Infinitely Many Touching Circles - Numberphile

Featuring Matt Henderson.
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[@numberphile]

Sponsor: www.kiwico.com/Numberphile Matt Henderson: twitter.com/matthen2 More videos with Matt Henderson: bit.ly/MattHendersonPlaylist

[@Typical.Anomaly]

What happens if the circles are circumscribed in the boxes >>under

[@bencanfield]

@@Typical.Anomaly I think they would be on the outside of the circle. What do you think?

[@antoniussugianto7973]

The AREA of infinitely many touching circles equals....

[@Bibibosh]

This has to be to most interesting mathematical things you should know!

[@Bibibosh]

What if the area is pi?

[@elnico5623]

I love how 3 times now they hit us with circle inversion

[@chaitanyalodha3948]

And it's still not enough!!

[@sillygoofygoofball]

I need MOAR

[@Syrange13]

They just think it's neat.

[@psmirage8584]

Like it's a fundamental property of Geometry.

[@minirop]

and how many times did they hit us with Pascal's triangle?

[@AMTunLimited]

I've seen many of these before, but the rectangle of area=1 is absolutely mind expanding. Absolutely nerd-sniped

[@EamonBurke]

It seems kind of obvious that is what will happen because when new circles are drawn on the uppermost space, they are arbitrarily placed precisely arranged in such a way that they will touch and not overlap. Because the rectangle with fixed area is acting kind of like a panto router it is merely translating this pattern to a smaller scale. The reason they are nested circles is because one corner of this rectangle is arbitrarily fixed and used to create the originating point of a radius. So it's kind of like, if I draw a bunch of circles that just barely touch and translate them into a rotational map of the same space, I will get a circular space of touching circles. Which is kind of like, yeah, of course you will.

[@AMTunLimited]

@@EamonBurke I meant the first circle from the line

[@WestExplainsBest]

Takes circles to a whole other level. Videos like this would enhance a secondary mathematics classroom.

[@Bronzescorpion]

@@EamonBurke What do you think arbitrarily means? Doesn't seem to me that you are using the word right, even the two words "arbitrarily" and "precisely" seem kind of a oxymoron when put together. The circles are not arbitrarily placed and neither is the fixation of the corner of rectangle. I would even argue that is the exact opposite. Both the circle and the rectangle follow a strict set of rules or deliberate thought in placement.

[@ShankarSivarajan]

@@EamonBurke "arbitrarily placed precisely arranged" Which is it?

[@ZachGatesHere]

This is one of the most genuinely mesmerizing videos on the channel so far, with Langton's Ant and Sandpiles being its only competition.

[@WestExplainsBest]

Takes circles to a whole other level. Videos like this would enhance a secondary mathematics classroom.

[@xleph2525]

Also Conway's Game of Life

[@bammam5988]

The logistic map

[@Triantalex]

false.

[@AliveInTwilight]

I love this "style" of episode - 2-4 similar but unrelated topics in one longer vid. It's like the Neil Sloane "amazing graphs" series of vids from this channel. Great stuff!

[@BigDBrian]

The animation just begs to draw the next horizontal line, one unit up. It should touch all the circles from both rows, and the 'origin', so you can imagine how it must be drawn fairly easily. I think it would look neat though.

[@HPD1171]

he did this at 5:40 but he drew a line four circles up. he just did not show the line only the resulting circle.

[@MajikkanBeingsUnite]

Apollonian gasket, you're thinking?

[@RowanAckerman]

I'm interested in what you would get if you drew circles below the line.

[@MarioDiNicola]

I'm interested in what the tightly packed configuration (hex) looks like on inversion...

[@jarrodfrench957]

I believe you'd just get smaller or bigger circles, whose touching on the right hand side of that original circle? That is, all parallel lines above the "original line" would be a smaller circle sharing one point with the "original circle" and bigger circles for the parallel lines below the original.

[@jonwoods6745]

I love seeing videos with Matt Henderson! Thank you all for what you do!

[@LilZombieFooFoo]

My brain melted at the rectangle. My goodness! Surprise circle inversion is the new "what is pi doing here."

[@wasabij]

Yeah, but seeing it in motion it's hard to not see! That is what I love about these animations: they give me a much better intuition of concepts the textbooks or my unfortunate tutors every could!

