Steiner's Porism: proving a cool animation  

13:45 "...real world applications?" Yes, arranging wire or cables within a larger circular cross section. For example, undersea power cables that might have three phases, as well as several smaller signal cables. Can be applied hierarchically !! Cheers.

I really like your tone, it reflects the bizarre "yeah I just thought this was cool" part of maths. I'm sharing this!

Using a sphere to transform circles is so satisfying .

This was an awesome video, thank you! Nothing's more noble to pursuit than neat GIFs!

at 8:17, the discussion is not complete. There are two more possible places the circles can go and still be tangent to all three circles (red, blue and white): one is touching the blue and white circles at their intersection and touching the red circle on the other side (from the inside). The other one is similar, going from the red and white intersection to touch the blue circle on the other side. To make the proof complete, these cases should be ruled out, too. (Not that it's hard, I just thought this should be mentioned as well.)

I love how from a simple definition of the projective complex line, we end up with such sophisticated diagrams. Homographies are so neat.

This gave me existential dread. Thank you.

I really appreciate how succinctly this video demonstrates the effectiveness of solving hard problems by first mapping them to simpler ones (in this case mapping a chain of circles of varying radii onto a chain of circles with the same radius).

Oh, ok, so "porism", has nothing to do with the fact that the circles look like _pores_ ? I'm slightly disappointed. The thumbnail certainly looked like a _porous_ disk to me! But no, it turns out it's related to the English verb "to pore" instead! Well, that's still interesting. Anyway, great video! I liked your presentation a lot. As soon as you said "mapping circles to circles" I thought of Möbius transformations, which I used in a _Basic Complex Analysis_ course. But I didn't know how they related to the stereographic projection! This is great! It's also nice to see the graphical version of the tangency argument, having already proved that fact analytically. All in all, thanks for this insightful video!

This is one of the best results in Projective Geometry, really nice video.

This is dangerously underrated

Porism comes from the Greek word πόρισμα (porisma) meaning conclusion or corollary!

This is super ineresting, thank you!! The arguments are really clear & I love the animations. The only thing I'd add is that the function is only tangent-preserving by coincidence, because you're looking at circles. More fundamentally, what your bijective function is *really* preserving is intersections (as can be seen because this is what your proof uses). Cheers!

THANK YOU, I'VE BEEN TRYING TO FIND SOMETHING TO EVEN DEFINE THIS STRUCTURE FOR LIKE 6 MONTHS. I didn't even know the name of what this was called in Mathematics, thank you.

Oh I absolutely adore your style of teaching. Image my surprise learning that this was a small channel considering that this was one of the coolest math videos I’ve seen

Beautifully explained and illustrated and that's a bizarrely simple proof leaving almost all the tools in the box and just doing some careful pondering.

Circle preserving maps in this example are one particular aspect of the field of conformal geometry, associated to the generalization of calculus with complex numbers. Yes, there are applications! Conformal geometry is an extremely important conceptual ingredient not just in modern quantum field theory, but ordinary phase transitions in everyday statistical mechanics. This should not be an indication that there is something special about our world to use it, but the ideas here are fundamental to any continuous geometry in plain mathematics.

I love that you actually drew that QED box in the end of the proof. I was somehow facinated everytime when I see that.

My guess was that you would use two circle inversions to map from the symmetric case. Now I need to go and check if that would work! Nice job with the video.

13:42 Best part xD

seems like this could be handy for designing eccentric ball bearings

You did an amazing job intuitively explaining the idea of functional continuity by the tangency preserving principle you mentioned!! Absolutely stellar

13:20 Haha, I was hoping for such an animation at the end. I'm not disappointed.

This was so damn satisfying on an OLED display

dude you are like 3blue1brown but more personal I love your content

Thank you! I have been musing about circle and sphere packing and this has given me lots of inspiration.

