Poncelet's Porism - Numberphile

[@Dalendrion]

Unlike other Numberphile videos, this one's left me unsatisfied. I still have no idea how the parallelogram explains anything. I still have no idea why the idea of connecting stars between two ellipses is interesting. What was proven? How was it proven? The delivery in this video isn't really there this time. That's a rare criticism for Numberphile.

[@jreinhart3382]

I agree.

[@dhayes5143]

Heartily agree. 'There was something cool proven, and there were cool proofs. And people found it exciting.' That's about as much as I got from it.

[@EtienneCharlier]

Same here. And listening to Brady's voice at the end of the video, I would not be surprised that he feels the same ☺️

[@soilnrock1979]

Totally agree. After watching the video I didn't feel any enlightenment. No idea what the parallelogramm was supposed to show/prove :-(

[@jamirimaj6880]

Trust me, the connection really is deep, he really did his best with the connect the sides to form a cylinder thing. It would take a knowledge on higher maths to understand why. Btw, one modular form of the elliptic functions is the one used by Andrew Wiles to prove Fermat's Last Theorem.

[@ButzPunk]

I wish this video had been twice as long. Felt like I was just getting into it when it ended.

[@ruinenlust_]

Most of Numberphile tbh

[@antoniussugianto7973]

Short and slim...

[@OldG4merDad]

I whole heartedly agree!

[@Triantalex]

??

[@KipIngram]

Ditto - this could have gone a lot deeper. As it is, the coolest thing is the name. Poncelet's Porism - that's just kind of hard to beat.

[@danielliu3322]

He failed to mention one important part of the theorem: the number of steps that it takes to get back to the point you started with is the SAME, no matter what point you started with. That's what makes this theorem so mind-blowing.

[@gregpotts9410]

Daniel Liu of course it’s the same, it’s a closed path

[@Alexagrigorieff]

Nothing mind blowing about it, if you think about moving the starting point around. Because all endpoints will also move smoothly, their number should not change.

[@Alexagrigorieff]

@First Last Think of it in terms of differential geometry. Any small change (delta) in the starting position will bring small change in the positions of every point. For the number of points to change, there should be a case when such delta would bring a unbounded change in position of one of the points.

[@eeplordsupreme]

​@@Alexagrigorieff Numberphile's appeal has always been explaining things to a general audience, yet you started off your explanation by appealing to differential geometry. You needing to appeal to diff geo (and there's obv nothing wrong with that) to best explain a numberphile video shows that they indeed missed the mark here

[@user-me7hx8zf9y]

@@Alexagrigorieff The proof by varieties in elliptic curves is more interesting

[@veggiet2009]

2:22 proof by animation

[@rcb3921]

Well they didn't say they were proving it -- quite the opposite (2:24). The animation is just describing what the theorem states.

[@veggiet2009]

@@rcb3921 it may not be exhaustive but it shows that for that combination of ellipses, starting at any point on the outer ellipse always ends up either back at the same spot or not. That's what I was meaning it was proving... It didn't prove that it would work for any sets of ellipses

[@johntavers6878]

my first thought as well! the animation shows how you might prove the theorem using continuity

[@Triantalex]

false.

[@v_aasu]

To everyone commenting that they're not smart enough to understand this or don't have the background, I don't really agree that that's the problem. We've seen many times professors on Numberphile videos present problems and demonstrate proofs in a very accessible / approachable way. I think this instructor assumed that we had too much knowledge and didn't connect things well.

[@dlevi67]

Um, no. The problem is that this is genuinely hard - or at least we ("humanity") don't know an easy way of bodging an explanation. Just like for Fermat's last theorem - people keep harping on about "the Taniyama-Shimura conjecture (now theorem)", but the connection between elliptic curves and modular forms (which are not "simple" in and of themselves, other than to define) and Fermat is not simple either, as soon as you go beyond the glib statement: "if you prove one, you prove the other". Simon Singh wrote a 400 page book which is interesting to read and accessible, but it still explains very little in mathematical terms.

[@Bollibompa]

And many, many videos completely omit the proof if it is too complicated ya doofus.

[@Reconsiderate]

> "I don't see the link...." > "yeah i haven't told it to you 🙂" (Every teacher's favorite transitional moment during a new lesson)

[@alexandertownsend3291]

I thought it was, "This is pretty easy so there is no point covering this in class. Do it on your own time". My teachers have done that many time.

[@dlevi67]

@@alexandertownsend3291 I think this case is: "this is so complicated that there is no point covering it unless you are doing a maths degree. Do _that_ in your own time".

[@alexandertownsend3291]

@@dlevi67 i think you areright.

[@TheZenytram]

Worst thing ever to do, this is a bad practice in teaching and why ppl still continue to do this.

[@alexandertownsend3291]

@@TheZenytram What is?

[@SwitchAndLever]

What I'm missing from this is why you couldn't just continue to add tangential lines though. It seems like they stop after an arbitrary N-amount of steps, whereas if N would approach infinity is it not likely that this would approach a valid solution where the lines do meet up at the origin point again?

[@schnauzpig]

That's what I was thinking. I hope someone can explain.

