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.
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.
@@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.
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!
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.
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.
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.
@@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.
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!
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.
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.
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
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.
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
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
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.
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)
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.
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.
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!
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 🍻
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.
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.
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.
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.
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!
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
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".
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.
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.
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.
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. ❤️
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 RUclips 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 :)
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 :)
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 :)
Wonderfully intriguing…TY. A side observation…these approximation only work in flat space, like on flat paper. In curved space, like the reality of Earth’s actual gravity well, Pi increases because it is further across the circle than the diameter calculated from the circumference/Pi (A tiny bit ok…before anyone gets excited). This is because space (length…the diameter) curves in time so travelling the diameter takes longer than calculated from a know speed of travel.
Well, there is maths and there is the real world. However, there is no need to adjust the value of pi just because when you try to draw a circle in the real world you don't end up with an ideal mathematical circle :)
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 :)
A part of the schwarz-lanturn as origami-model can be crafted into a simple frameless paper-kite fold the paper into 8-10 bands for the lantern-structure with valley folds and add the diagonal mountain folda in a way, that they start at the corners and center of the top Edge and divide the valleyfolds with their own cross-points alternating in two halves and ome half between two quaters. This makes 1/4 of a 8-8 lantern or 1/5 of a 10-10 lantern Attach a kite-string to the seccond cross-point along the middle axis from one curved edge and some paper-strips as a tail in the middle of the other curved edge.
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.
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
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.
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.
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.
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 :)
Split-complex numbers relate to the diagonality (like how it's expressed on Anakin's lightsaber) of ring/cylindrical singularities and to why the 6 corner/cusp singularities in dark matter must alternate. Dual numbers relate to Euler's Identity, where the thin mass is cancelling most of the attractive and repulsive forces. The imaginary number is mass in stable particles of any conformation. In Big Bounce physics, dual numbers relate to how the attractive and repulsive forces work together to turn the matter that we normally think of into dark matter. Complex numbers = vertical asymptote. Split-complex numbers = vertical tangent. Dual numbers = vertical line. These algebras can be simply thought of as tensors. Delanges sectrices can be thought of as opposites of vertical asymptotes. Ceva sectrices as opposites of vertical tangents, and Maclaurin sectrices as opposites of vertical lines. The natural logarithm of the imaginary number is pi divided by 2 radians times i. This means that, at whatever point of stable matter other than at a singularity, the attractive or repulsive force being emitted is perpendicular to the "plane" of mass. In Big Bounce physics, this corresponds to how particles "crystalize" into stacks where a central particle is greatly pressured to degenerate by another particle that is in front, another behind, another to the left, another to the right, another on top, and another below. Dark matter is formed quickly afterwards. Ramanujan Infinite Sum (of the natural numbers): during a Big Crunch, the smaller, central black holes, not the dominating black holes, are about a twelfth of the total mass involved. Dark matter has its singularities pressed into existence, while baryonic matter is formed by its singularities. This also relates to 12 stacked surrounding universes that are similar to our own "observable universe" - an infinite number of stacked universes that bleed into each other and maintain an equilibrium of Big Bounce events. i to the i power: the "Big Bang mass", somewhat reminiscent of Swiss cheese, has dark matter flaking off, exerting a spin that mostly cancels out, leaving potential energy, and necessarily in a tangential fashion. This is closely related to what the natural logarithm of the imaginary number represents. Mediants are important to understanding the Big Crunch side of a Big Bounce event. Black holes have locked up, with these "particles" surrounding and pressuring each other. Black holes get flattened into unstable conformations that can be considered fractions, to form the dark matter known from our Inflationary