What's the curse of the Schwarz lantern?
ฝัง
- เผยแพร่เมื่อ 8 มิ.ย. 2024
- Second coop with Andrew. This time it's about the Schwarz lantern a very famous counterexample to something that mathematicians believed to be obviously true. A 3D cousin of the famous pi = 4 paradox.
00:00 Intro
00:39 Troll math: the pi=4 meme
02:25 Archimedes chops off corners
05:51 Archimedes boxing of pi
07:40 Schwarz lantern
16:59 Area formula
17:12 Schwarz pi = 4 memes
20:17 Folding flat
21:12 Andrew's VR experiment
22:24 Soda can --- lantern
23:26 Thanks
25:00 Andrew's Christmas tree
The pi = 4 troll math meme on reddit:
/ troll_math_pi_4_crosspost
Good discussion by GoldPlatedGoof
static.nsta.org/pdfs/QuantumV...
The Vi Hart video:
• Rhapsody on the Proof ...
Closely related, the Staircase paradox:
en.wikipedia.org/wiki/Stairca...
Good discussion of the history of this sort of paradox
tinyurl.com/3pe2eaav
Great wiki page on the Schwarz lantern:
en.wikipedia.org/wiki/Schwarz...
Schwarz's original paper:
archive.org/stream/gesammelte...
Diamond-Cut Can
www.toyo-seikan.co.jp/e/techn...
Lots of nice ideas (coffee cans, soda can creasing, collapsing cylinders, patent for effectively collapsing cans)
beachpackagingdesign.com/boxv...
The IKEA looking lantern features in the paper: Application of paper folding technique to three-dimensional space sound absorber with permeable membrane: Case studies of trial productions
tinyurl.com/223948tz
Great article in Quantum magazine (there is also an article about Gabriel's horn in the same issue):
static.nsta.org/pdfs/QuantumV...
Andrews Geogebra app in action:
Schwarz Lantern in Geogebra 3D & AR: • Schwarz Lantern in Geo...
Schwarz Lantern christmas tree: • Schwarz Lantern christ...
Geogebra 3D AR mode on an iPad how-to: • GeoGebra 3D Mobile App...
Schwarz lantern origami with Deepali Karanjavkar:
• How To Fold Yoshimura ...
Crushmetric water bottle (really impressive can crushing inspired toys):
www.crushmetric.com/
Hydraulic press vs. metal pipe (square and circular)
• Crushing Long Steel Pi...
Mel Brooks: "May the Schwarz be with you"
• "May The Schwartz be w...
Music today: Flat Rock by Bennett Sullivan
T-shirt: Google hohoho cubed t-shirts for lots of different versions ((ho)³ A case where (ab)³ ≠ a³b³ :)
Enjoy!
Burkard
This is mathematical proof that cutting corners gives bad results ! ;-)
If you let someone like Archimedes do the cutting it's fine :)
@@Mathologer Yeah but I heard his disciples never got the angles right...
Unless you're painting
You completely reversed the conclusion. Cutting corners is the only thing that worked. 🤨
@@ModestJoke. en.wikipedia.org/wiki/Pinking_shears
That final example of constructing a lantern from a soda can was awesome.
Also a lot of fun to do. Even if you just use your fingers all this remarkably easy :)
It really was 🙂👍🏻.
The cicadas :) Nice and crisp video as always. And big thanks to Andrew!
This time of year my brain filters the cicadas out - it was only when Burkard pointed it out that I thought “oh yeah, they’re noisy today.”
And you’re welcome - my input to this one was a lot less than the last.
Best math channel on youtube by a total landslide, there's simply no competition. Keep up the amazing work Burkard! but I do hope that you don't work too much on youtube to the detriment of your own research (or worse that you burn yourself out); you're doing such a great and important service to math by inspiring so many future mathematicians and showing them ways to think mathematically that almost no degree would ever teach them, but it is crystal clear that you yourself are more than capable of contributing every-bit-as-important new results to mathematics - it is my opinion, and I'm sure also many others', that you are one of the best mathematical minds alive in the world today, and that you can achieve something _big_ (yes, *big* _big_ , but we know it's not winning a million dollars that you care about :P)
Merry christmas! - we wish you the best combination of luck and continued improvement in skill, in choosing your mathematical battles; may you make major breakthroughs in the new year or beyond and push your field forward. I have zero doubt that you'll do us all (and most importantly, yourself) proud!
Perhaps an honorary Field Medal would be justified...
This might be the best math channel, but if it is, it's not by a landslide. Eg 3 blue 1 brown is also really good. And there are a few more.
