Epic Circles - Numberphile
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- Опубликовано: 21 сен 2024
- The man who loved circles (Objectivity): • The Man Who Loved Circ...
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Pappus chains, circle inversion and a whole lot more in this EPIC video with Simon Pampena.
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Find yourself someone who looks at you the way this guy looks at circles.
This is true love. It's beautiful.
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.
this is a 26 minute video of a man trying to find the relative radius of a circle, and he is very happy too.
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 🤣🤣🤣🤣
“If you kiss in real life you have to kiss in the inversion too. Exactly”
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.
What does circle inversion have to do with an eclipse?
You should really ask for consent before touching tangentially.
13:52
And don't get me started on the kissing circles...
Say no Mohr...!!
@@SriRamDasariChandra - nudge nudge
Ever been kicked in the tangetiallys?
8:35 "Gee you've got good instincts" is my favorite part of the video, the dynamic between these two is hilarious
Im thinking Simon originally thought I'd be an oval
@@conkrcstf6405 are you?
Funny to note that 1/95th is also the portion of this demonstration I've understood.
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.
I'm glad the whole thing came full circle in the end
+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.
For anyone wondering, the general formula for the area of the nth blue circle (and the nth number in the sequence) will be 1/(15+4*n*(n-1))
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.
Just imagine trying to solve the initial problem, and you think,
"quadratic mirrors"
and it works
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.
This video has instilled me with the very bizarre experience of knowing exactly what you are doing whilst also having no bloody clue what the fuck you are doing.
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 just adding an arithmetic sequence.
15 + 8 = 23.
23 + 16 = 39.
39 + 24 = 63.
63 + 32 = 95.
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."
11:59 He said 'Circle inversion!' with the same level of happiness how Hulk said 'Time travel!'.
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
I can't pretend that I understand it, but I do so much enjoy this guy's enthusiasm!
He is absolutely loving it!
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!
this guy scares me, he's eroticly in love with maths.
You might say he's a numberphile
@@FairyNuffMuffin2 badum tssss
Beautifully explained. I was excited as a child when the purple cirlcles began to align! hehe.
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!!
I just love how happy Simon is the entire time, this is a man who truly loves what he does
He's just high
Professor: "Show your work"
Me: "NO."
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.
I like the part when he said circle
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
That brown paper should be framed and hung on a wall. Beautiful!
Beautiful construction.
I suspected the final result from the other denominators: 15, 23, 39, and 63.
From 15 to 23 is 1*8. From 23 to 39 is 2*8. From 39 to 63 is 3*8. From 63 to 95 is 4*8.
But it's worthwhile to watch him construct the inverted circles and enjoying it.
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 😅.
I have no idea what I just watched but now whenever I close my eyes all I see are circles.
LittleMikey this is related with hyperbolic geometry.
LittleMikey Regrets?
derci ferreira Euclidean geometry is not the same as hyperbolic geometry
I don’t understand....
The editing was really nice. I'm sure that explanation took forever in real time. Really cool stuff.
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 RUclips 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
I've watched this 4 or 5 times now, and I really feel like I understand it. I use geometry at work constantly (I make custom stairs and handrails, nothing but triangles, circles and the occasional ellipse) and the more I watch this the more I know that this doesn't help me to do with anything with my job, but I love it anyway.
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
"This is epic. This is seriously epic... This is absolutely epic."
It is epic
Please purchase this guy a compass which you can mount pens and markers in.
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 ^^
When you watch and listen to someone like this guy who is SO passionate about something, you cant help but become interested. I love people like this. I wish all the teachers of our children could have this kind of drive.
He keep using the word "simple", but this is the inverse of simple
I love hearing smart people talk it makes me feel smart
This isn't circle inversion. This is circle INVASION.
nice one
SO? Now I can find radius of every circle I wish? Even if it'll be the 9999 circle? mhahahahahahah, absolutely power!
@1:24 From now on when I wish to use the word: "kissing" I will substitute the expression "touching tangentially".
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)
1/15, 1/23, 1/39, 1/63, ...
15, 23, 39, 63, ...
the difference between each number forms a pattern:
8, 16, 24
the next difference would be 32
63 + 32 = 95
the next number in the series is 1/95
the series continues:
1/95, 1/135, 1/183, 1/239, 1/303, 1/375, 1/455, 1/543, 1/639, 1/743, 1/855, 1/975, 1/1103, 1/1239, 1/1383, 1/1535, 1/1695
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
13:52 *vigorously rubs hands* “This is the reason why I came.”
Read this exactly when he says it.
23:40 Pirate does math
Not as exciting as a pirate becoming a student union president.
d:^)
LOLOL
best comment so far
AAAAARRRRRRRRRRR
New drinking game, everytime he says circle, take a drink
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
Touching tangentially sounds naughty.