I ended up pausing at 1:20 and trying to prove using circle inversion. Supposing we have one working example, choose one white circle, "W1". Draw a new circle "A" which inverts the white circle to itself, the red circle to itself, and additionally the blue circle to itself. Now, select a white circle, "W2", adjacent to the first, and similarly construct "B" which inverts that white circle to itself and also the red circle to itself and the blue to itself. If we reflect "W1" over "B", the resulting circle "W3" will be tangent to the red and blue circles (since those map to themselves), and also to "W2" since "W1" was tangent to "W2" (which is being mapped to itself). If we reflect "W2" over "A", the resulting circle will similarly be tangent to the red circle, the blue circle, and "W1". We can continue reflecting any new white circles over "A" or "B" (whichever one didn't generate them in the first place) to create "W4", "W5" and so on. The numbering doesn't state their spatial ordering, but as you can see, they'll form a chain since each new one will be linked to an old one. Due to their tangents, each such new circle must correspond to some white circle in the original diagram. Because the original diagram formed a loop, this process must come to an end with some finite number of white circles. Furthermore, we can invert the circles "A" and "B" over one another repeatedly to generate "C", "D", "E" etc., which are circles fixing the red and the blue circle, and additionally fixing "W3" or "W4" etc. IE, each white circle will have a corresponding circle which maps it to itself in addition to mapping blue to itself and red to itself. Allow me to name the collection of "A", "B", "C" etc., calling it "R". The thing to note about "R" is that each member is a symmetry of the entire figure, mapping it to itself. Because all the members of "R" map the whole image to itself via circle inversion, the members of "R" must all intersect at the intersection of "A" and "B", since if they didn't, it would violate the symmetry. The intersection of "A" and "B" is actually two points, one inside the red circle, and one outside the blue circle. Let's put a new circle around the second intersection point, keeping it small enough that it's entirely outside the blue circle. I'll dignify this new circle with a color - it's the green circle. If we invert all members of "R" over the green circle, something interesting happens. Because they intersect the center of the green circle, they necessarily become straight lines instead of circles. Now invert the rest of the figure - the white circles, the blue circle and the red circle - around the green circle. We know we'll end up with white circles in a chain, still tangent to the new blue and red circles. We also know our new straight lines are tangent to our new white circles. But what's really notable is that the new straight lines will be symmetries of the new white circles, since circle inversion itself preserves symmetries expressed using circle inversion. But if our new figure is symmetrical over a collection of lines, then actually, it's rotationally symmetrical, and all the circles are the same size. So if we rotate the white circles in our new figure, by any amount we want, and then use circle inversion over the green circle to map them back onto the old figure, then we get a working chain, and can create the desired animation. I like this proof because its structure is a good match for the question; the stereographic proof loses me a bit when it starts trying to show the red circle can be put anywhere. (The original stipulation was that we can choose the red circle's size, not that we can choose its distance from the edge.) However, the proof in the video gives a good sense of why any size of red circle will work, and mine doesn't. Things I omitted: Why can we construct "A" and "B"? Why can we trust that symmetries defined by circle inversion get preserved by circle inversion?

I feel like you took what was in my head and animated it in this video. And wow, those animations were gorgeous.

I find the statement that something is “always or never” very funny for some reason

Great explanation and animations. Well done! I thought it was going to be some kinda elliptical cross section thing, and I think that's kinda what it turned out to be (I'm not to heavy on the math terms).

Very nice! Enjoyed the video from start to end.

In greek math bibliography, πόρισμα (> porism) actually means corollary.

BRUH WHEN YOU ROTATED THAT SPHERE 10/10

We could use this to design non-concentric planetary gears, their mass would be excentric and perhaps a neat application of this method could lead to a particular variable CAM design. As an engineer I can actually see it through an application, we would have to wonder tho if it's going to be practical enough to go through the effort of designing it

This video is simply amazing!! Can't believe this didn't win. Do keep up the good work and make more math videos you are really good at this.

god, what a wonderful "aha!" moment

Very interesting, I love this! I've actually spent some time doing constructions like this by hand; all of the circles beyond the third (first two are given, third one is trivial) require you to solve a special case of the Problem of Apollonius to construct them correctly (i.e. without just eyeballing it). There are 8 solutions to the problem of Apollonius in general, but the fact that some of your starting circles are already tangent to each other is going to reduce the number of solutions that exist (I think to just the two you show, but I'm not certain if there are more hiding somewhere). That last circle - which is tangent to four other circles - is a fun extra that you don't get by just solving the Problem of Apollonius; figuring out the proper construction to make it would be fun I think! The stereographic projection is also something I use a fair bit in my actual research (crystallography / TEM); always fun to see it in a new context! Bonus: if - after constructing Steiner's Porism - you solve a different special case of the Problem of Apollonius (this one has a name: the inner Soddy circle) for each of the triplets of circles in your Steiner construction, you could construct something very much like an Apollonian gasket (neat fractal). It's a little bit different though, so it needs a different name. Since the Apollonian gasket is a porous sort of thing, I propose we should call this new construction: Steiner's Porous Porism!

Great video Mr. Newton. Made your ancestors proud. Beautiful use of the stereographic projection.

new favourite channel

What a wonderfull channel. I really like your voice presentation and scripting. It is clear crisp and entertaiing. To too complex and not too dumbed down.

Great video! I love this particular solution using projections. My initial thought upon seeing the animation was that it looked like an optical illusion where the small circles were further away and the big ones were closer. Seeing this intuition of projections being used to solve the problem is very satisfying.