@ No shade to 3b1b of course, but it really is a landslide
It is certainly one of the best math youtube channels.
I think I would put 3Blue1Brown on top, but Mathologer is definitely in the top 3.
I have to say (and I am hard to impress ) this is one of the best high level explanations I have seen. The amount of work involved in putting this to air both at the animation level and the level of explanation are truly top drawer. But as at least one other viewer has pointed out, don't burn yourself out on this stuff at the expense of your own research.
Glad you are impressed. I've been trying to keep the burn out under control by only making a video every four or five weeks. Works reasonably well :)
@@Mathologer If i take u literally, then please take care. I dont have a solution for you since i don't know u or your life that well. But I hope u do well. And as always love your content. Please take care.
@@MathologerI’m just as impressed! I think you could make a video showing how you make these amazing animations.
As you were finding the tiny areas and adding them up, I couldn't help thinking that essentially the Jacobian is involved. If you don't get the Jacobian right, your area won't be right. That is, the Jacobian is what tells you how to get areas right. And it is also related to your little spikeys.
Jacob's Lattice! 😁 Not to be confused with Jacob's Ladder.
I really like this comment.
@@Novastar.SaberCombat
or Jacob's Lettuce
20:13 cant we do this to get an equation for the perimeter of the ellipse?
@@lautamn9096
It is easy enough to find the perimeter of any given ellipse to any accuracy you might want. It is just finding a closed form formula we can't.
This feels like a deliberate deep dive into the slightly shoddy limit taking at the end of the previous video. Brilliant!
This was one of the reasons I suggested the Schwarz lantern to Burkard as a topic. I think I said “what the hell is surface area anyway?”
The Cyclides argument from mathologer #100 was necessarily shoddy, as properly defining surface area requires a lot of machinery (multiple integrals and vector cross product is one standard way) that is beyond the scope of a Mathologer video. Also it relies on maths that is 2000 years after Archimedes.
At that point in mathologer #100 we already know 4πr² fact. The cyclides are to give the “gut feeling” for 1 sphere = 4 circles that the 3b1b video sought.
Well spotted, that's exactly the original motivation for this video :)
Fascinating. With the crushed cans and cylinders at 22:00, it's like the can or cylinder is trying to retain its original surface area while being crushed smaller, so it tends towards a Schwarz lantern shape.
For what you said at the end, I didn't feel that anything necessary to understanding was left out. Perhaps because the "staircase diagonal = 2" idea is familiar to me, I actually felt there was more explanation than necessary, but perspectives may differ so don't take that as a criticism.
If, if only I have had a math teacher like you. Merry Christmas, dear Sir!
Merry Christmas :)
@@Mathologer Thank you! And, yes, keep on, please. Sincerely Yours, Peter.
Be careful what you wish for...
I did, for college algebra. Only time I've seen people show up early for a math class and head for the front rows. He was Dutch and had been in a concentration camp as a kid during WW2. The stories he could tell! My college calc prof was pretty darn good too. I ended up taking a semester of calc I didn't need just because I enjoyed his classes - and I'm not a math person. Good teachers make a lot of difference.
You'll probably appreciate the recent Action Lab video "This material can un-crush itself" that shows a simple application of the Schwartz Lantern geometry
Just ordered some of their pens. But what I really want is that nice tumbler and the chair (which currently don't ship to Australia :(
The problem with the sphere seems to be a matter of curvature. The angles of triangles with flat curvature will always add up to 180 degrees no matter how small they become, and so there will always be extra area at the points.
Really? There is a platonic solid with triangular faces you know.
The thing is that, if the straight line assumption overlaps, then we always measure a bit more. Because, we know that the sum of the two sides of a triangle is always greater than the third one. If our assumption overlaps then it gradually leads to a fractal.
It's too bad that I had teachers who complained that I was too slow and would never be good at math.
Interchangeability of nested summands (or integrands) is usually glossed over and to me this video/result emphasizes the rigor and caution to be exercised when performing such. Thank you!
Cool paradox this also shows why the curve area length formula has to be secant lines instead of just horizontal and vertical. Approaching something doesn’t always mean that they have the same properties
Secant lines work inside, or tangent lines outside.
The problems start to settle in when your curvy curv happens to be a fractal. Where the more dots you use on your segment, the lenght continues rising without converging. So not all curvy curves have a lenght, so I conjecture that only the ones that don't contain fractals can have lenght (i.e. curves where for any two there exists a dot that admits a tangente line)
I would LOVE a series of videos on measure theory and conceptions of length in fractaline structures. I've been trying to solidify my intuition of some of the bizarre units of measure come upon by manipulation of formulae, like units in m^±½. I have a feeling such a series of videos by you would help me to connect some of these abstract dots in my mind.