Epoch. Sectrices are inversely related, as they deal with dark matter being broken up, not added like the implosive, flattened "black hole shrapnel" of mediants. Ford circles relate to mediants. Tangential circles, tethered to a line. Sectrices: the families of curves deal with black holes and dark matter. (The Fibonacci spiral deals with how dark matter is degenerated/broken up, with supernovae, and forming black holes. The Golden spiral deals with black holes being flattened into dark matter during a Big Bounce event.) The Archimedean spiral deals with black holes and their spins before and after a reshuffling from cubic to the most dense arrangement, during a Big Crunch. The Dinostratus quadratrix deals with the dark matter being broken up by ripples of energy imparted by outer (of the central mass) black holes, allowing the dark matter to unstack, and the laminar flow of dark matter (the Inflationary Epoch) and dark matter itself being broken up by lingering black holes. Delanges sectrices (family of curves): dark matter has its "bubbles" force a rapid flaking off - the main driving force of the Big Bang. Ceva sectrices (family of curves): spun up dark matter breaks into primordial black holes and smaller, galactic-sized dark matter and other, typically thought of matter. Maclaurin sectrices (family of curves): dark matter gets slowed down, unstable, and broken up by black holes. Jimi Hendrix's "Little Wing". Little wing = Maclaurin sectrix. Butterflies = Ceva sectrix. Zebras = Dinostratus quadratrix. Moonbeams = Delanges sectrix. Jimi was experienced and "tricky". Jimi was commenting on dark matter. How it could be destabilized by being slowed down, spun up, broken up by lingering black holes, or flaked off. (The Delanges trisectrix also corresponds to stable atomic nuclei.) Dark matter, on the stellar scale, are broken up by supernovae. Our solar system was seeded with the heavier elements from a supernova. I'm happily surprised to figure out sectrices. Trisectrices are another thing. More complex (algebras) and I don't know if I have all the curves available to use in analyzing them. I have made some progress, but have more to discern. I can see Fibonacci spirals relating to the trisectrices. The Clausen function of order 2: black holes and rarified singularities are becoming more and more commonplace. Doyle's constant for the potential energy of a Big Bounce event: 21.892876 Also known as e to the (e + 1/e) power. At the eth root of e, the black holes are stacked as densely as possible. I suspect Ramanujan's Infinite Sum connects a reshuffling from the solution to the Basel problem and a transfer of mass to centralized black holes. Other than the relatively small amount of kinetic energy of black holes being flattened into dark matter, the only energy is potential energy, then: 1 (squared)/(e to the e power), dark matter singularities have formed and thus with the help of Ramanujan, again, create "bubbles", leading to the Big Bang part of the Big Bounce event. My constant is the chronological ratio of these events. This ratio applies to potential energy over kinetic energy just before a Big Bang event. Methods of arbitrary angle trisection: Neusis construction relates to how dark matter has its corner/cusp singularities create "bubbles", driving a Big Bang event. Repetitious bisection relates to dark matter spinning so violently that it breaks, leaving smaller dark matter, primordial black holes, and other more familiar matter, and to how black holes can orbit other black holes and then merge. It also relates to how dark matter can be slowed down. Belows method (similar to Sylvester's Link Fan) relates to black holes being locked up in a cubic arrangement just before a positional jostling fitting with Ramanujan's Infinite Sum. General relativity: 8 shapes, as dictated by the equation? 4 general shapes, but with a variation of membranous or a filament? Dark matter mostly flat, with its 6 alternating corner/cusp edge singularities. Neutrons like if a balloon had two ends, for blowing it up. Protons with aligned singularities, and electrons with just a lone cylindrical singularity? Prime numbers in polar coordinates: note the missing arms and the missing radials. Matter spiraling in, degenerating? Matter radiating out - the laminar flow of dark matter in an Inflationary Epoch? Corner/cusp and ring/cylinder types of singularities. Connection to Big Bounce theory? "Operation -- Annihilate!", from the first season of the original Star Trek: was that all about dark matter and the cosmic microwave background radiation? Anakin Skywalker connection?
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).
The background noise is a cicada rubbing itself in hopes of convincing another cicada to join in, because Andrew evidently lives far enough south that the wildlife has not realized it is winter.
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.
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.
Hey, I got a nice idea. We have learnt from this video that in order to measure the length of a curve, we have to put several points on the curve and then to join every consecutive points by segments. But, I think if we draw tangent segments from every point, the measurement will be more accurate. And it is the way by which Archimedes measured π
As I said Archimedes used both ways: Tangents to get an estimate greater than pi (22/7) and inscribed polygons to get an estimate less than pi (223/71)
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.
Essentially inside pi over outside pi does not equal 1. As the two circles approach each other, the wall gets thinner, the distance between them approach zero and pi over pi approaches 1 .
I am not calculating volume I am calculating area. Volume is not a problem. Doesn't matter how you refine with edges going to zero the volume will always come out correctly :)
Despite being woefully lacking in math skills, I am nevertheless enthralled by mathematical concepts. The Mathologer never fails to mesmerize and entertain me. 👍😊
I think it would be neat to make an animation of a reflective lantern getting more and more bands and converging to occupying the space of a cylinder while reflecting the surroundings very differently from a cylinder.