[@CursedKyuubi]

THIS is the beauty and simplicity of Math. Our numbers give reasoning and make it complex, But behind those smoke and mirrors of numbers and variables, is geometric beauty. Amazing work Matt Henderson. And discoveries/explanations like these is why Numberphile is an OG of the youtube Math community

[@Hello-pz6hb]

"Infinitely Many Touching Circles" sounds like a cool band name.

[@HanzCastroyearsago]

I have an idea

[@dave2.077]

they play abastract punky rock

[@irokosalei5133]

That sounds like a track from Explosion in the Sky

[@Slyzor1]

Yeah, if they play nerdcore

[@EDDhoot]

or mathcore

[@eliaspoulogiannis]

This reminds me an epic older Numberphile video with Simon Pampena where he manually draw the circles

[@davidgillies620]

Apollonian gaskets are cool. They have a connection to Ford circles, which I think have been covered in another video, and thence to Farey sequences and the Stern-Brocot tree (ditto).

[@matthewellisor5835]

Well, my Ford does need a new set of head gaskets. Where can I find that brand? :D I'd say that I'm sorry and I couldn't help it, but I won't lie to you.

[@luizchagasjardim]

When I saw the constant area thing, I immediately shouted "that's inversion with extra steps". Very cool way to introduce this concept.

[@unvergebeneid]

What does the circle on the right map to, the one formed as a limit of all the increasingly smaller circles? And what happens when you go below the line? Do you just get a mirror image of the pattern within the circle?

[@bobby_tablez]

This is what i need to know

[@csours]

Below the line would be outside the main circle. The circle formed as a limit of the other circles is the sides of the triangle that is formed from the circles drawn above the line

[@sinisternightcore3489]

Looks like it's drawing four infinite rows of circles and topping the fourth row off with another straight line.

[@alonamaloh]

Below the line you'll get mostly another copy of the circle but going to the right instead of to the left of the origin. I'm not exactly sure what the two rows of circles just below the line will map to. I should make this picture.

[@Bodyknock]

Even though the picture at 8:01 makes it look like at the limit there’s a big empty circle on the right, that’s just where the simulation stopped. If you think about it though, if the lower right corner of the rectangle is the origin (0,0), then there are circles above it touching all points of the form (0, 1 + (2k+1)/2) for k = 1,2,3,…. So the height of those rectangles is 1 + (2k+1)/2, which makes the width the reciprocal, and as k grows to infinity that width approaches 0. So as more and more circles are added you are getting circles that intersect the x-axis at points closer and closer to the origin, which means there isn’t a big empty gap on the x-axis, it’s filled with an infinite string of circles approaching that pivot point.

[@tapenoisecafe]

4:42 the funny

[@TheGreatAtario]

I feel so emotional. These circles are just so touching.

[@kitconnick427]

I never want these Matt Henderson videos to end, I love them, thanks for bringing him to my attention!

[@kdawg3484]

I need to see either an epic math fight or an epic math collab between Grant Sanderson and Matt Henderson.

[@MathFromAlphaToOmega]

Here's one way of seeing why the intersection of the circles and lines is a parabola: Say the center point has a circle of radius r intersecting one of the lines. Then that intersection point is r units from the center. But since the circles and lines are moving right at the same rate, the intersection point is also r units left of a certain line. That line is the directrix, and the center point is the focus.

[@jacobspillane5028]

The circle inversion reminds me of the tanks in Bubble Tanks, must've been the method they used.

[@crackedemerald4930]

Bubble tanks is great

[@MaverickReynolds]

Dude I swore I was the only person thinking of that

[@yaseenshaik67]

This is the channel that can make anyone fall in love with mathematics💯💯❤

[@WestExplainsBest]

Takes circles to a whole other level. Videos like this would enhance a secondary mathematics classroom.

[@divyanshsrivastava824]

But our schools is teaches maths in a way like they are sst , Because of there teaching some people hate maths either maths is a subject no one can hatee

[@tiberiu_nicolae]

The concentric circles becoming a cone blew my dimension challenged brain

[@gertjan1710]

Try some Lorentz transformations next

[@BooBaddyBig]

Circles, ellipses, parabolas, hyperbolas are all referred to as 'conic sections' and are produced from second order powers of x and y.

[@MrAdzielinski]

I remember seeing and attempting the original inverted circles video. Glad to see math is still fun

[@stanimir5F]

Everytime I hear about "circle inversion" I get a flashback from the Simon's laugh in "Epic Circles" at 21:50.