Thank you for this video on limits and the potential for arbitrary endpoints in their calculation. This is fascinating!
Measure theory is so cool, it makes integrals properties more natural, because you redefine and extend the definition of an area or length, in things way more practical that only the basic definition of the area of a rectangle
@@FreeGroup22 I love extensions!!
Christmas came a little early this year! Thanks for the gift!
In my family the 24th is the day you give presents (a German thing) and so early for the rest of the year but not for me :)
I'm an artist. I'm working on a Carravagio style of painting. When I look at your channel, I see the enthusiasm you have for mathematics. It's like an artist. There's that thrill of discovery that can captivate you for weeks. Great show. Keep up the great work.
Wieder super!
Frohe Weihnachten, lieber Burkard!
Fröhliche Weihnachten :)
Some energy absorbing crushable elements in cars, called 'crush cans', are designed to buckle like the tin can shown, even to the point of having the first band of triangles pre buckled to make sure that the buckling progresses in the correct manner.
Ah, very good, something I did not come across during my pre-reading. Also maybe check out this hydraulic press crushing a pipe :)
Nice - next time I crash my car I will have a look for this.
2:10 I saw the sqrt(2) version that uses the hypotenuse of a triangle with the other sides equal to one.
Happy holidays!!!
You can find a nice discussion of the history of this sort of paradox here
tinyurl.com/3pe2eaav
By approximating a straight line with half-circles you can similarly prove that pi "equals" 2.
that is an awesome christmas t-shirt!!!!! happy holidays Mr. Loger!
"Are you going to sit around and drink beer all afternoon?"
"I'm building Schwarz Lanterns."
I notice you linked GoldPlatedGoof's video in the description! That was the explanation that first got the original pi = 4 meme to really click for me
Yes, a pity he stopped making videos. Great explainer :)
As always, the presentation was great and the closing music was awesome!!!! Thank you so much!
Usually, I test out a few pieces of music but this time it was love at first sight :)
Logarithmic Time self-defining relative-timing ratio-rates of reciprocation-recirculation Singularity-point nothing in Eternity-now Superspin Relativity of No-thing, the Conception of Existence/Everything.
Merry Christmas, thank you for your fabulous teaching videos.
(ho)³. :)
Its due to channels like yours that makes youtube videos worthwhile to watch.
Glad you think and say so :)
My intuition on the initial paradox was thus: even in the limit, there's a point of the initial square not on the circle. That point will always not touch the circle, so that's where rhe extra perimeter comes from.
I like the conjugate explanation that having the normals of the approximating curve not getting closer to the actual curve also explains the discrepancy.
It's math, it's ok. Physics not so much.
So, we do need to make physics better :) lim A->infinity where A is a point in any "surface" infinitesimal point, you'll just measure measurement instrument. Also one may argue there's unlimited angles to be measured in space, but same problem same math same measurement problem. Troll wins. Make it 1-dimensional you get strings so out of my pay-grade.
As you were making more bands (but keeping the points around the perimeter fixed), I couldn't help noticing that the triangles were getting 'more horizontal'. So the areas were not decreasing as much as they 'should' for more bands. So in the limit, each band becomes more like a flat ring (of zero thickness) that you're stacking on top of each and of course a 'ring' has a fixed limit of area depending on the points you've picked around the perimeter.
Then you explained how the 'normal spike' for the triangles has to behave to get a good approximation and I realized that controls how the 'points' versus 'bands' relationship has to behave. Has to be such that the 'spikes' approach being normal to the true surface. Very nice explanation and very nice graphics.
This is a beautiful clarification. Thank you and Happy Christmas
Merry Christmas :)
Merry Christmas Mathologer!
Merry Christmas :)
I’m not a huge math person but this was really cool; thought I was going to watch a couple minutes to see what the “crisis” was and ended up watching the whole thing. Thanks!
Fascinating video, loved the animations!
Many thanks!
Good explanation. And an excellent Xmas shirt too!
Wonderful as usual Burkhard, thank you. It struck me while watching you measure a curve that really this is just relative to the scale of the ruler and that all measurements (at least of this nature) are relative to some agreed upon standard and are not absolute. And Pi of course is simply the ratio of the circumference and diameter of a circle. As circles are always the same in terms of their shape then pi is always the same. Merry Xmas!
Merry Christmas :)
Waouw ! Merry Christmas and happy new years ! This is a wonderful explanation ans so intuitive ! But now I ask myself, how do you do the same in higher dimensions ?