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
Rewatching this a year later and man there are just too many great lines (both in terms of geometry and dialogue ;)
trequor Hah..yup. Shalom
Me too!
"Yo Dawg, I heard you like circles so I put circles in your circles" :o
+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!
For all you commenting about adding 8s repeatedly to the denominator, the most important thing you have to realize is that noticing a pattern does not amount to a proof. The techniques he showed in the video (applied a bit more rigorously, albeit) certainly do. That's the important part. He proved the result, and in quite a marvelous manner.
+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.
my head just exploded
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
It's been 4 years since I first watched this video. I was in high school. Numberphile gave me so much passion for mathematics that I'm now in university. Now, 4 years later, I can finally understand this video.
I thought it would be 95 because the difference between 15 and 23 is 8, 23 and 39 is16, 39-63=24, so 95
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.
Epic circles but every time Simon says circle it speeds up
Video only last 2 minutes
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? 😂
What did I just watch.
something... _sniff_ ...beautiful....
Genuinely one of my favourite videos ever on this platform. The pure joy is infectious
For sure!
"We're gonna do this quite rough, Brady, if that's alright."
Fueling lemons.
This guy's enthusiasm is the best =)
Brady, this video isn't too long, at least not for me. I have always had a love for circles, so this is right up my alley. It looks like this video could have been two hours long. I would have still watched it multiple times.
26:29 *Laughs in Mathematician*
Rusty Shackleford : )
just wow. that structure is truly beautiful. so many amazing properties.
From the thumbnail, I thought it was Matthew Santoro with a wig on.
+STaSHZILLA420 Lol, me too.
+STaSHZILLA420 I thought a much younger Bob Ross
you beat me to it!
KILLER KEEMSTAR
To those of you saying that 95 is the obvious answer because it continues the pattern, that is an insufficient answer. There are infinitely many ways to continue this sequence and achieve different numbers for the next term.
For example, at the start of the video, I noticed that adding the first two denominators gives 15 + 23 = 38, which is 1 less than the next denominator (39). The second pair of denominators added together gives 23 + 39 = 62, which is 1 less than the next denominator (63). By this logic, the next term should have a denominator of 39 + 63 + 1 = 103.
So using this method, the continuation of the sequence yields 1/103 rather than 1/95.
Thank you. My thoughts exactly.
Its because you did wrong fool
@@mikebarnes7441 Either you didn't watch the video, or failed to understand the comment.
@@maxiom7476 you're wrong
@@mikebarnes7441 wow is this the language of gods ?
I love how the spiral of circles inverts to identical circles in a straight line
Guys, I'm not kidding when I say this, I did the procedure while watching the video, and it was the most satisfied I have ever been! This is just a wonderful way to show connections between different parts and areas of mathematics. I love these videos, please keep doing the world a favor by making videos like this!
I wish I had the programming skill to write some drawing software for this!
It would allow you to draw a circle of inversion first, then you could draw whatever shapes you liked inside or outside it.
I'd want to see what happened if I drew squares or triangles... they'd probably come out really weird and distorted.
**gasp** and then someone would write the 3D version, with a sphere of inversion. And you could pop cubes and pyramids and stuff into it or around it... it would be like a freaky hall of mirrors on acid.
"If two lines touch at infinity...well this is kind of tricky stuff here" My favorite part of the video. :)
5:58 - "So the radius is 60." -> Every math teacher: "60 what? Apples? Bananas?"
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).
Brady, You should sell these brown papers at a charity auction or something along those lines.
Syrio Forel escaped and became a mathematician.
***** BUT NOT MINE!
my formula is:
1 diveded by (15+(X*8)) and X is the number of which circle you wanna no the ratio... and the first one is Nb=0
I was wondering as to how many people figured out at one glance that the radius of the inverted blue circles was in fact R/16. He totally glossed over that detail. It is a fun little exercise to confirm that it is indeed the case.
Also, following this derivation, it is not that difficult to arrive at the general form for the nth circle radius. This has to be one of the sweetest numberphile video I have watched!
Dude is by far the best Numberphile guest.
I love this guy so much, he's one of my favorite people that Brady interviews. Everyone he interviews obviously loves what they do and is amazing, but Mr. Pampena's enthusiasm is infectious.
This is the firat time I truely craved the brown paper as art for my wall.
Of course, you can give the next number in the sequence immediately, just by noting that the denominator increments by 8 more each time. 23-15=8, 39-23=16, 63-39=24, and so 95-63=32 --> 95 is the next denominator. But nonetheless, this was just an incredibly fun video, and our man's enthusiasm is just amazing and contagious.
ok, now add the 3rd dimension...... I dare you.
People seem to stop when they get to three.
Where's the CPGgrey "first" comment? :P
Most of the videos I can follow...But this, yikes!
