The fact that this series of videos exists is really encouraging. The way the videos are set up, fostering math dialogue in the comments and having a challenge question at the end - is an excellent way to communicate otherwise-intimidating math concepts and intuition to the masses! You have earned my subscription!
My solution for the pants was to invert the top of the pants out and down to the floor, and invert the legs back up through themselves, so that the pants are inverted but also upside-down.
I'm pretty sure that you can stretch a human shape into a t-shirt shape. Nostrils become the sleeves, mouth becomes collar, and "aboral" end becomes the bottom opening.
You technically have the outer and middle ear separated by the ear drum, with the middle ear connected to your sinuses via the Eustachian tubes. Since these are dead ends they act like the "cup part" of a coffee cup and do not count as holes.
Yeah, that's what I was thinking. I'm also pretty sure that the bladder is separate from the intestines. So that would leave only four holes going into the digestive system. Making a human a genus-3 surface, same as a t-shirt.
I spent a while at the beginning of the video. I paused the video as soon as I heard "if you were wearing really stretchy pants, could you remove them without lifting your feet." My answer is no, but let's see if I'm wrong lol. I was stuck for a while, but soon realized that the pants can be reduced to essentially a rectangle with two holes, where the two holes come from the two pant legs. The waist part can be expanded and flattened to the floor in the shape of a rectangle. The rectangular part isn't so important, but it is important to note that the pants are homeomorphic to something with two "holes". So from there, I envisioned standing with my left foot in one of the holes, and my right foot within the other hole - While the rest of the rectangular part lay around my feet, on the floor. From here, I guess you could lift the other hole up and over to the other foot, but from there, with the two holes stacked on top of one another, there is no way to remove it from both legs. And I think another thing worth mentioning is that, while the pants have two holes essentially, our entire figure (with out legs, hips and floor making a sort of triangle) has just one hole. The arms and torso, etc., do not contribute to any additional holes. Now if we take what we've learned from VSauce, we know that the body has a couple of "Through holes," such as the gastrointestinal tract. If we include this as a part of our solution space, MAYBE we can remove the pants! Let's see if I'm wrong! Edit: That wasn't even the right question. ;[
There seems to me to be an evolution across the PBS Digital series, where the subjects vary from the very deep to the very accessible, but doing so by being "accessibly deep" and "deeply accessible". That is, seeing the mystery in common everyday observations, and finding clarity in deep and obscure theory. It's like a boxer working an opponent from low to high and back, but in this case seeking ways to impart knowledge rather than punches. Thanks!
Under the standard definition of convergence, the sum of (-1)^n diverges. But it converges under a broader definition of convergence called Cesaro convergence. en.wikipedia.org/wiki/Ces%C3%A0ro_summation
Nope. Usually with any youtube math video, you need to dig a bit deeper to get to the actual rigor of what's going on.
7 ปีที่แล้ว +2
With the logic that 1+1-1+1-1+...=-1/12 you can also say that sqrt(-1)=infinity or -infinity, depending on how you execute the approximation algorithm.
I think you have your series mixed up. The series (-1)^n=1+1-1+1-1+1... converges to 1/2. The sum of all natural numbers, 1+2+3+4+5+... converges (through 'analytic continuation') to -1/12.
0:00 After careful consideration: yes, by pulling the waistband over one's head, closing it and treating the garment like a pantleg tube. Somewhat acrobatic. 2:35 Seeing this part helped me visualize how the loops swap when just pulling the original depiction inside-out. 2:50 Oh, well there it is. 8:21 Well, that's more efficient.
To the question posed at 11:15, if you add all of the terms defined by the sequence 1/n^p, the resulting series will converge as long as p>1, no matter how close you get, and it will diverge if p≤1. In a sense, 1/n is the series on the cusp of converging.
Jup, that's what I was thinking about as well. It confused me that 1/1+1/2+1/4+1/8+1/16 .... does not generalize to 1/n^2 but (1/2)^n. So I am not sure whether op means the geometric series or the p-series.
That series is on the verge of converging, but it's a wide verge. The series 1/n for n = 1, 3, 5, ... diverges more slowly so it's arguably "more convergent" than 1/n for n = 1, 2, 3 ... And you can always find a series that diverges even more slowly. So in addition to an infinite sequence of series that are "almost divergent", there is an infinite sequence of series that are "almost convergent". It seems to me that the ideal answer would be a series for which convergence is undecidable. I have a memory, possibly a false one, of learning about such a series many, many years ago.