Awesome video!!! Beautiful animations. Got me thinking why not use rectangles they look much tamer. : )
Best Christmas present I could ask for
These kinds lf problems are my favorite, solvable in a flash with no thinking at all because i can see the limit being formed for all kinds of cases in all kinds of dimensions. And it takes but a second to see the angle and area correlation.
For the lanterns there are two substitutions that effect the angles, we need all the angles to go to 0 or 180 depending on how you measure angles to get the minimum surface area, just like with the line segment, but now in higher dimension. If you have the verticle band substitution increase all the angles or some of the angles, but the substitution around the circumferance always reducing the angles, to do both and get a limit that is identical to the object approximated, the substitutions that always reduce the angles between triangles to reduce the angles the vertical substitutions increase for each step, such that all the angles reduce towards the limit, then we can end up with a smooth surface and not an infinitesimally wrinkly one 🍻
Yes and this is why they make air filters with those unusual buckling patterns. But can you spot the black hole singularity. The seperations of matter, the fluctuating entropy, the dispersion evolution of our universe.
As always such a great video! Lately I've been thinking a lot about what makes good science divulgation and I still don't have an answer, but I'm sure your videos have every needed ingredient.
As for this video in itself, I think it is the best simple and direct explanation to this "paradox" I've ever seen (and I look for a lot if them when I was studying differential geometry).
The video make me even laugh at 21:57 😜
Of course making people laugh is more important than anything else :)
I've been watching your videos for years and respect you as a creator, so I did not expect you to post a thirst trap near the end of this video.
TH-cam tells me that you've been subscribed for 6 years. Good to know that I am able to still surprise even long term subscribers like you :)
I liked it all. Even your TShirt. Don't burn out and have a great year!🎄
Thank you! You too!
When I was a child, my dad bought a bunch of stuff from an auction, and among it was a large-diameter (maybe three or four inches; I don't know my artillery history) shell casing, such as from an artillery gun. But not an intact shell casing, quite -- nope, this one was half crushed, into precisely the type of lantern pattern you show here, but crushed vertically, parallel to the straight sides of the cylinder, to the very extreme limit of the strength of the material. It was made of. Steel? Bronze? Brass? I don't know my artillery composition, either.
Someone once expressed the opinion that this casing had been crushed by getting caught in the ejection mechanism when fired, which I suppose is possible (though I don't know my artillery mechanisms), but It seems to me that the crush pattern was much more uniform than I would think an accident would produce. Surely wouldn't an accidental crushing produce some bit of bias, a tilt to one side or some other non-uniformity in the bucklings? Or does the math prohibit that and force an all-or-nothing symmetry? So someone else suggested it was a deliberate artwork, such as you have begun to produce in your final footage of that soft drink can which you score with the piece of wood. (Thank you for that, incidentally, I've been looking for a start on that technique most of my life. Score the can as you have done, then press it from above, and you can make quite an attractive little vase for some pretty flowers, exactly the way my mother used that shell casing in my youth.)
I'm also surprised you can't compensate for the buckling inward of the triangles by simply calculating and applying a correction factor in determining their contributions to the surface area of the cylinder. You know the normal vector along which the surface area must point, and you should be able to calculate the normal vectors to each triangle in the lantern pattern, and from the supply a (sine theta?) correction factor to each off-normal triangle's surface area contribution. Is it simply that that technique was outside the scope of this video? I find it hard to believe you wouldn't at least mention it in passing. Or is there some other mathematical dragon that arises to defeat the valiant knight who attempts this correction?
One could even envision cutting the surface of the cylinder as though to unwrap it, then examining the zigzag/seesaw pattern formed by the cross-sections of the lantern pattern at each point in its back and forth cycle, then figure out some way to correct for that, as a whole, and apply that correction to the summation over all the triangles. Now, damn it, either you need to make a video discussing this, or I'm going to have to try it myself.
Would be interested in seeing your shell casing :) You may also be interested to check out one of the TH-cam videos of metal pipes being crushed by hydraulic presses. As for the last point you make, everything about the Schwarz lanterns is completely determined by the two parameters and at some point I flash their area formula. It's quite easy to analyse what's possible in terms of refinements just using this formula. I recommend you also have a look at the very good wiki page dedicated to the Schwarz lanterns :)
Frohe Weihnachten nach Down under und vielen Dank für's Video!
Fröhliche Weihnachten Ö=
@@Mathologer , lustiger Space Invader >> Ö=
Great video, as always. I just thought you should mention that although those two constructions don't converge to the actual length/area, they do converge to the area/volume, respectively. I think stating this point may help some people who are still confused.