I feel like this would make a neat GIF

this video has infiltrated the maths community and I love it, hope you see more stuff.

Besides cool animations, this might be useful for stimulating fluids of various temperatures and pressures, with the clines being represented by the tangents. The absolute position of the sphere hosting the stereographic projection would be the balance between the various masses.

4:33 Okay how have I never stumbled upon this gem of a video sooner

Only one vid? I'm disappointed... looking to see some more excellent stuff coming from you!

I love the animation, you need to make more.

Dude this is so cool.

Great video mate. Would be awesome to see more

That animation is so beautiful...

this is super neat. the animations are really helpful, and satisfying. the only thing that could possibly be improved is the audio.

Very nice mate

What i saw there is ideal pump, keep it up

This is phenomenal! Great job, great explanation, I hope you keep making videos :) I'm subscribing!

This video reminded me what I loved about teaching math. I kinda miss it.

This is definitely gonna blow up

This was awesome dude, keep it up. Love the style.

Wow this was such a great video. I easily understood everything you said, really great demonstrations. I wish I could like this video 5 times because you definitely deserve it.

Wow this blew up for you huh? Good for you! It reminds me of 3blue1brown 😁

It's cool to have a competition thing named after me

that would be a cool thing to put on a loading screen

It s so amazing! Thank you!!!

I enjoyed this very much! Keep up the amazing work and have a wonderful day!

Thank you for your hard work, this was illuminating

Great video, complex concepts explained easily

There's some neat logic involving the incidence structure of the mobius plane hiding under some aspects of this proof. Also, treating the outer circle as the boundary of the Poincare disk, we can get a nice corollary out of this involving what radii for circles in the hyperbolic plane give rise to chains of horocycles. Really, it's neat stuff all around.

Incredible good animations!

Amazing! Thanks for sharing this great animation and ideas! Hope to see more videos from you!

5:20 Oh. My. That _is_ elegant. I'm all a-flutter.

This is very cool!

I cannot believe my wild guess that this would be a Möbius transform was correct. That's what happens when one remembers exactly one thing from complex analysis.

Watching this video in front of my friends make me look smart :D

Really cool video!

I couldn't finish the video because of the sound it's too low. nice work and good explanation as far as i watched. next videos pay more attention to the sound plz. good luck!!

That is sooo cool!

disregarding all transformations, there are countably infinite arrangements of circles that link up like that, as the number of circles must be an integer.

Very enjoyable and easy to follow. Nice work!

Another real world application would include extending the understanding of how mapping spherical geometry to the plane will unavoidably distort the shapes on the sphere no matter which translation is used (hence why the Mercator projection distorts things at the pole). This way when moving the area of interest to the South pole will minimise the distortion to what is important.

loving it! thx

5:13 Amazing! 🤯

Great video, awesome channel. While watching i wondered if you could use this to build an excentric roller bearing. i might try it in the future

Maybe next time up the volume a bit? I'm at max and medium sounds overwhelm the vid :(

So the concentric circles setup is the 'canonical' form and all other valid setups (where the chain is linked) are just projections of the canonical form and anything that isn't a projection is invalid (the chain is not linked). This seems very obvious after watching the video 😅 so I think your explanation did its job.

Very nice! It's a gem!

Beautiful, simple, and elegant -- both the mathematics and your explanation. Really great video on a really nice piece of math, and animated really well.

There is another application: two parallel lines, 2 circles, can we fit a finite number of circles in between to "connect" them and the lines? This is extreme case, and can be useful in construction by well chosing the sizes of the supports, thus minimizing the number of rods keeping things connected or smth This can definitely have a use

Excellent video, congratulations!

3:01 I think that's part of the Riemann zeta function critical line Kappa

No, this is definitely not topology. In topology there is no such this as distance. Circles are very much defined using distances. This is geometry.

I would say the study of transformations is geometry as per Klein's Erlanger Programm.

4:36 nice thing to write in an exam :-D

This is how pros place their towers in Bloons tower defense

How about a 3D-Graph. Z shows the amount of circles. X shows the proportional size between inner and outer circle Y shows the proportional offset of the inner circle to the outer circle How would it look like? A lot of seperated walls? Stacked areas? What if the axises would be swapped?

anouther great SoME

I expected porism to mean pores like holes.

never know this, but I can make off center bearings now base on that, just to figure out if there is round off numbers that happens to fit. this can make a presume very stable boring bar.

maybe there could be applications in mechanical engineering? like making some sort of gearbox from such a configuration?

Dunno if it would be of any practical use, but you could make a bearing with an offset centre...

Cool cool. Conformal transformation. Can you do something with Descartes Theorem?

To be fair, they really do make "neat GIFs"!