Convergence cannot be undecidable if you know the underlying sequence that you add together. Of course, you cannot decide whether sum_{i=1}^{n}a_i converges for n to infinity, without knowing what the sequence (a_i)_{i in IN} is. But as long as you know that, there are only two possible cases: - There exists a real number R such that sum_{i=1}^{n}a_i comes arbitrarily close to zero for larger and larger n (and for any given tolerance epsilon, you can find a natural number n such that the difference is at most epsilon _for every natural number larger than n_). - For every real number R there exists a given tolerance epsilon such that the distance of the sum up to n to the number R will stay bigger than epsilon _for infinitely many numbers n_. The way the natural numbers work, you always have one of those two cases up there. The only way to not have infinitely many n with distance bigger than epsilon is to only have finitely many n (duh!), and finitely many n have a largest element. After that largest element, all others will have tolerance smaller than epsilon, or the largest element wouldn't in fact be the largest element for which the distance is bigger than epsilon.
+Franz Luggin I am not sure you know what undecidable means. The question whether an algorithm terminates or not has also only two possible answers but this problem is undecidable. In general, the question whether a series converges or not is indeed not decidable, simply for the reason that there are uncountable many series.
As I said, if you know what the sequence a_i is, i.e. if there is one sequence given, you have enough information to decide what the answer is. There are convergence tests that work even though you do not know what the limit is.
By the way, genetics has ties to knot theory which is related to topology. Bacteria use special turing-like "algorithms" to untangle their own DNA when it gets knotted. They do so using a mathematically minimum number of DNA slices and how they do it exactly is not only not known by scientists but also requires solutions to problems in knot theory that we still don't know yet. (correct me if I'm wrong - my information is not completely upto date. Here's some fun: I thought of a good idea... find a set of 4 characters in the ascii extended chars that can represent the gene sequences A T G and C so that when negated, the image is it's opposite (A T) (GC). Here's what I came up with: A = • (black circle in a white box) T = ◘ (white circle in a black box) G = ○ (black ring in a white box) C = ◙ (white ring in a black box) If you express the Genes rather than atgtactgtca, but as it's RNA/inverse-RNA pairing is it would be in nature (nature's redundancy plan) then inverted blackwhite, equals it's flipped image... cggatttagcgagtaattctacgagaatagcgactgtaagtacggacttggcaagtaatt gcctaaatcgctcattaagatgctcttatcgctgacattcatgcctgaaccgttcattaa ◙○○•◘◘◘•○◙○•○◘••◘◘◙◘•◙○•○••◘•○◙○•◙◘○◘••○◘•◙○○•◙◘◘○○◙••○◘••◘◘ ○◙◙◘•••◘◙○◙◘◙•◘◘••○•◘○◙◘◙◘◘•◘◙○◙◘○•◙•◘◘◙•◘○◙◙◘○••◙◙○◘◘◙•◘◘•• By the way, this may work better with a monospace font like Lucida Console. Just bored and looking for things to entertain on the weekend. :-)
I'm in 10th grade and I prepared a project on Topology just a day before the Science Project Exhibition. I explained it in such a way, so much mathematical way that even the research scientist failed to understand it.
For the interlinked rings and the double donut, you can also start by shrinking the big loop until it's just a short conduit between the rings; for convenience, you can rotate the rings to put the conduit at a point where the circles cross visually in the illustration. Then it's just a matter of straightening out the rings so they point in opposite directions.
For the pants, I first pulled the waist down to my ankles, then pulled the ankles up to my thighs; this turns the pants inside-out, but now they're upside-down. However, since they're topological pants, this is easy to fix! Your legs and the planet you're standing on form a ring shape like a donut; just rotate the pants around that ring till they're right side up! First the entire Earth goes through one leg, then your upper body goes through the other leg.
Here's how I'd hang the picture from two nails: Run the string clockwise over the left nail, then counterclockwise around the right nail, then counterclockwise around the left nail, then clockwise over the right nail. This is a 3-strand braid with two strands stretched straight and turned into nails. It's also symmetric.
Great session. Im terrible at all arithmetic disciplines,but I enjoy watching. Love the pants. Quite flattering. The colours well matched the form. I'll keep watching until I start to understand.