Edit(s): Typo.
Yes, you are absolutely right, in terms of areas in the 2D case and volumes in the 3D case nothing can go wrong, as long as the approximating curves "converge" to the target curve.
Your animation of sliding a straight rule along a curve reminded me of an antique tool that predates the tape measure. I know it as a walking rule, others as a stepping rule. It is a disc with circumference 12 or 24" with a handle on the centre of the disc. It could be used to measure quite well from a wood workers POV. Craftsmen working with wheels or long beams used them extensively. I just thought you might find this interesting.
My father used to be a civil engineer. I still remember using a similar wheel ruler to measure the length of a street :)
Thanks for the nice video Christmas present! Looking forward to more good things.
Well, I still got a couple of lifetimes worth of great topics lined up ... :)
So for the solitions that work, at the infinite iteration the curve is continuous everywhere and also differentiable (hence why they are smooth).
But for curves that are not differentiable (hence the buckling), the approximation fails? In fact now that I think of it, they're examples of curves that are continuous everywhere but differentiable nowhere.
That's why I keep going on about smooth :)
Willans' Formula for primes:
2 to the n part = vertical asymptote and p-adic numbers. 1/n part = vertical tangent. Factorial part = vertical line. These tensors from differential calculus determine singularities in stable matter as represented as primes.
7:41 whoops, I reflexively thought Schwarz's first name would be Cauchy :D
:)
A great way to start Christmas morning - thank you. Also i really like many of the teashirt designs. Could you not put the digital design files for all your t-shirts on sale in your online store so that people could buy and then send to an online t-shirt producer to get both their prefered t-shirt and design. It would expand your market. Best wishes for the new year.
Most of the t-shirts I wear in these videos are not my own designs. Apart from that happy to share a printable file.
Hey there! As the year comes to a close, I wanted to take a moment to send my warmest wishes to you. May the upcoming holiday season bring you joy, relaxation, and cherished moments with loved ones. May you find inspiration and creativity in the coming year, and may it be filled with success, growth, and exciting new opportunities.
Your content has been a source of knowledge and entertainment for many, and I want to express my gratitude for the valuable content you've shared. Thank you for being a part of this online community and for your dedication to your craft.
Wishing you a peaceful and joyful holiday season, a fantastic New Year, and continued success in all your endeavors. Stay safe and keep up the fantastic work!
That's great ! Thank you very much :)
Merry Christmas, Mathologer!
Merry Christmas :)
Andrew Wiles proved Fermat's Last Theorem using "a special case of the modularity theorem for elliptical curves". That was step one! He got us all to the "top of the hill". What an amazing view there is to be had!
I cannot get into that math, dealing with stability, very easily, as I have to mind how it relates to sound silencing technology - just the same as I did back in the '90s when I talked about this on the POP3 server at UW-Parkside. But, I CAN get into how this relates to Ramanujan's Infinite Sum and some of its 12 parts.
Step two is using hyperbolic curves, along with dual, split-complex, and complex numbers as tensors (and triangular positioning and angle trisection methods), to help prove the Riemann hypothesis and its close connection to Big Bounce events. I may need to have some Target Motion Analysis mathematics declassified. I cannot say for sure on this, but would be a fool to not anticipate a potential clash or partnership with the Navy over this "Ancient Bubblehead Knowledge".
Another great video, I always watch on Sunday. Of course today its Christmas Eve. HO HO HO Merry Christmas
I loved this. The final part where you showed the angles add to 2π was both incredibly obvious and utterly amazing all at once. It blew my mind. And the result being that you can fold such a lantern from a single intact flat rectangular sheet ... I have no words. But I have a new party trick. ❤️
The segment about finding the length of the curvy curve reminded me of the Infinite Coastline Paradox (the finer the measurement units used, the closer the answer goes to Infinity as you get in to the finer fractal-like nooks and crannies of a country's coastline) and how a smooth waveform of sound is digitised in to 'samples' (eg: the 44.1KHz sampling rate of standard CD & DVD audio being basically 2x the upper bound of human hearing with some extra buffer room for flaws to have a clean listening experience).
DVD actually uses 48kHz, which gives a bit more headroom over a CD, though not by much.
The right one is the SMALLEST one, so long as the path is always strictly "not inside" the circle.
I like your shirt. I didn't think I was going to watch a calculus video, but I love it.
Glad you like it!
That was way more interesting than I expected. Thnks!
Glad you liked it!