But shiny ray-tracing is sooo 1990s. AR is where it's at today, baby. .... and I'm not just conveniently making an excuse for me doing the easier of the two options. 🙂 (Mmmm - Actually AI is where it's at, so maybe I should just ask chatGPT to write me some povray code.)
There’s a more familiar example of inconclusive limits, but with the variables converging to a finite value, rather than infinity. This example relates to a value for 0⁰. If we consider x^y (x to the power y) as both x -> 0 and y -> 0, the limiting value depends on *how* you make x and y converge to 0. If you let x-> 0 a lot faster than y -> 0 then the limit is 0. For example x = 2⁻ⁿ, y = 1/sqrt(n). You can also make the limit 1. For example x = 2⁻ⁿ, y = 1/n². Or x = 2⁻ⁿ, y = -1/n gives limit 2. There can’t be multiple values for this limit. So we say that the limit does not exist, meaning that 0⁰ does not exist. Sometimes we give it special status as “indeterminate” - within certain formal working it can behave as if it has a value. This requires care. The obvious instance of this: it can be convenient to formally assign x⁰=1, no matter what x is, such as with polynomials and power series. What makes this ok is that the x^y limit has y=0 always, since the xⁿ expressions in polynomials are only for integer n.
One observation on the “curvy curve”: yes, the measured length gets longer and closer to the actual length as the intervals get closer to zero - but if they actually get to zero, you’re just summing an essentially infinite number of zeros. So the length gets closer and closer and then drops off a cliff. 😊
@Mathologer Yup! It's the point where the semantics become mathematically significant. As a word enthusiast who enjoys maths, I've always liked that sort of curiosities. 😃
To me, it seems like this argument about area of the lantern going to infinity due to the increase of points and planes is similar to there being an infinite amount between the numbers of 0 and 1. An finite infinity, I guess, which sounds reasonable to me. The lantern doesn't increase area infinitely, it just increases the measurable area to an infinite precision within a finite amount. (edit) Huh. I've seen some vids of crushing tubes of various materials and wondered about why the material crushed the way it did. Thanks for adding that part at the end of this video.
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.
Interesting. I haven't seen anybody use ... since my calculus prof used it to do calculations in his head which we were supposed to somehow keep up and follow as he blasted through formula after formula. I did however sense how the convolutions added surface area by adding facets without reducing the cylinders height. It was a bit intuitive if your gifts include spatial aptitude even without the math. Thanks, this was quite fun.
From the idea this is related to measuring the length of a coast line. The finer grained you measure it the longer it gets. Or the surface area of the intestine, If you take the length and the circumference and multiply you don't get the 200-300 m² scientists approximate it to.
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 :)
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.
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.
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.
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.
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.
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!
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.
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.
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
@@Anokosciant I love extensions!!
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.
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.
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 :)
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 :(
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. :)
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)
Wieder super!
Frohe Weihnachten, lieber Burkard!
Fröhliche Weihnachten :)
It's too bad that I had teachers who complained that I was too slow and would never be good at math.
This would have made for a great Halloween-themed video, with ”cursed lanterns” 🎃.
"Are you going to sit around and drink beer all afternoon?"
"I'm building Schwarz Lanterns."
Its due to channels like yours that makes youtube videos worthwhile to watch.
Glad you think and say so :)
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.
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 :)
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.
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!
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 🍻
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!
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 :)
2:07 And all the engineers rejoiced!
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.
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.
Merry Christmas Mathologer!
Merry Christmas :)
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.
Frohe Weihnachten nach Down under und vielen Dank für's Video!
Fröhliche Weihnachten Ö=
@@Mathologer , lustiger Space Invader >> Ö=
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.
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 :)
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
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".
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.
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 :)
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.
RUclips 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 :)
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.
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. ❤️
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 RUclips 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 :)
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 :)
This is a beautiful clarification. Thank you and Happy Christmas
Merry Christmas :)
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 :)
Wonderfully intriguing…TY. A side observation…these approximation only work in flat space, like on flat paper. In curved space, like the reality of Earth’s actual gravity well, Pi increases because it is further across the circle than the diameter calculated from the circumference/Pi (A tiny bit ok…before anyone gets excited). This is because space (length…the diameter) curves in time so travelling the diameter takes longer than calculated from a know speed of travel.