Why would anyone want to hang a picture using two nails so that if either breaks the picture falls? Half the point of using two nails is so that if one fails there'll still be something holding the picture on the wall until it can be fixed.
I'm not really sure how to explain that hole disappearing. I was gonna say the hole was just an illusion, but it's not. Confusing. Anyhow, to get intuition about how this thing works, try this: Press your thumb and middle finger together. On other hand, lock your other thumb and middle finger with first one by doing the same, so your thumbs and middle fingers now form 2 circles and without releasing your thumb and middle finger on either hand, your hands are stuck together. Now because these two rings are connected by your hands, arms and chest, you actually have topologically equivalent situation to the starting position there.(If you're re-reading this because you think you found an error, instead of thinking chest as the connecting piece, lock your elbows together so you get smaller loop) What this morphing she described does, can be explained by simply this: Connect your elbows and wrists, pull your thumbs, sort-of trying to break free from their interlocked status. You now notice that you have managed to smoothly join your hands together and form the exact same shape as in the video.
Hey thanks for your reply! And yeah i see it now. I suppose you can straighten out the thicker curve and then shrink it so that the two rings are touching where the thicker curve connected them.
Another answer to the second question asked at the end: If a_n is a positive sequence whose series diverges to infinity, then there is always an asymptotically smaller sequence b_n which still diverges. By asymptotically smaller, i mean that b_n/a_n -> 0. The same is true for convergent series just in the other direction. This shows that there isn't really any edge for these things.
My solution to the pants problem was to turn the left.leg/Earth/right.leg loop into a torus. The pants became a toroidal sheath with a hole in it. I then stretched the hole all the way over both toroid and sheathe. Inside out pants!
The simplest case I can think of is a sphere and a bowl being topologically identical. If you've ever handled a ball that is completely deflated so that it collapses in on itself, it forms a bowl shape. Add air, and it takes on a familiar sphere shape (at least approximately).
i can do this with regular pants/jeans, turn them inside out, but in the end they lose some of their purpose. you pull down the upper part, then pull the insides of your legs up to your waist. now you still "wear" your pants but they are inside out, without lifting a foot.
I was thinking to just nail one of the nails in deeper into the wall so the head of the nail is under the second one (because you put em close n shit), so when you take it out, the other one comes out without you doing anything :P
the question about border between finite and infinite series, well, you can distort the sequence 1/n into 1/n², but the obvious one is to let the power change continuously. what I'm going at is the zeta function = 1 + 1/2^s + 1/3^s + ... if this is what we choose, then actually s = 1 the harmonic series *is* the edge. because choosing any power slightly larger than 1 gives us a finite sum, and from 1 and below the series diverges to infinity. (for values less than 1 or 1+imaginary part, zeta is no longer defined as the series)
That reminds me of a part of topology called Non-Orientable Manifolds, and it specifically reminds me of möbius loops. You can connect two möbius loops (one left-handed and one right-handed), each one having only one edge, and the combination of them will produce a shape that requires four spatial dimensions to exist called a Klein Bottle. It has no edges and only one side. In three spatial dimensions it intersects with itself in a given location, but in a fourth spatial dimension this never happens.
When iterating ƒ(x) = x² + c with certain real values of c, the number will bounce around infinitely between 2 and -2 never coming back to the same value. I just love that. ^_^ This is very strongly related to the Mandelbrot set.
There was a proof I saw somewhere that for any series that diverges to infinity, there is a sequence that diverges slower. There is similarly a proof that for any convergent series, there is a series that converges slower.
Genus Various: M. C. Escher's wood block prints included several in which the subject was two intertwined, but completely disconnected, worlds. Typically they would have different themes, such as light and dark or summer and winter. Anyway, both those worlds and the negative space surrounding them have so many holes, and (what is perhaps more interesting) they are quite constrained from any meaningful simplification without unacceptable crossing of boundaries. So that's my example, and it's weird like Escher.
I love this. Because I am highly interested in the topology of the universe, and the other fields that are around us must also have topologies and would be mind annihilating to keep in 3D, probably.
Hi! I loved this video. Back in the day, I couch surfed in the home of a UC Berkeley mathematician at the time when Grisha Perelman announced his proof of the Poincare Conjecture. It was all "yeah, we'll see" at the time. What ever became of it? I remember a hastily called meeting at the Uni.
