "here's an observed formula for the nth case" doesnt prove anything.

How does it constitute proof? He noticed the pattern matches the circles. That no more constitutes proof than noticing it matches a formula. So his "proof" is no better than simply saying the nth number is 1/(4*n^2-4*n+15). All his "proof" actually amounts to is the equivalent of saying 1/(4*n^2-4*n+15) is 1/95 when n=5. He has failed to show that this pattern should correspond to the circles rather than simply matches. And that is something that it would be impossible to do as does not fall into the realm of proof.

The issue here is that the question was introduced in a form of find a pattern question. And then he started drawing circles. So technically all those "adding 8s" are correct answers to the question.

get your "mood" comment outta here