[@OnatBas]

The sound effect at 1:15 scarred my ears.

[@jacklardner8229]

Yess more Matt Henderson content

[@lennywintfeld924]

Wonderful! Astonishing.

[@gillfortytwo]

This is the coolest demonstration of conic sections I've seen!

[@irwingalvarez]

This is super cool. I'd love to spend a day with this guy just asking him what else he finds interesting . Also @ 5:00 giggity

[@saranchance5650]

Very cool. The music matched things well

[@DeadJDona]

5:20 what happens if you draw a circle _under_ the line?

[@riuphane]

I believe it would create a circle outside the original one... But that's just using my basic understanding and intuition, not actually tested

[@iteerrex8166]

So cool! Yet another use of the pantograph. 👍

[@DougMayhew-ds3ug]

All of these animations are beautiful, and the circle inversion one reminds me of the holographic principle. Brilliant! I am left wondering about variations or extensions of these themes into still higher forms.

[@remek_ember]

This brings back memories, I follow Matt's tumblr since the beginning. Those were the days lol. I loved his animations!

[@matthewsaulsbury3011]

Wow, this is amazing and fascinating! 👍🏼😀

[@JocelynDaPrato]

Thx Matt, really nice!!! Could you do an animation that plunge into the "infinit" circles ?

[@qzbnyv]

I know this is pretty subdued. But it’s cool. Thanks for sharing! The sounds effects help too btw

[@lidular]

The original circle inversion video "epic circles" is probably my favourite numberphile video.

[@ccbgaming6994]

“Infinitely touching circles” I haven’t heard that since my old college days…

[@punpcklbw]

Wicked stuff, seeing how simple the circle tracing algorithm is. Not as intricate as Apollonian gaskets, but definitely shows some interesting patterns.

[@henrymarkson3758]

Matt Henderson, the master of understatement

[@WZaDproductions]

I wrote a paper inspired by the original circle inversion video, I was fascinated by it and the whole new perspective it gave me on math!

[@KaiCyreus]

i was about to ask if that was circle inversion i was seeing there, turns out it's an even easier way to think of the process, so glad that it was clarified ☆

[@MecchaKakkoi]

A 3D version (and so on)? Great vid! :)

[@aryst0krat]

I love him explaining the ripple in water with a computer behind him featuring a ripple in water as its background.

[@sonaxaton]

The Epic Circles video is one of my favorite Numberphile videos ever, cool to see a new take on it!

[@witzman]

Epic circles is the best video on the internet. Mindblowing.

[@gabor6259]

Math continues to amaze me.

[@certainlynotthebestpianist5638]

When I saw this line drawing a circle, my brain immediately shouted "PTOLEMY'S THEOREM!" with Prof. Stankova's voice. Oh, that one, it's still my favorite video I've watched in my whole life!

[@ArtSeiders]

Thanks!

[@sethgilbertson2474]

Beautiful!

[@eplumer]

OK, that was a cool interpretation and visualization of circle inversion

[@blahsomethingclever]

Wow the first maths video I didn't understand. And have to watch again and take out a notepad. Thank you. I mean it

[@Lodekac]

Amazing!

[@recklessroges]

It tickles my brain that the infinite circles inside of the original circles just emerge from the outside circles through the area constraint.

[@glenneric1]

Have you ever tried to trace out the fourth rectangle point? It looks like it might be making some cool leaf pattern.

[@thomasbirchall9047]

I'll be honest, I didn't have a clue what was going on. I just liked the animations

[@glenneric1]

I've never seen this. Very cool.

[@johnchessant3012]

Indeed I have not looked at circles the same way since that "epic circles" video.

[@VidyaBhandary]

Awesome !

[@ASOUE]

Matt’s voice is so soothing.

[@sayeager5559]

Longtime fan of Numberphile circle videos.

[@deliciousrose]

Funny that this video uploaded after I watched circle inversion video. I was looking for more info about Pappus Chain since it appears in some of Alphonse Mucha's works.

[@adrianpadalhin854]

I'm really mesmerized by the "Infinitely Many Touching Circles" part - very, very beautiful pattern. I'm really curious what it would look like if you instead used one of the corners of the fixed-area rectangle to inscribe circles on a non-square grid (triangular or hexagonal?). Would you get the same pattern? I wish I could try this out myself, but I'm not a programmer... sigh.