[@HasekuraIsuna]

Me too! Surely there must be "stars" with arbitarily high number of bounces?

[@timotay22]

Could it be that some ratio pf geometric parameters is irrational? Like how a rectangle with side 1 and sqrt(2) couldn’t be patterned together with some of them tilted vertically and some horizontally?

[@paulpantea9521]

You can add as many as you want, but it never comes back to the origin. Another way to think about it is using rotations. If you start on a circle and rotate an amount that is a rational fraction of 2pi, then you eventually get back. But not if you start with an irrational amount. Even if you do it infinitely many times, you never reach back.

[@Megaranator]

I think that you would just get infinitely closer to them touching each other

[@kshiwram]

And he still didn't explain how the parallelogram relates to the ellipse hmm

[@whoeveriam0iam14222]

yeah it makes a donut and the sides of the parallelogram make 2 circles on the donut and what does that mean???

[@drewkavi6327]

This channel does not aim to explain maths, it aims to popularise it. To understand the mathematical connexion between the two a maths degree would be your best bet.

[@qrubmeeaz]

It gets kinda deep. Go read up on this stuff.

[@RealLifeKyurem]

If you want to read up on this, look up fundamental polygons. They’re used a lot in topology.

[@srednadahlberg]

Yeah. It’s really frustrating when the math behind the topic is too advanced to explain, even to the interested. Sort of ruins the video a little bit. But I guess that’s what I get for forgetting all my university math.

[@dudewaldo4]

Really feel like we didn't get to the meat of this one!

[@turtlellamacow]

"I don't see how the ellipses correspond to the parallelogram." "Well, if you look at the points on the ellipse, and the points on the parallelogram, there's a correspondence." I don't even understand what the elliptic curves have to do with the ellipses, apart from sounding similar!

[@willnewman9783]

At 5:07, he writes down a subset of the pairs of points on the ellipses. The pairs of points will have dimension 2, so this subset must have dimension 1. I am guessing this subset is an elliptic curve.

[@LuisManuelLealDias]

Worst teacher ever. "Yeah this stuff is exactly like that stuff" "Huhhh how?" "Because. Cool right? Also, doughnuts. Even cooler, right? Let me sit here by myself looking damn smart. Because."

[@quicksyss]

he said right off the bat he wasn't going to get into the proof, only talk about the problen a bit; i think that would be better suited for a different channel

[@kostantinos2297]

@@quicksyss He did say he was about to explain how the parallelogram approach is relevant to finding ellipses whose tangents return to the starting point, but as far as I'm concerned nothing of what he said afterwards made much sense.

[@jashshah1033]

Right... I didnt understand much of what he explained

[@Munkingly]

You should demand your money back

[@ThaSingularity]

7 minute video

[@PassionPopsicle]

I think this is the first time I've watched a Numberphile video and felt LESS interested after watching

[@azhakabad7498]

One of most underrated topics in mathematics!

[@x_gosie]

One but not many. That's the math there.

[@R2Cv1]

"mmkay" "yeah" "okay"

[@antman7673]

Those old mathematicians, some were such galaxy brains.

[@DustinRodriguez1_0]

Loads of them, if not most, made lots of their discoveries when they were teenagers or early 20s. If a teenager suggested a groundbreaking theorem today, their teacher would just tell them 'thats not in the book, it must be wrong.'

[@thekingoffailure9967]

Throughout my highschool pre-calc type courses the most common complaint was "Why do I have to solve the question THAT way when this way is better for me?" Not to mention the science questions that were "too high level" for the teacher to bother explaining, even slightly.

[@zwz.zdenek]

@@DustinRodriguez1_0 It's more like all the low-hanging fruit had been picked. Nowadays, you have to invent a new branch of mathematics and use it in a 100+ page long work to stand a chance.

[@lonestarr1490]

@@zwz.zdenek Not necessarily. The PHD thesis of John Nash is 26 pages long and includes 2 citations. And you can understand large parts of it even with just high school level math. Fundamentally new ideas do not have to be overly complicated. They just have to be fundamentally new.

[@TrebleWing]

Well we only ever hear of the most amazing ones most often since they are the ones that made the progress and were worth remembering. I'm sure in 500 years time, the amount of figureheads that are lifted up as the best mathematicians will be talked about in the same frequency ratio compared to their contemporaries.

[@Yossus]

This is cool but it made no sense to me. Let's go down the wiki hole!

[@dlevi67]

The wiki pore. It's a porism.

[@Yossus]

@@dlevi67 take your upvote and get out

[@dlevi67]

@@Yossus Gone. Through a pore. ;-)

[@vladimir520]

Lol I remember doing this with a ton of Numberphile videos. Teaches you a lot about math in general :)

[@sadrevolution]

The joy of the post numberphile video week!

[@michaelhird432]

Hey it's this guy again, i like him

[@stormsurge1]

He is litt

[@nikoli6601]

I love his handwriting

[@onecanina]

At 5:15 he is really enjoying! He is definitely down for a math challenge

[@Ripen3]

@@onecanina haha, you think he's cool dont you..

[@Triantalex]

??