I remember a pictorial essay in Esquire mag about prisoners figuring out a way to take their pants off while wearing ankle shackles. They did pick up their feet!
1. yes but not in public-slide one pant over your body and bring the other pant back; 2. and do the hokey-pokey and turn yourself about-that's what topology is all about; 3. if you require using the waist opening, slide the pant over the whole Earth instead.
Fun stuff, I just finished BS in Math a few weeks ago and my last math course was an independent study in Algebraic Topology! Are you familiar with the recent research in Homotopy Type Theory? Essentially, you apply the intuition of topology to the notions of term and type. Types are recognized as spaces in the same sense as a topological space, and terms with a certain type are regarded as points in those spaces. For example, the natural numbers are points in the space Nat. Under this scheme, equality of terms (a=b) is a path between terms (a path starting at a and ending at b). Also regarding the question about the harmonic series, I'm reminded of the Kempner series: basically, if you remove terms from the harmonic series which contain any particular string of digits in the denominator in any particular base (originally, any term in base-10 containing a "9") the series converges. Given this and the incredibly slow rate at which the harmonic series diverges to infinity, it's always felt to me like the harmonic series itself exists on a sort of cusp between divergence and convergence, at least in the traditional (non p-adic) sense of convergence.
[05:22] how do you contract a circle to a point-without condensing its finite interior → 0 (you'd be able to do 'most-anything' by merging 0-width-points and expanding them out).
I don't know if this is helpful, but I'll try: The intuition behind those loops is that they start somewhere on the surface of the shape. It could be any point really, but let's mark that point with X. The idea is that, through point X, you feed loop, so that you hold both ends of the rope, then from your hands, each end of the rope connect to the point X, and then they go doing loop things on the surface of that shape. And the intuition with that circle shrinking away is that for sphere, I can simply pull both ends of the string and take the rope back to me. But in case of Donut shape, it forms a loop, so I can't pull it back without letting go from one end or the other(which isn't allowed). The rules are that, I can move the point X around freely(there are some rather mild restrictions to this in some edge cases, but for purposes of this video, it's completely free decision on your part where the point X is). I can also pull both ends of the rope, or give more rope, but I can never let go of either end of the rope. If given these rules, I manage to go from one loop to another, then these loops are considered the same. So in case of sphere, every loop can be made into "rope completely pulled back in", and every loop can likewise made from "rope pulled back in" position by releasing some rope, so every loop is the same. In donut case, you can see how there are different kinds of loops which we cannot make into one another. Like, no matter how much you pull or release the rope, two loops and one loop can't be made into one another.
Romaji, it does get to 0. Consider the map f(r, t) =(r*cos(t), r*sin(t)) for r between zero and one, and t measuring the angle. For r>0 this is a circle, but for r=0, f(0,t)=(0*cos(t), 0*sin(t))=(0,0) so is just a point, regardless of the angle t. This is is an example of a homotopy between two maps.
When Kennedy said, "I'm a doughnut" he reminded that (wo)man's digestive system is like the hole in a doughnut, that we are topologically equivalent. Come to Berlin and run Axel Flinth's lecture!
I figured I could just drop my pants, and then pull both legs up through the waist until they're inside out. You never said I had to be wearing the pants. Only that they had to end up inside out, and I couldn't move my feet.
This made me think of math proofs. Where you can extend (bend) any of the current proofs to (shapes) to create a new figure (new proofs and branches of math), but they must all be topological equivalent, for else it is not the same. So no matter how much you extend math it is always the same, just we learned more ways to bend it.
About the question at the end of the video: I think that in real life there are no infinite numbers, because infinity is a cycle with n-numbers and we think that it has no end and we do not know what is equal to n. The difference between them is that the first set has a bipolar separation: known and not known, and the second one is n-polar: the known, not known, and the order-of-cycle index, which in the example is shown as the power of the number. Thanks for attention, sorry for my English.
A bit late, just saw your 11 May 2017 episode. Here are two 4D (or perhaps 11D?) topological shapes with unknown topological complexity which I am pretty certain are topologically equivalent, at least on a coarse grain scale. 1. The entire universe right now. 2. The entire universe ... just now. "Sketch Proof": Use the laws of physics time cobordism to deform one into the other, assuming a scale at which smoothness assumptions of general relativity apply and that no new Black Hole singularity was created in the intervening seconds.