Well, there is maths and there is the real world. However, there is no need to adjust the value of pi just because when you try to draw a circle in the real world you don't end up with an ideal mathematical circle :)
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 :)
A part of the schwarz-lanturn as origami-model can be crafted into a simple frameless paper-kite
fold the paper into 8-10 bands for the lantern-structure with valley folds and add the diagonal mountain folda in a way, that they start at the corners and center of the top Edge and divide the valleyfolds with their own cross-points alternating in two halves and ome half between two quaters.
This makes 1/4 of a 8-8 lantern or 1/5 of a 10-10 lantern
Attach a kite-string to the seccond cross-point along the middle axis from one curved edge and some paper-strips as a tail in the middle of the other curved edge.
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.
The right one is the SMALLEST one, so long as the path is always strictly "not inside" the circle.
7:41 whoops, I reflexively thought Schwarz's first name would be Cauchy :D
:)
Best Christmas present I could ask for
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 :)
So, this is like the cylinder version of honeycombs, where conjoined circles collapse i to hexagons. Neat
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 ... :)
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!
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!
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 liked it all. Even your TShirt. Don't burn out and have a great year!🎄
Thank you! You too!
"I see your Schwarz is as big as π!"
May the Schwarz be with you.
For the people who don't get this one: ruclips.net/video/pPkWZdluoUg/видео.html :)
I like your shirt. I didn't think I was going to watch a calculus video, but I love it.
Glad you like it!
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.
The "Crushmetric" water bottle look just like that!
Wonderful, got to get myself one of those :)
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 :)
Split-complex numbers relate to the diagonality (like how it's expressed on Anakin's lightsaber) of ring/cylindrical singularities and to why the 6 corner/cusp singularities in dark matter must alternate.
Dual numbers relate to Euler's Identity, where the thin mass is cancelling most of the attractive and repulsive forces. The imaginary number is mass in stable particles of any conformation. In Big Bounce physics, dual numbers relate to how the attractive and repulsive forces work together to turn the matter that we normally think of into dark matter.
Complex numbers = vertical asymptote. Split-complex numbers = vertical tangent. Dual numbers = vertical line. These algebras can be simply thought of as tensors. Delanges sectrices can be thought of as opposites of vertical asymptotes. Ceva sectrices as opposites of vertical tangents, and Maclaurin sectrices as opposites of vertical lines.
The natural logarithm of the imaginary number is pi divided by 2 radians times i. This means that, at whatever point of stable matter other than at a singularity, the attractive or repulsive force being emitted is perpendicular to the "plane" of mass.
In Big Bounce physics, this corresponds to how particles "crystalize" into stacks where a central particle is greatly pressured to degenerate by another particle that is in front, another behind, another to the left, another to the right, another on top, and another below. Dark matter is formed quickly afterwards.
Ramanujan Infinite Sum (of the natural numbers): during a Big Crunch, the smaller, central black holes, not the dominating black holes, are about a twelfth of the total mass involved. Dark matter has its singularities pressed into existence, while baryonic matter is formed by its singularities. This also relates to 12 stacked surrounding universes that are similar to our own "observable universe" - an infinite number of stacked universes that bleed into each other and maintain an equilibrium of Big Bounce events.
i to the i power: the "Big Bang mass", somewhat reminiscent of Swiss cheese, has dark matter flaking off, exerting a spin that mostly cancels out, leaving potential energy, and necessarily in a tangential fashion. This is closely related to what the natural logarithm of the imaginary number represents.
Mediants are important to understanding the Big Crunch side of a Big Bounce event. Black holes have locked up, with these "particles" surrounding and pressuring each other. Black holes get flattened into unstable conformations that can be considered fractions, to form the dark matter known from our Inflationary Epoch. Sectrices are inversely related, as they deal with dark matter being broken up, not added like the implosive, flattened "black hole shrapnel" of mediants.
Ford circles relate to mediants. Tangential circles, tethered to a line.
Sectrices: the families of curves deal with black holes and dark matter. (The Fibonacci spiral deals with how dark matter is degenerated/broken up, with supernovae, and forming black holes. The Golden spiral deals with black holes being flattened into dark matter during a Big Bounce event.) The Archimedean spiral deals with black holes and their spins before and after a reshuffling from cubic to the most dense arrangement, during a Big Crunch. The Dinostratus quadratrix deals with the dark matter being broken up by ripples of energy imparted by outer (of the central mass) black holes, allowing the dark matter to unstack, and the laminar flow of dark matter (the Inflationary Epoch) and dark matter itself being broken up by lingering black holes.