[@onthecover5042]

Alternative title: How to draw a weird star

[@zoobie2000]

Seems a bit smug. I know I'm not as clever as him but there didn't seem any attempt to explain properly.

[@stephentrueman4843]

some people shouldn't teach lol

[@techwithwhiteboard3483]

you wouldnt get eliptic integrals well not in a 20 30 min video but now know an interesting result

[@Chris-hz8lj]

To be fair we didn’t come here for a documentary length video.

[@giladzxc17]

I know a similar theorem, where if you have to circles where one is inside the other, and you draw a chain of circles where the first one is tangent to both original circles and the next one in the chain is tangent to both original circles and the previous one in the chain. Then if the last one meets the first one, after n circles, then you can choose any circle that is tangent to the 2 original ones and the last one will meet it, too after n circles. However, this is easily proven by inversion, not at all by elliptic curves.

[@yeoman588]

I played with this in GeoGebra a bit and discovered that concentric circles display a pattern. With an interior circle of radius 1 and an exterior circle of radius csc(pi/n) with n ≥ 2, the number of points you'll touch as you bounce around cycles through [n/2, 2n, n, 2n] (starting with the degenerate case where the circles are the same radius and you can't leave the starting point).

[@OliVersed]

Whatever this Porism is that I supposedly have, let's find out

[@jounik]

I think this needed some motivation for introducing the parallelogram, like how the double periodicity of elliptic curves defines a lattice in the complex plane or something like that.

[@judychurley6623]

A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem.

[@NatCo-Supremacist]

this video should've 15 minutes long tbh

[@iowain8623]

So, you explained where the parralelogram comes from (kind of, not really), and that whether you can get back to the point you started at depended on being able to add a vector within the parrallelogram and get back to the same point. This vector is also "associated" with the ellipses is some way that is not explained. Makes perfect sense to me.

[@proloycodes]

😂😂

[@JM-wx8ik]

I felt like there could have been more detail here at the end. Oh, sorry maybe I’m not smart enough. 😒

[@manueldelrio7147]

Great stuff!! Please do more on elliptic curves and elliptic integrals!

[@bentaye]

From the parallelogram, I didn't understand a thing

[@screwhalunderhill885]

It's just another way to represent the elliptic curves.

[@paulpantea9521]

A new Numberphile video always brings joy to my heart, and a ray of sunshine for the day.

[@jhnoor9705]

For the past two years, I have been studying an area of math known as googology. Googology is basically the study of very large numbers and the notations that is used to express them. When you study googology in depth, you can see that the so-called scientific notation which we usually use to express large numbers is actually incredibly weak in comparison to many of the commonly used notations in googology such as Knuth's up-arrow notation, fast-growing hierarchy, Bird's Array Notation and Bashicu Matrix System, although it may not seems weak at all to an average person. This is mainly because we don't need numbers much larger than those that can be made using exponents in real life. For example, the mass of Sun is approximately 2*10^30 kg and the number of subatomic particles in the universe is approimately 10^80. Below I will show you the formal definition and some examples of expression in Knuth's up-arrow notation: a^^^^...^^^b with n uparrows = a^^^...^a^^^...^^a^^^...^^a... ...a^^^..^^a with (n-1) uparrows between each successive a's Where a^b is the same as a raised to the power tower of b First, we will take a review of addition, multiplication and exponentiation. We had all studied in school that addition is repeated counting, multiplication is repeated addition and exponentiation is repeated multiplication, mathematically: a+b = a+1+1+1...+1 (repeated b times) a*b = a+a+a...+a (repeated b times), and a^b = a*a*a*a...*a (repeated b times) Now, we will start with the double up-arrow operator, which is better known as tetration. a^^b (read this as "a tetrated to b") = a^a^a^a^...^a (b times) (a power tower of a's b terms high) Keep in mind that exponents are always right-associative, so a^b^c^d is the same as a^(b^(c^d)) For example, 2^^3 = 2^2^2 = 2^4 = 16 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65536 2^^5 = 2^2^2^2^2 = 2^2^2^4 = 2^2^16 = 2^65536 = 10^(log10(2)*65536) using rules of logarithm = approx. 2*10^19728 (a number WITH 19729 DIGITS!!!!!!) 3^^3 = 3^3^3 = 3^27 = 7,625,597,484,987 (approximately 7.6 trillion for short) 10^^^3 = 10^10^10 = 10^(10 billion) = 1 followed by ten billion zeroes Now do you see how powerful tetration is in comparison to scientific notation? But that's not the end of the story. Just like how tetration is repeated exponentiation, pentation is repeated tetration, which is normally denoted as triple up arrows (^^^) In summary, the n arrow operator is repeated (n-1) arrow operator After that, I suggest you to learn the definition of fast-growing hierarchy, which is basically like this: f_n(a) = (f_(n-1))^a(a), when n is a successor ordinal, or in other words, f_(n-1)(f_(n-1)(...(f_(n-1)(a))...)) (nested n times) When n is a limit ordinal, f_n(a) is defined as f_(n[a])(a), where n[a] is the n-th element of the fundamental sequence of the ordinal n For the definition of successor and limit ordinals, you can search it yourself in Googology Wiki Now, here's an equation for you. Given that 1/f_x(100) is the amount of DNA I inherit from my mom, try to find the ordinal x. The ordinal x here is known as my DNA ordinal Here's the approximate value of x in Bashicu Matrix System: (0,0,0)(1,1,1)(2,2,2)(3,3,3)(3,3,0)(4,4,1)(5,5,2)(6,6,2)(7,7,0)(8,8,1)(9,9,2)(10,9,2)(11,9,0)(12,10,1)(13,11,2)(13,11,2)(13,11,1)(14,12,2)(14,11,1)(15,12,2)(15,11,1)(16,12,0)(17,13,1)(18,14,2)(18,14,2)(18,14,1)(19,15,2)(19,14,1)(20,15,2)(20,14,1)(21,15,0)(22,16,1)(23,17,2)(23,17,2)(23,17,1)(24,18,2)(24,17,1)(25,18,2)(25,17,0)(26,18,1)(27,19,2)(27,19,2)(27,19,1)(28,20,2)(28,19,1)(29,20,2)(29,19,0)(30,20,1)(31,21,2)(31,21,2)(31,21,1)(32,22,2)(32,21,1)(33,22,2)(33,21,0)(34,22,1)(35,23,2)(35,23,2)(35,23,1)(36,24,2)(36,23,1)(37,24,2)(37,23,0)(38,24,1)(39,25,2)(40,25,2)(40,25,1)(41,26,2)(41,22,1)(42,23,2)(42,23,2)(42,23,1)(43,24,2)(43,23,1)(44,24,2)(44,23,0)(45,24,1)(46,25,2)(47,25,2)(47,25,1)(48,26,1)(49,27,0)(50,28,1)(51,29,2)(52,29,2)(52,29,1)(53,30,0)(54,31,1)(55,32,2)(56,32,2)(56,32,0)(57,33,1)(58,34,2)(59,34,2)(59,34,0)(60,35,1)(61,36,2)(62,36,2)(62,36,0)(63,37,1)(64,38,2)(65,38,2)(65,38,0)(66,39,1)(67,40,2)(68,40,2)(68,40,0)(69,41,1)(70,42,2)(71,42,2)(71,42,0)(72,43,1)(73,44,0)(74,45,1)(75,44,0)... ... For the definition of Bashicu Matrix System, you can search it up yourself in google So can you please help me to analyze my DNA ordinal?