[@hatredlord]

It's not the same pattern, naturally, but i don't think the difference would be visible without it being pointed out: Consider that every time circles touch, they do so in both "worlds". A triangle pattern above means you have each circle below touching 6 others, rather than 4 as shown. An hexagonal pattern is just the triangular pattern with some gaps, unless i misunderstood you.

[@gtziavelis]

Numberphile has a video called "Epic Circles" that is related to this concept, from a while back.

[@kitlith]

at roughly 5:40 they go "and what this really is is circle inversion" and throw up the other videos they've mentioned circle inversion in before, including epic circles

[@frankharr9466]

Inversion was the first or second thing I thought of. That was cool. The curl was pretty right before it became a circle.

[@travpots6318]

Good job

[@rcookman]

Cool love it.

[@yoyoyogames9527]

love some circle inversion :D

[@box9283]

What's better than touching circles on weekends?

[@ishanv08]

:O

[@carlosdelossantos5115]

8:14 "hands-on", and immediately I see something that looks like a wooden handcuff, holup XD

[@MushookieMan]

I believe it gives a rotated, translated, and possibly mirrored inversion. In ordinary inversion, points drawn on the circle of inversion map to themselves.

[@williammundy6562]

Seems like a nice visual proof / demonstration that the arc of an infinitely large circle is a straight line.

[@rudiklein]

The more I watch these video's, the more confused and humble I get.

[@bomberdan]

I didn't know the guy from "You" was on Numberphile videos!

[@JavierSalcedoC]

remember reading about Apollonius of Perga as a kid, how trying to solve a military problem (stacking shields to make walls) turned into one of the first fractals

[@NathanChojnacki]

this is beautiful

[@Traceuratops12]

I immediately thought of the circle inversion video when the circle animation first came up.

[@gigglysamentz2021]

So many awesome things ♥o♥

[@eri4108]

I draw such circles every time when my teachers start to complain about us. Now I know how to improve that, thanks guys!

[@ygalel]

Yep. Circular inversion was my point in life where I realized that the world as we see is simply subjective and depending on other perspectives thing may look different for an identical object.

[@aaronsmith6632]

Fascinating! Would be neat to see circle inversion of tesselating hexagons. Does this work in 3D, with len*wid*hgt = 1? Reminds me of hyperbolic mapping.

[@kyriet9303]

Hey Matt, at first I thought you are Daniel Sharman (the actor) 😄 Nice vid, interesting topic 👍

[@bunga_raya96]

Incredible

[@AtlasReburdened]

Would this map into 3d? Would fixing a corner on a constant volume cuboid and drawing a plane with another produce a 3 sphere drawn by a third? And would similarly drawing an array of 3 spheres above the plane propagate the volume of the one below the plane with infinitely many touching 3 spheres?

[@b.a.r.c.l.a.y9701]

oh that sounds really cool id wanna see that

[@Xbob42]

Based on the beginning of the video I was actually waiting for him to shift perspective at some point.

[@madeanaccountjus2say]

this guy out here inventing circles and I can't even tie my shoes

[@sooryanarayana3929]

Now start selling merch with these circles

[@SquirrelASMR]

More pls

[@TheMeaningCode]

Is there a way to determine what the inside area is of all the small circles? Do they equal the area of one of the two larger circles?

[@zaubergarden6900]

the wonderful feeling when you get the maths

[@Veptis]

I love the beauty of those visualization. Perhaps one could use all of their code to training a neural network that does text generation. And see what comes up. The dataset is probably way to small - and even larger datasets probably don't do enough for current methods. I am trying the same with webGL shaders from Shadertoy and that has 22k shaders, which might not be enough either.

[@scottanderson8167]

Infinitely many touching gorillas.

[@stevesalt8003]

Why are them circles so satisfying? I was craving more circles.

[@studyforyou6794]

Good work 🥰🥰

[@ved9402]

Did you notice that the set of circles being constructed above line reminds me of Pascal's triangle

[@frab8061]

How can we get the outline of the final circle, that's formed by the infinitely many tiny circles at the end? Is this like a line at a certain height? I hope my question is not too confusing

[@xxnotmuchxx]

what if u draw circles below the flat line? also, what is the area of all the circles inside the big circle?

[@AaronQuitta]

The touching circles remind me a lot of the Poincaré disk model of hyperbolic space.

[@dregoth0]

So, if you can use this to circle the square, can you do the inverse to square the circle?

[@josemanuell1838]

How/where do you learn to program these animations??