Delanges sectrices (family of curves): dark matter has its "bubbles" force a rapid flaking off - the main driving force of the Big Bang.
Ceva sectrices (family of curves): spun up dark matter breaks into primordial black holes and smaller, galactic-sized dark matter and other, typically thought of matter.
Maclaurin sectrices (family of curves): dark matter gets slowed down, unstable, and broken up by black holes.
Jimi Hendrix's "Little Wing". Little wing = Maclaurin sectrix. Butterflies = Ceva sectrix. Zebras = Dinostratus quadratrix. Moonbeams = Delanges sectrix. Jimi was experienced and "tricky".
Jimi was commenting on dark matter. How it could be destabilized by being slowed down, spun up, broken up by lingering black holes, or flaked off. (The Delanges trisectrix also corresponds to stable atomic nuclei.)
Dark matter, on the stellar scale, are broken up by supernovae. Our solar system was seeded with the heavier elements from a supernova.
I'm happily surprised to figure out sectrices. Trisectrices are another thing. More complex (algebras) and I don't know if I have all the curves available to use in analyzing them. I have made some progress, but have more to discern. I can see Fibonacci spirals relating to the trisectrices.
The Clausen function of order 2: black holes and rarified singularities are becoming more and more commonplace.
Doyle's constant for the potential energy of a Big Bounce event: 21.892876
Also known as e to the (e + 1/e) power.
At the eth root of e, the black holes are stacked as densely as possible. I suspect Ramanujan's Infinite Sum connects a reshuffling from the solution to the Basel problem and a transfer of mass to centralized black holes. Other than the relatively small amount of kinetic energy of black holes being flattened into dark matter, the only energy is potential energy, then: 1 (squared)/(e to the e power), dark matter singularities have formed and thus with the help of Ramanujan, again, create "bubbles", leading to the Big Bang part of the Big Bounce event.
My constant is the chronological ratio of these events. This ratio applies to potential energy over kinetic energy just before a Big Bang event.
Methods of arbitrary angle trisection: Neusis construction relates to how dark matter has its corner/cusp singularities create "bubbles", driving a Big Bang event. Repetitious bisection relates to dark matter spinning so violently that it breaks, leaving smaller dark matter, primordial black holes, and other more familiar matter, and to how black holes can orbit other black holes and then merge. It also relates to how dark matter can be slowed down. Belows method (similar to Sylvester's Link Fan) relates to black holes being locked up in a cubic arrangement just before a positional jostling fitting with Ramanujan's Infinite Sum.
General relativity: 8 shapes, as dictated by the equation? 4 general shapes, but with a variation of membranous or a filament? Dark matter mostly flat, with its 6 alternating corner/cusp edge singularities. Neutrons like if a balloon had two ends, for blowing it up. Protons with aligned singularities, and electrons with just a lone cylindrical singularity?
Prime numbers in polar coordinates: note the missing arms and the missing radials. Matter spiraling in, degenerating? Matter radiating out - the laminar flow of dark matter in an Inflationary Epoch? Corner/cusp and ring/cylinder types of singularities. Connection to Big Bounce theory?
"Operation -- Annihilate!", from the first season of the original Star Trek: was that all about dark matter and the cosmic microwave background radiation? Anakin Skywalker connection?
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 background noise is a cicada rubbing itself in hopes of convincing another cicada to join in, because Andrew evidently lives far enough south that the wildlife has not realized it is winter.
Yep, Australia :)
Did not know about this „proof“ until half a minute ago! Pure beauty… 😅
Definitely a must know :)
"I see your shwartz is as big as mine."
- Banksy
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 :)
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, I got a nice idea. We have learnt from this video that in order to measure the length of a curve, we have to put several points on the curve and then to join every consecutive points by segments. But, I think if we draw tangent segments from every point, the measurement will be more accurate. And it is the way by which Archimedes measured π
As I said Archimedes used both ways: Tangents to get an estimate greater than pi (22/7) and inscribed polygons to get an estimate less than pi (223/71)
Nice. I was waiting for a graph of p versus b with color coding to show convergence.