[@aaryunik]

Happy to have found a Numberphile video while randomly browsing KZbin!

[@bonob0123]

unsatisfying non-explanation. 6:03 brady's reaction summarizes it: ".....okay."

[@Ripen3]

Word, I have the impression he was sort of hoping for more as well.

[@douro20]

The Wikipedia article really understates the importance of Jacobi's contributions to mathematics. Without him we wouldn't have functions for elliptic curves, or multivariable calculus, or partial differential equations.

[@adamwulf]

I _think_ i understand the relationship between the ellipses and the parallelogram: the sides of the parallelogram are built from the circumferences of the ellipses. So ellipse C and D will build a parallelogram with sides of C and D. Then the start of the first line in the ellipse is the corner of the parallelogram, and the end point in the ellipse is the same distance along the opposite side of a the parallelogram.

[@Onekick92]

This wasnt very informative tbh

[@julesk1088]

I wonder if we can connect/there's connections between Andrew Wiles' proof of Fermat's last theorem and the proof for Poincelet's Porism, since both use elliptic curves despite being wildly different areas of math. The parallelogeam thing was so cool, I got the connection and just love the idea of making this circular thing an easily understandable parallelgram analogy. Breaking complex things into simpler analogies sounds like a great tool to study complex things.

[@GolumTR]

Yes, Poncelet’s Theorem is equivalent to the existence of a group structure on elliptic curves. As this video pointed out, Jacobi showed that the addition theorem for elliptic curves provides a solution for Poncelet’s Theorem. Lesbesgue went the other way around, using Poncelet’s closure theorem to prove the addition theorem for elliptic curves

[@rickseiden1]

It's too bad that Poncelet was born so long ago. Doctors know how to treat Porisms now.

[@EmdrGreg]

; - )

[@vivekram6362]

I feel the video quality is better in this video than the rest...Love Numberphile 💚

[@meatlemonade9938]

i somehow read the title as poncelers pattern, which means we're off to a great start

[@erdenwurm7208]

5:25 that move is the gesture for "qed"

[@jamieg2427]

😂😂

[@chrisgriffith1573]

Ok, on the longer of the two sides of the eclipse, if one side has a more acute curve than the other, in other words the ellipse is non-symmetrical, (not expressed by a parallelogram) then the points will not meet as the outer expression is not totalled by 360 degrees, but one side is acute to the other. This can further be exaggerated by two points of perspective on the ellipse, not just one. I might not have explained that right, but I can illustrate it visually... comments don't allow for that very well.

[@EebstertheGreat]

Apparently "porism" is now used mostly to refer historically to a few specific theorems. Occasionally, the word is used for new theorems which state the conditions under which a particular problem has no solutions, or under which it has infinitely many solutions.