18:06 "... these two circles are lines ..." was about where slipped out of consciousness
But why is it 1/15? :(
At 22:17 he just states that the blue circle is R/16 without any proof or reasoning. :-(
All the touching and kissing, I got lost and ended up touching myself :(
Did you tangentially touch yourself?
Zachary Mitchell Very, very tangentially...
I am not at all exaggerating when I say that this video was what put me on the path to becoming a math major. Thank you
Started losing me around the 15 minute mark when you were talking about the placement of the inverted circles. Maybe I just didn't pay enough attention, but I feel like it wasn't properly explained for someone who's never seen this before.
If you raise each term of that sequence to the power of -1, then add 1 to it, then divide it by 8, you get the Fibonacci sequence starting at the term of 2.
lol i thought the same thing
I thought of Fibonacci numbers at first also, but the denominators seem to follow triangular numbers (times 8 then plus 15) which is how I predicted the 95 in the beginning.
@@linkAKAdude the first few denominators also seem to have a wierd pattern: the nth term is the sum of the two previous terms +1, but that doesn't work for 1/95 though, is there a particular reason for that or just a coincidence?
Are you all serious? Or speaking nonsense just to be funny?
@@whywelovefilm7079 We're simply pointing out weird patterns, most of them are probably useless but asking questions is how maths work
To all those saying, "Just add multiples of 8 to the denominator," this video solves the problem in a manner ancient mathematicians like Euclid would have done. Back then, they saw numbers arise from geometry, not the other way around. Basically, this video solves with geometry, not algebra.
Ben Summers thank you for this. I solved algebraically in about a minute of paused staring at the sequence. I was so happy that I got it right, unlike with the -1/12 thing.
I've learned something new today:
Complex maths makes me cry.
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.
Mathematics: surprisingly erotic...
I'll have to watch this several times more...
The next term will be 1/135. This is because the formula for the sequence is a quadratic reciprocal. an=1/(4n^2 - 4n + 15)
And this is how Alfa Romeo came up with their wheel design
Now Brady! Look Brady!
Boy this guy surely is enthusiastic about his circles!
I've never rooted so strongly for someone to draw a good circle before now
Watched this when I was still young at 2016, so I didn't understand much, thinking that it wasnt that impressive. Watching this video again made me realize how amazing this actually is. Dayum.
"This is the best time I've done it."
"That's nice."
dang it... 1/94 sooo close XD
rechecked my calculation and made an ERROR OMG DX
15 - 23 - 39 - 63 - ?
differences between those numbers:
8 - 16 - 24 - ?
differences between those:
8 - 8 - ?
assume the last one is also 8:
8+24=32
32+63=95
answer=1/95
just a guess, no clue why it works o.0
Also, if you look at the correlation between 15,23,39,63 you notice that the numbers increase with ascending multiples of 8...
15+8=23
23+16=39
39+24=63
63+32=95
I thought the title of the video was the guys's name: Eric Circles
I'm not that bright lol
Where is CGPGrey's first?
+Niklback1 Yeah I just started listening to HI.
At the bottom just keep digging
That was an interesting but horribly inefficient method of solving that. Took me a few seconds to answer.
15+8=23
23+16=39
39+24=63
63+32=95
I saw a simple pattern quickly and that made this really easy.
I just started the vid and the question to your answer is 1/95th this is bc 23-15=8 then 39-23=16 and then 63-39=24 this is going by 8 each time so 24+8=32 and 32+63=95 so the answer is 1/95th
That's what I did too. I could make him a nice iterative function that would save a lot of paper and time :) but of course that would just spoil things
+Benjamin Burgess All the circle equations was unnecessary work lol
+Benjamin Burgess your equation would be ax=8x+7
+Greg master It isn't unnecessary, it is beautiful.
question to your answer
(wut)
This is why mathematicians+graphing programs=LOVE.
I'm sure it's the same feeling as when Lotus 1-2-3 came out for the accountants.
It's like a calculus with circles. Circulus!
2:26 It's a big touch-fest today. 0.o
Do you know how hard it is to get four circles to kiss?
+George Wang It's not straight, that's for sure! ;)
+Ozibiey circles have no gender...
I have no idea what this means, the process is long as fuck and doesn't seem to work anything out that couldn't be measured with a ruler
Beautiful! I love explanations where I get lost, but end up understanding.
Math is art...
23-15=8, 39-23=16, 63-39=24, 95-63=32. pretty easy to find the pattern here. I got the answer without spending countless hours killing myself staying up all night trying to figure out the circles shit which he only partially described.
i hope one day i can be as passionate about anything as that man is about circles
There's something really gratifying, satisfying and fulfilling about seeing you folks do mathematics in these comments. It feels like you're using your keyboards to the extreme. Flexing almost. While having fairly enlightening, very intelligent, nerdy conversations! And as a nerd myself, i love it!