Awesome video!!! Beautiful animations. Got me thinking why not use rectangles they look much tamer. : )
I'm not very good at math but I understand everything.
You are a very good teacher. Thank for the video.
That's great, mission accomplished :)
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 :)
Another great video, I always watch on Sunday. Of course today its Christmas Eve. HO HO HO Merry Christmas
You are actually calculating the volume of a thick cylinder! Gradually getting thinner.
Essentially inside pi over outside pi does not equal 1. As the two circles approach each other, the wall gets thinner, the distance between them approach zero and pi over pi approaches 1 .
I am not calculating volume I am calculating area. Volume is not a problem. Doesn't matter how you refine with edges going to zero the volume will always come out correctly :)
Thanks for this Christmas present! 🎁
You are welcome :)
Fascinating video, loved the animations!
Many thanks!
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.
Despite being woefully lacking in math skills, I am nevertheless enthralled by mathematical concepts. The Mathologer never fails to mesmerize and entertain me. 👍😊
Great, mission accomplished :)
I think it would be neat to make an animation of a reflective lantern getting more and more bands and converging to occupying the space of a cylinder while reflecting the surroundings very differently from a cylinder.
Yes, that would be nice :)
But shiny ray-tracing is sooo 1990s. AR is where it's at today, baby.
.... and I'm not just conveniently making an excuse for me doing the easier of the two options. 🙂
(Mmmm - Actually AI is where it's at, so maybe I should just ask chatGPT to write me some povray code.)
I find it interesting that approximating to infinity gives different answers for different methods.
There’s a more familiar example of inconclusive limits, but with the variables converging to a finite value, rather than infinity.
This example relates to a value for 0⁰.
If we consider x^y (x to the power y) as both x -> 0 and y -> 0, the limiting value depends on *how* you make x and y converge to 0.
If you let x-> 0 a lot faster than y -> 0 then the limit is 0. For example x = 2⁻ⁿ, y = 1/sqrt(n).
You can also make the limit 1. For example x = 2⁻ⁿ, y = 1/n².
Or x = 2⁻ⁿ, y = -1/n gives limit 2.
There can’t be multiple values for this limit. So we say that the limit does not exist, meaning that 0⁰ does not exist.
Sometimes we give it special status as “indeterminate” - within certain formal working it can behave as if it has a value. This requires care.
The obvious instance of this: it can be convenient to formally assign x⁰=1, no matter what x is, such as with polynomials and power series. What makes this ok is that the x^y limit has y=0 always, since the xⁿ expressions in polynomials are only for integer n.
That was way more interesting than I expected. Thnks!
Glad you liked it!
One observation on the “curvy curve”: yes, the measured length gets longer and closer to the actual length as the intervals get closer to zero - but if they actually get to zero, you’re just summing an essentially infinite number of zeros. So the length gets closer and closer and then drops off a cliff. 😊
The same could be said to be true for just about everything we do in integral calculus :)
@Mathologer Yup! It's the point where the semantics become mathematically significant. As a word enthusiast who enjoys maths, I've always liked that sort of curiosities. 😃
To me, it seems like this argument about area of the lantern going to infinity due to the increase of points and planes is similar to there being an infinite amount between the numbers of 0 and 1. An finite infinity, I guess, which sounds reasonable to me. The lantern doesn't increase area infinitely, it just increases the measurable area to an infinite precision within a finite amount.
(edit)
Huh. I've seen some vids of crushing tubes of various materials and wondered about why the material crushed the way it did. Thanks for adding that part at the end of this video.
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.
Merry Christmas, Mathologer!
Merry Christmas :)
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.
Interesting. I haven't seen anybody use ... since my calculus prof used it to do calculations in his head which we were supposed to somehow keep up and follow as he blasted through formula after formula. I did however sense how the convolutions added surface area by adding facets without reducing the cylinders height. It was a bit intuitive if your gifts include spatial aptitude even without the math. Thanks, this was quite fun.
Not many people seem to know about the lanterns these days. Also, I could not find much on RUclips which is a bit strange.
From the idea this is related to measuring the length of a coast line. The finer grained you measure it the longer it gets. Or the surface area of the intestine, If you take the length and the circumference and multiply you don't get the 200-300 m² scientists approximate it to.
You are absolutely right there and at some point I was actually pondering whether I should also delve more into that part of the story :)
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 :)
Merry Christmas!
This is a good day for the lantern to shine bright...