[@compellingpoint7802]

Neat video. The idea of a 'porism' is the concept that some truth about the nature of things can be derived from existing truths. The world is not just random, it has an underlying structure and logic to it. Truths that are true by definition imply other truths as well. For example: Definitions like "a triangle is a figure with three straight sides" or "a circle is a plane curve enclosing a space" are true in all possible worlds (they have necessary truth) but they also imply other propositions as well. For example: The proposition "if a figure has three sides then it must be a triangle" is true in all possible worlds (it has necessary truth). But this proposition also implies that any shape with three straight lines is a triangle. All shapes, however complex, can be broken down into line segments of various lengths and angles. In particular the circle can be broken down into an infinite number of triangles. And this is essentially what Poncelet's Porism states: everything can be broken down into something else. For example, a triangle can be seen as an infinite number of triangles. A circle is nothing but a series of lines and points. This is a general property of the universe. Things are not only made out of other things, but they can be reduced to simpler forms. We may have already discovered most of these 'simpler' forms in our own investigations, but there will always remain new and more complex types of phenomena that we might discover. Is this porism also true in all possible worlds? That is an interesting question. I don't have any particular insight into it, and my knowledge of mathematics is limited. However, the more primitive forms of some scientific laws seem to follow from other less fundamental grounds. For example, Newton's equation F = ma is a law that describes the relationship between force and acceleration. It follows from other fundamental laws of physics such as E = mv^2/r , which in turn follow from even deeper insights about nature. Your take?

[@hakesho]

I (and a professor I had as an undergrad) always used "porism" to describe results that were like a corollary of the proof of some larger theorem. By this I mean, in the same way that a corollary is a result that follows quickly from a larger theorem, a porism would be a result that follows quickly from the way you proved that larger theorem.

[@ReynaSingh]

Amazing. Life is just the perfection of math playing itself out.

[@anastasiagoold4975]

The first mathematician to figure out an accessible way to talk about elliptic curves to people that aren't mathematicians should probably win a nobel

[@AlanTheBeast100]

Wikipedia: "A porism is a mathematical proposition or corollary. In particular, the term porism has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem." If someone talks like that at a party, don't worry, he won't be at the next party.

[@elvest9]

Rick Moranis has found a new career in maths.

[@carltonleboss]

Your Schwartz is as big as mine

[@TheAlps36]

Honey, I think I shrunk my ellipse

[@bryanhaney7930]

I knew it. I'm surrounded by ellipse-holes.

[@drossword]

I was thinking Ed Helms.

[@bluerizlagirl]

If you shrank the kids by compressing their molecules into the empty space that makes up most of them, wouldn't they still weigh exactly the same? Like 20kg. for the lightest one?

[@NoodleCollie]

My surname genuinely is Poncelet, and it's cool to find out a bit about what strange, arcane maths some of my ancestors were getting up to.

[@masonhunter2748]

There’s also Olivier Poncelet

[@nHans]

Thank you for giving me a chance to use a Latin phrase from a handful that I know: *_non sequitur_* 😂 Sure, you didn't say so explicitly, but you clearly implied that people surnamed Poncelet, including the mathematician mentioned in this video, are your ancestors. Aah, the satisfaction of doing to others what your math and logic tutors did to you innumerable times!

[@asymptotichigh5]

Porism is like a corollary, but instead of following from a theorem, it follows from the proof of a proposition, lemma or theorem

[@businessguide6219]

Your videos are splendid! Thank you for giving me a new perspective to contemplate!

[@julesk1088]

I think it's so awesome how you're channel is about business but you watched (and commented) on this math video

[@fcolecumberri]

It would be nice if you could go deeper into elliptic curves

[@nickjacobs422]

The rhyme and reason to the universe is No. is is -i. -8. -6. -5. -3. +-0. 1. 1. 1. 3, 5, 7, 9, 11, 13, 15...8i It’s how we count.

[@EmdrGreg]

Not that there HAS to be, mind you, but I am curious: Is there any area of engineering, physical science or applied mathematics where these ideas play a role? I agree that the maths is beautiful in its own right.

[@MichaelDeHaven]

I don't know specifically about poncelet's porism but, the tool of elliptic curves is very useful for cryptography.

[@EmdrGreg]

@@MichaelDeHaven Cryptography! That is something I would never have guessed. Thanks, Michael. Something to look up.

[@bluerizlagirl]

The idea of "going around several times and always eventually getting back to where you started" crops up in cryptography. It doesn't matter how many times you have been round, just how many steps it takes to get there.

[@altafhossain7793]

I have a feeling ... is there anything to do with irrational numbers ??Else why we cant those tangents wont meet even after an infinite steps ???

[@revenevan11]

Exactly. Like adding sqrt(2) to itself over and over, you can get arbitrarily close to an integer but you'll never have the sum = an integer, because it's an irrational number. Or integer movements around the circumference of a circle never getting back to the point you started because pi is irrational. I don't know exactly where it is in this ellipse problem but something definitely smells of irrational numbers!

[@tehdarkneswithin]

@@revenevan11 when you look at the parallelogram picture its quite evident. If you extend that corner vector, every time it intersects one of the horizontal lines, it loops back around. This can also be viewed as a horizontal translation. If the horizontal translation is not a rational multiple of the total width of the parallelogram, then it will not loop back.