The image at the left at 16:20 actually shows p=20 and b=20 instead of p=10 and b=10. The case p=10 and b=10 is shown at 20:24.
The Hydraulic Press channel shows many examples of thin walled tall "cylinders" folding into triangles when under extreme vertical pressure.
Yes, also one of the first things that Andrew pointed out to me. Lots of fun to be had with hydraulic presses :) That reminds me that I should put some links in the description of this video.
The circumfrence of a circle is the sum of all points where the line through them is perpendicular to the duameter. The shrinking square always has 4-3.14159 excess where its line is not perpendicular to the diameter
As it stands what you say here does not make any sense. However, I think you probably got the right idea, something along the lines of what I am discussing at the end of the video :)
For higher dimensions we just need the limites of any and all angles to go to 0 or 180 for the area to be miniminzed and non wrinkly, so you can just look at the substitutions seperately and garuantee a nice limit by overpowering the bad substitutions that increase angles by a larger substitution per step or something lile that for the other kinds of substitutions that effect that spesific angle affected by the naughty kind of substitution, then chrismas can be saved.
Triangulating a smooth surface allows combinatorics to apply since it makes the surface into a simplicial complex. As mentioned, similar geometries in the form different triangulation can give different invariants. For example, as explain in this video the surface area which is invariant the under Euclidean motions is different while another invariant the genus remains 0.
From algebraic geometry different sheaves define different coverings of topological spaces with different geometry and different invariants. Depending on how the sheaf of the smooth space is defined changes the invariants.
I once advised Dr. Amy Mainzer of how to cross-reference Euler's Identity with conic sections to learn about stable particles, although I did not specifically quote or understand Euler's Identity at the time back in 2009 (or about).
Bubbles. I always use an electron, an incandescent light bulb, or similar shape to understand this connection: newly formed dark matter has a bubble at its center bulge. Circles and the conchoid of Nicomedes correspond. Compressed atomic nuclei (or neutron stars) - more bubbles, at their contact points. Elliptical part. Delanges trisectrix. The inflection point is parabolic and corresponds to the birth of a black hole and the regular trifolium/the rose with three petals. The ring/cylinder singularity is hyperbolic and corresponds, in this viewing of gravitational increases over time until a Big Crunch - with respect to conic sections, to many, many black holes and the Maclaurin trisectrix.
It is no coincidence that advanced mathematical proofs often talk about elliptical and hyperbolic curves.
The movie "Dark Matter" had a similar concept. The part where the student is boiling water and exclaims, "bubbles".
So I take it (adding a few bits about fractals that I know) that when using rectangular corner-cutting to approximate a circle, you approximate the area of the circle, but the perimeter of your approximation is a fractal curve that has infinite right-angled discontinuities which are not present in the smooth circular curve. Fractal curves have "fractal dimension" of *at least* 1 - they can e.g. be area-filling curves that have an area rather than a length, or space-filling curves that have a volume. Fractal surfaces have fractal dimension of at least 2 and can be space-filling surfaces. Either way (and including all the cases with non-integer dimension), arguments about measures being preserved/increased/whatever as the approximation is improved break down when the measure (length or area) ceases to apply, so you can't then take those measures from fractal approximations and apply them to the non-fractal thing being approximated.
On the other hand when subdivision forms smaller and smaller in-between angles, in the limit you have infinitely many zero degree angles - the discontinuities have disappeared so that the infinite pieces (lines/triangles/whatever) form a smooth, continuous curve/surface.
There's a different non-convergence problem (Runge's phenomenon) which I find interesting when approximating smooth curves using high-order polynomial curves, or when approximating smooth surfaces using high-order polynomial surfaces. The normal solution is using piecewise curves/surfaces using low degree polynomials for each piece, though usually based on a single specification for the full curve (e.g. B-spline) rather than explicitly working out the polynomials for each piece. Converting into a connected sequence of simple polynomial curves (e.g. Bezier curves) is easy enough but unnecessary. Subdividing into linear B-splines into lines is the linear-polynomial-pieces special case that (like linear B-splines themselves) no-one uses. I mention this because I'm now wondering if the Runge's phenomenon convergence failure indicates another kind of fractal, with infinite extreme "oscillations" rather than infinite discontinuities.
"So I take it (adding a few bits about fractals that I know) that when using rectangular corner-cutting to approximate a circle, you approximate the area of the circle, " Yes nothing goes wrong in terms of area. In fact, since the area formula of the circle also features pi, we can also use this weird approximation to calculate pi.
"I see your shwartz is as big as mine."