[@jdmarino]

How do you know the lines never meet their starting point? Maybe you just didn't iterate enough times.

[@srednadahlberg]

I suppose the proof ... proves it?

[@yeoman588]

As I understand it, if each bounce takes you an irrational fraction of the way around the ellipse, you will never wind up at a whole number of revolutions and thus will never reach your starting point. Though you may be able to get arbitrarily close with enough iterations.

[@bluerizlagirl]

Think about a degenerate case with circles, and start with the inner one fixed. You can always draw a regular polygon, with any given number of sides, having each side tangent to the original circle; and you can always draw another circle through its vertices. The polygon need not be convex; non-convex polygons are equivalent to convex polygons whose edges have been extended to meet at new vertices (obvious example in 5-point star). The outer circle will be close to the inner circle with convex polygons, and get even closer as the number of sides increases (all converging when the inscribed polygon has an infinite number of sides, i.e. is a circle). But the extended lines push it away. Some shapes are spikier than others. Now for any given inner circle, every possible polygon gives you an outer circle; but not every possible outer circle necessarily has a polygon that would generate it from that inner circle.

[@yousorooo]

“How do you know adding positive numbers together will never yield a negative number? Maybe you just didn’t add enough times.”

[@alfeberlin]

Normally you can _compute_ when they should meet again, and that boils down to finding the greatest common divisor of two numbers. So if we use two integers or two rational numbers there, this will always have a result which means that they will always have a closed loop. But if the numbers are irrational, then this does not have a result which in turn means that the star line will never close.

[@loneoceans]

I enjoyed the 'hand-drawn' look of the animations, very nice!

[@acentasecond3721]

exactly it looks so much better

[@stefanospapaiordanidis8297]

Porism is from the greek word "πόρισμα" (pronounced porisma), which means deduction.

[@ChrisConnett]

I really hope there's more about this coming on Numberphile2 ;)

[@tommythecat4961]

It's amazing how an apparently simple, even trivial question, requires complex math to be properly understood and (sometimes) solved! I don't know how many times it happened to me, you're just playing around with numbers or shapes and suddenly, like out of thin air, a strange property manifests itself... It's the closest thing to magic!

[@KuraIthys]

It's funny, because I'm reminded that in the past there was a subset of philosophy called the 'philosophical arts'. Which, broadly speaking, is philosophical stuff that has some kind of actual application. Now, most of these 'arts' are still with us, even though we don't call them that anymore. Guess what they were? Mathematics, Science, and... Magic. Of those, magic turned out to be the least credible or useful. But they all share this history of being a special subset of philosophy. (and if you think about what 'magic' is when you examine what documents we have on the subject, it's full of weird formulas and notions that doing a specific set of actions would have a specific outcome. In other words, it is indeed a bit like an amalgamation of science and mathematics, but without anyone bothering to check if these things actually hold up or not. And there's some grey areas; Alchemy is literally the precursor to chemistry, just with more dubious goals and less understanding of what was going on.)

[@dlevi67]

@Immanuel Kant The condition is actually quite simple and very precise: Let K and C be nondegenerate conics in general position which neither meet nor intersect. Suppose there is an n-sided polygon inscribed in K and circumscribed about C. Then for any point P of K, there exists an n-sided polygon, also inscribed in K and circumscribed about C, which has P as one of its vertices.

[@dlevi67]

@Immanuel Kant Surely general results are more interesting than highly specific ones? I can tell you that x^2 -5x +6 = (x -2)(x - 3), but the quadratic formula is more interesting, and Galois's insight on the symmetries of polynomial roots is even better. (Oh, and you can extract a lot of information from the existence or otherwise of a Poncelet polygon, and the criteria for determining whether that exists - some of that work has been done already. But without the initial insights that some conditions exist you'd be left with "for this pair of conics, it works; for that pair it doesn't". For someone that uses "Immanuel Kant" as a user ID you seem to be unusually interested in generalisations.)

[@thevastuniverse246]

Video in a Nutshell... So if I was i was in a room which is an ellipse, and I have a small elliptical table with donuts.... If I start of from the end of the room and continuoisly grab a donut as I bounce from the walls... If I come back to the place from where I started... Then I could have chosen any other place at the end of the room... And the same thing would imply!!(for the exact same configuration of the room) Amazing!

[@kalpanarms9597]

I see what you did there....XD

[@Cernoise]

He said conics but only used ellipses… could this also work with hyperbolas or parabolas?

[@sweatymetagross3704]

So this was the DVD logo screen of the 1800s

[@nickolaos00]

Porism means conclusion. It is indubitably a Greek word, and it's frequently used in mathematics too.

[@narakarrarr6191]

I reckon it's to do with whether the ratio is irrational or not. All rational ratios will loop back on themselves eventually, some irrational ratios won't though like the golden ratio, e and pi. I think root(2) and such do though.

[@xiaokangzhang3077]

This is an elementary attempt at justifying a potential link between the ellipse and the parellogram. Suppose you start at a point (P0) on the outer ellipse and draw a tangent to the inner ellipse and it lands on another point (P1) on the outer ellipse, call this R1. Now use this new point and repeat this process as mentioned: call the subsequent line formed R2. Now instead of form R2 this way, consider the tangent at P1 and reflect the ellipses around that line. Extend R1 through the reflected image of the ellipses and the line hits the reflected outer ellipse at a point (P2) whilst still being tangent to the reflected image of the inner ellipse. Call this line P1 - P2, R2'. Now R2' is the exact reflected image of R2 over the tangent line at P1 (verifiable). Note that P0 - P2 is a straight line. So instead of thinking of starting at an initial point and repeating the "line -drawing" process between ellipses, think of drawing an extended straight line by reflecting the outer and inner ellipses across a tangent line. Lemma: if after n finite iterations of reflections across the tangent lines, if the resulting image of the ellipses are "parellel" or an exact translation of the original image of the ellipses, then the star or 'curve' is closed. This is because as demonstrated in the drawings, the points that fall on the outer ellipses form a clockwise/ counterclockwise rotation, and if it forms a closed loop, the points traverses a multiple of 2 pi radians, which will similarly be 'reflected' (no pun intended...) in the transformations of the tangent method. Now I imagine that a relationship of the factors between the two ellipses can be encoded in some sort of parallelogram whose side lengths/angles etc reflect that relationship preserving the implication of the continuous straight line when mapped from the ellipses to the parallelogram, such that if the vectors in the parallelogram re intersects with its original point, points on the ellipses form a closed loop Now if his mapping between the two is indeed possible, then note with the parallelogram if a vector indeed intersects with its original point, then shifting the starting position of the vector (but perhaps not its angle relative to the base of the parallelogram, would still mean the the vector aligns with itself after n iterations (draw an infinite aligned row of parallelograms i suppose for the proof). This may translate into to stating that if two ellipses forms closed loops, then varying the starting point would still ensure a closed loop, assuming the equivalence in the mapping. May be way off idk xD

[@aresharesh8671]

To me, it looks like there's some way to distort 2D space so that the ellipses become concentric circles and tangent lines are preserved. From there, the sequence of tangents returns to the same point iff it moves a rational part of the circle around each time (forming a regular star).

[@Keldor314]

I think you might be on to something here. I'll also note that you can convert an ellipse to a circle using an appropriately chosen linear function. In fact, we can transform an ellipse into any other ellipse using a linear function. Could there be a relation between the coefficients of the function transforming one ellipse into the other with whether the tangents return to the starting point?

[@aresharesh8671]

@@Keldor314 A linear function wouldn't work in the general case, since it wouldn't be able to correct the fact that the ellipses aren't concentric. In fact, no bijective function on the plane would work since it has to map the distinct ellipse centers to the same spot. Maybe the function we need only acts on the toroidal area between the ellipses.

[@Keldor314]

@@aresharesh8671 I was thinking of a seperate linear function for each ellipse, and then stitching them together somehow. Though interpolation between two linear functions is a quadratic, IIRC. Still might be feasible though. It might also be possible to define some sort of geometry where ellipses form an equivalence class, and the tangent line properties we need are invariant. In fact, Poncelet's Porism strongly suggests the existance of such a geometry, but of course that doesn't help us when we're trying to prove it in the first place.

[@Keldor314]

@@aresharesh8671 It might be possible to represent the problem directly in terms of the underlying geometry of conic sections. That is, a cone with two planes forming the ellipses, which is then projected to a viewer plane. What can we say about the properties of the curves our tangent lines in the viewer plane with respect to their projection back into the cone?

[@Keldor314]

@@aresharesh8671 Could we use the tools of projective geometry?

[@ngwoo]

This is maybe the only numberphile video where I feel like I came out of it understanding less of something.

[@sadrevolution]

Short and sweet., and mind-blowing. I have two questions: i) aren't the lines on the parallelogram curves when they are on the surface of the torus? and ii) how does the complex plane come into play? I guess I'll cheat and follow up with iii) does the answer to ii explain i?

[@akankshyamahapatra6352]

That animation made the porism feel like magic

[@jonathanlevy9635]

A Porism is a relation that holds for an infinite range of values but only if a certain condition is assumed

[@whitslack]

I felt smart when I saw the drawing with the two conics and the two tangent lines and immediately thought of elliptic-curve cryptography and then was validated momentarily later when Daniel mentioned that the proof uses elliptic curves.

[@Caspar__]

I think it's great that you get to have insight into such a complex field in this short of a timespan.

[@PC_Simo]

I could definitely see the proof also using rational and irrational numbers 🤔.

[@maty2faty204]

what would happen if it was two circles

[@steveb1243]

So, in summary, there was someone called Poncelet and he did some stuff and it's a bit like other stuff but we won't go into the stuff itself or the stuff that it's like here.

[@krautbrain]

Sounds like a paradox. If you do in infinity numbers of times you should come back to the start because you cant get stuck in a loop if its not coming back to the start. A loop would say it comes back to the start but without a loop shouldnt you sooner or later come back to the start?