- Banksy
what is interesting is that this curse is also a blessing: for battery as well as for chemical catalyst, you want the biggest surface per volume. in this case you have to think the opposite
Great video, I still have question though: towards the end, you gave us the condition with the normal vectors, which guarantees that our approximation for the the cylinder, has a surface area which aprroaches the correct one. I find it somewhat analogous to a uniform convergence condition, but I think that uniform converge is usually a sufficient but not a nesscery condition to exchange limits. So contining that reasoning, is there aome sort of a lantren such that the normals don't behave nicely, but the surface areas still aproaches the correct result? I don't mind changing the surface we are trying to approximate from a cylinder, to something else.
Nice connection. A lot of the ways we handle swapping limits, infinite sums or integrals is to insist that our maths is well-behaved.
An example, answering your question: a cylindrical lantern where the patch over which normals “misbehave” get smaller and smaller. For instance if there were b-1 “nice” bands and 1 “naughty” band (too skinny for its own good). Then the nice bands would have triangle normals converging to cylinder normals, the naughty band wouldn’t so you wouldn’t have uniform convergence of normals. However the nice bands cover most of the surface, so the area still converges.
It’s seasonally appropriate for nice to outweigh naughty. Santa approves.
I agree, thank you. I also finds a connection here with uniform convergence. When I studied at uni, my country's equivalent of real analysis, we saw the sequence of functions: f_n(x) = x^n on the [0, 1] interval, as an example where the limit of continuous and even differentiable functions is not continuous, We then saw how it does not converge uniformly to its limit function (because for each n, the function x^n approaches 1 on the right, and even though that convergce becomes slower as n increases, it does not matter, because what we do here (by definition), is to first look at the entire closed interval, calculate the sup |f_n(x) - f(x)| and then take the limit as n -> infinity). But despite that, the limit function is integrable, even riemann integrable, and we can still exchange integration and limit and get the correct result.
Did not know about this „proof“ until half a minute ago! Pure beauty… 😅
Definitely a must know :)
Great episode. And I loved the outro music!
I wonder how (and if) this would translate to understanding 3D Modeling software calculating object data. I mean there are vertices, surfaces and normals, how the norlmals work (and why is it important), why a surface with flipped normal mess up the shading on a curved object (and what exactly does that mean) and why that does not happen with a flat surface. Why is it better to model in quads instead of triads, yet the render engine breaks all quads into triads during the final render. This video really tickles so many question, thanks for this one.
Yes, lots of different interesting rabbit holes to dive into :)
Nice. I was waiting for a graph of p versus b with color coding to show convergence.
Sir, I was reading about the Chinese remainder theorem and found it very interesting. I tried to find the proof of the same on TH-cam but couldn't find a satisfactory one. I request you to make one video on it.
I found this very enjoyable and thought provoking. Speaking of provoked thoughts - the original problem posits a cylinder with diameter 1 and height 1 giving a surface area of pi. However as soon as one starts creasing the cylinder to create rings (even or especially an abstract mathematical cylinder) perforce the indents will shorten the height of the cylinder implying that as the number of rings increases infinitely, the height shrinks to zero leaving just a circle. Now I am not sure if I just disproved the lemma by showing that it can not exist or just found an other solution.
"However as soon as one starts creasing the cylinder to create rings" in my original construction of a lantern I don't actually crease the cylinder. That's just filling in the triangles between a bunch of points on the cylinder surface.
Now at the end of the video it turns out that there is a totally different cylinder (different height and different radius) that can be creased to give the same lantern. The whole creasing business cylinder > lantern is not dynamic, that is different lanterns do not continuously transform into each other.
Anyway, very important to be supersorted in this respect :)
So, this is like the cylinder version of honeycombs, where conjoined circles collapse i to hexagons. Neat
It is similar to the Coastline Paradox, but in this case the problem is simpler to visualize as it comes down to trying to approximate a smooth surface with a combination of convex and concave shapes, which will always have a larger area than a smooth surface.
That's the business I hint at at the end of the discussion, the difficulty of making sense of length and area in the case of buckling curves and surfaces and in particular fractal ones :)
Shoreline paradox coming in hot
Excellent, as always
It just occurred to me that π
Thank you for the video.
Reminds me of how I used to crush soda cans by hand. I'd pinch 3 spots near the top, and 3 spots near the bottom, creating something like a 3 point 1 band lantern, and then twisting the two ends would allow the can to be crushed easily into a pancake.
Got to try your method at some point. A lantern-creased can also squashed down nicely into a pancake. There even exists a patent in which cans get "lanterned" to make them pancake squashable :) Have a look at this article beachpackagingdesign.com/boxvox/pseudo-cylindrical-concave-polyhedral-packaging
@@Mathologer Thanks for the link, that's a great article on a truly fascinating subject!