[@ZacJelke]

Is there some sort of arbitrary limit as to how many tangent lines you can use? If you haven't closed yet then that means there are still unique tangents to draw that likely would eventually close.

[@ancientmodernist9391]

I'm not a mathematician so can someone help with a basic question: How do we know that an end point truly coincides with a start point? I assume the point is defined by values in the curve function but what if the values are irrational numbers? Or do we define the start as the origin and therefore zero? Thanks.

[@yousorooo]

The math deals with variables instead of actual numbers, so it doesn’t matter whether the number is irrational or not. It applies to all numbers.

[@apoolplayer278]

Those curves are called conics. Actually, they are defined by a function. If they are 2dimensional, their functions contain x and y, to some power, could be x squared over 2 plus 3 times y cubed equals 1. When a line starts from a point of the conic and is tangent to the other conic, that can be spelled in the form of some equations, and to have that line come back to itself means to prove that 2 things, with x and y in them, are equal.

[@charlieangkor8649]

could we always call all variables in math C, just written in different fonts? it would make things so much less confusing

[@Simonsays7258]

Porism is a direct consequence of a proof as a corollary is the direct consequence of theorem.

[@Uddaybhaikabaap]

Numberphile is just awesome Makes my day Love from math enthusiast ❤️❤️

[@CodingTutorialsAreGo]

I presume, though it's not made explicit, that given a pair of ellipses, the position of one within another is irrelevant. Does one even have to be within the other?

[@dlevi67]

Not in Poncelet's original formulation of the theorem; however if the conics are ellipses they cannot be totally external to each other, because otherwise it is obvious that there is no polygon inscribed in one that is also circumscribed to the other...

[@theultrapixel]

If the line never returns to the starting point by doing this but you go forever, do some choices of starting point/direction have a smaller minimum distance from the starting point it ever reaches than others?

[@-YELDAH]

not as Litt as i expected but still interesting

[@jadegecko]

Do the paths that never link up correspond with resonance in similarly-shaped orbits?

[@JSLing-vv5go]

Interesting question. Go research it and get back to us!

[@alfeberlin]

I guess it's like the arrow path in the paralleogram if the side length relationship wrt to the angle of the arrow is irrational. Then these arrows will also never match up again.

[@nopetuber]

The best part is like 10 seconds at the end! Please expand!

[@jokusekovaan]

Aren't there infinitely more ellipse pairs that don't connect like this (if it relates to the distance of the vector in the second picture)? Making the probability of being able to connect the lines of a random pair zero? So you would have to construct them, by reversing the process, somehow.

[@user-pb3pp6pu1g]

Sound of Pen on paper is giving me a chill

[@mvmlego1212]

I feel like I've just found Michael Knowles' secret, long-lost brother.

[@3blame]

ooo that's interesting. if you were to have a smaller circle inside of a bigger circle that shares the same center point, what is the ratio of the circles' radii that would allow Poncelet's Porism to occur?

[@Narokkurai]

A porism is basically the same thing as a corollary. When you make a proof, and then as a natural consequence of your proof, you find another neat little mathematical truth, that's a porism.

[@jannegrey593]

Perhaps something about holes? Porus and Poros in Latin and Greek respectively. Since it is anglicized here it could be about this I guess? EDIT: Of course those words in Ancient languages mean passage rather than hole, but it is close enough.

[@kostantinos2297]

Porism is still used in Greek (as "porisma" - πόρισμα) and usually means conclusion, something inferred from facts. I'm not a mathematician but in the math I've taken it's essentially interchangeable with "theorem".

[@jannegrey593]

@@kostantinos2297 Thank you for that information! Broadening my knowledge is very important to me. They didn't explain it in the video, whether it was called like that by Euclid or if the theorems were called like that in Ancient Greece. But this closes the case for modern use of the word.

[@EmdrGreg]

In the case of a never-closing solution, can we know anything about the clustering of tangents on the inner figure? Do they turn out to be basically evenly spaced over large numbers of iterations with an ever smaller maximum gap, or is it possible that for some solutions the tangents cluster in some systematic way?

[@JSLing-vv5go]

Another excellent question. Maybe whether the figure closes is not even the interesting question. What if the tangents favor some portions of the ellipses over others. If so, why? A very interesting subject for future research!

[@anthonycannet1305]

What is the relationship between the lengths of the sides of the pareallelogram, the angle of them, and the vector with the two ellipses? If it’s intuitive then the vector represents the tangent line in which case wouldn’t all points eventually get back to the start if they aren’t limited by a number of transitions?

[@LamNguyen-zy4zp]

Is he talking about modular forms when he talking about the parralelogram? I have never studied eliptic curves but know about the basic outline of Fermat last theorem's proof.

[@rhoddryice5412]

This one had terrible camera work from about 5:15. impossible to follow when the brown sheet goas in and out of focus and the reflections.

[@Omni_ro]

I tried this with circles and it works

[@leefisher6366]

But... if you don't come back to where you started, what's stopping you drawing more tangents until you do, which is going to be inevitable (even if you draw infinitely many tangents before getting there)? I'll keep watching the video, because it may explain this at some point. I'm only at 2:12 at the moment. Nope, this wasn't addressed, so any insight please?