Havent heard about that drama - but will be happy to hear about it :) Thank you!
1:30 - ok now i know why i didnt bother taking notice of that problem - its not a problem for a phycist - the diagonal of a square is still SQRT(2) - they could prove the perimeter is less than pi that way and that would be a contradiction.
The Schwarz lanterns used to be taught as part of multivariable calculus courses. It's really all about figuring out what works and does not work when you are playing with infinity :)
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Mandelbrot: length of the boundary of a country depends on the size and scale of the measuring device.
If you used that four green plus to red equation but swapped green for light and red for dark, or green for magnetic and red for magnetically attracted, then you would be explaining the propagation of laser light, not just a silly cylinder.
I really got a lot out of this video. Thank you very much and merry Christmas!
"If you used that four green plus to red equation but swapped green for light and red for dark, or green for magnetic and red for magnetically attracted, then you would be explaining the propagation of laser light, not just a silly cylinder." A link s'il vous plait :)
Thanks!
Was the concept: sometimes while surface area is bound, length can approach infinity?
Definitely part of it :)
Fascinating. Do I understand correctly that the side surface of the limiting cilinder lantern equals 2 Pi r h x sqrt(1+ (Pi / 2)^4 x k^2), whereby k = limit of b / p^2? If yes, does that mean that there is a correspondence between the angle of the spikes and k? Something like angle = 2 arctan (k)?
Have a look at this www.cut-the-knot.org/Outline/Calculus/SchwarzLantern.shtml
Minor correction: The cylinder on the left at 16:20 has 20 bands and 20 points, not 10 and 10 as stated.
I did not expect to ever see the trollface in a mathologer video 😂
Glad I still manage to surprise people every once in a while :)
How could I miss a new mathloger video??? I need to see it rn lol
Edit: Such lectures never fail to amaze me ^^
That's great :)
Fun Video! What language is on the first can? I can see some Hebraic nuances in the block script, but it's not Hebrew.
It's Japanese :) That was actually my first attempt at turning a soda can into a Schwarz lantern (during an expedition to a Japanese restaurant)
@@Mathologer That's a surprise! I love Japanese food. Wow--for a first try, you sure look like a pro! 😄
Merry holidays, Buckard, Andrew, and everybody else who reads this! :)
Appropriately seasonal celebratory greetings to you too Willem. The world always needs more universal good will - we're all in this together.
Merry Christmas!
This is a good day for the lantern to shine bright...
Are those "spikes" or are they actually Christmas lights decorating your curve? Also, I feel like I've seen this trick a few times, where letting two sequences go to infinity depends on "path" you take in the indexing space.
"Also, I feel like I've seen this trick a few times, where letting two sequences go to infinity depends on "path" you take in the indexing space." Absolutely, you get this sort of phenomenon a lot in multivariable calculus. (I just grabbed the following from somewhere on the web) A classic example of a multivariable function with different limits along different paths is the function f(x, y) = (x*y) / (x^2 + y^2) as (x, y) approaches the origin (0, 0). If we approach the origin along the x-axis (y = 0), the limit of f(x, y) as (x, y) approaches (0, 0) is 0. However, if we approach the origin along the path y = x, the limit of f(x, y) as (x, y) approaches (0, 0) is 1/2. This demonstrates that the limit of the function f(x, y) at the point (0, 0) depends on the path taken to approach that point.
Stray thought after just the intro (up to where Schwarz has been introduced): consider a simple straight line of length 1. Approximate it, in the first instance, by the other two sides of an equilateral triangle on it. This approximation's length is 2. It lies within sqrt(3)/2 of the line, attaining that bound at one point. Repeatedly: construct a line parallel to the original at half the distance from it of the furthest point(s) of the path from it; fold the parts of the path beyond that down so those parts previously furthest from it are now on the line. This does not change the length of the path since, in each case, we flip an equilateral triangle about the edge of it that isn't part of the path. At each step we halve the distance from the original line of the most distant part of the path. So the path "converges" to the line, but has twice its length. So you don't need circles or pi to get the original pi = 4 paradox: you can do something exactly equivalent to show 2 = 1.
Sure, maybe check out some of the relevant links in the description of this video :)
I actually do a crude Schwarz lantern to my energy drink cans when I crush them to save space. So I immediately knew where you were going with it, although I had never connected it to differentiation before.
I use a can crusher device. Flatter than a pancake, no math required. Your way sounds like more fun!