Another good answer to the question about the edge of infinity (regarding the sums, 1+1/2+... and 1+1/4 + 1/9+ ...) is the following: The sum of 1/n diverges. But 1/n^2 converges. The sum of 1/nlog n diverges. But the sum of 1/(n log^2 n) converges. The sum of 1/ (n log n loglogn) diverges. But the sum of 1/ (n log n (loglog n)^2) converges, and so on. So if you take the product n times log n times loglog n times logloglog n times ... loglog...n (k times). The sum of its reciprocals diverges. But if you square the last term, it converges. So the edge is sort of n log n loglogn logloglog n ....
Superglue your shoes to the floor; While this step isn't strictly required, it helps for proving the authenticity of the solution. Saw circles into the floor around where your shoes are attached. Pop out the pieces of floor and pull the legs through past each respective wooden circle. Now that the pants are separated from your body, turn them inside out. Then slide your feet out of the shoes and put the pants back on. Now pay thousands of dollars to fix your floor.
So I tried taking off underwear without lifting my feet from the floor. I suddenly need new underwear, and I have an increased appreciation for thought experiments... 😂
For the question posed at 11:30, I posed the same question on reddit's r/math. It's proven that there can be no such function for Sum(1/f(n)) such that any function asymptotically bigger converges and any function smaller diverges. However, there's a really interesting boundary where nlog(n)log(log(n)) diverges but nlog(n)log(log...log(n))^2 converges. However, that's not rigorous enough to be an answer. Here's the post: www.reddit.com/r/math/comments/5h6ck0/smallest_function_whose_reciprocal_partial_sums/
The fact that two shapes can be continuously deformed into each other implies that they are topologically equivalent, but the reverse is not true. Two topological spaces are said to be topologically equivalent if there is a continuous bijection between them whose inverse is continuous. For instance, all knots are topologically equivalent [to a circle]. It would be good to be precise in a channel about math.
If the person wearing the pants were infinitely tall but the earth was still considered finite, you could still turn the pants inside out if you can pull it around the world.
If you take the 1/n in the harmonic series, and raise the n to any power greater than 1, even if it is only 1 plus an arbitrarily small epsilon, then the resulting series is convergent.
Challenge problem: A Hole in a Hole in a Hole is actually the same thing as a bowling bowl with its 3 holes piercing through. But I only know that because I cheated and watched Numberphile.
![ILLSLICK - KILLSHOT REMIX [FULL VERSION]](http://i.ytimg.com/vi/RIK8RrqIAzE/mqdefault.jpg)
The fact that this series of videos exists is really encouraging. The way the videos are set up, fostering math dialogue in the comments and having a challenge question at the end - is an excellent way to communicate otherwise-intimidating math concepts and intuition to the masses! You have earned my subscription!
This is why coffee and donuts go so well with each other, they are topologically similar
that only works for the mugs, although the thought of coffee in a shape that's homeomorphic to a donut is quite funny
My solution for the pants was to invert the top of the pants out and down to the floor, and invert the legs back up through themselves, so that the pants are inverted but also upside-down.
Matthew Giallourakis That was my thought also. She should have specified that they should be "on" like normal.
Yes, wearing your pants inverted and upside down is a terrible trend! We must stop it before it even starts.
Yeah, you don't even need crazy clown pants for this. I can do it with normal fitting sized gym shorts.
Matthew Giallourakis same
I thought of it in that way as well!!
See numberphile's "a hole in a hole in a hole"
That guy was high on math, plus 3-handle beer mug ftw
Three hole donut plus three handle coffee mug equals a slightly weird time at the office.
I'm 99.99% convinced that the guy in that video is a mad scientist, er, mathematician.
Mathematician pick-up line: "let your pants be topological"
I want to hang that picture on my wall. My two favorite PBS series hosts.
Is this it? Is this how you turn a sphere inside-out?
same area of maths. same level of weirdness.
The Brony Notion you bet!
+The Brony Notion groan, so cringy
+HerebyOrdinary He's quoting that video. Come on, dude.
HerebyOrdinary I've seen it. Very interesting stuff.
I'm pretty sure that you can stretch a human shape into a t-shirt shape.
Nostrils become the sleeves, mouth becomes collar, and "aboral" end becomes the bottom opening.
Don't forget the ears. BTW that is one disgusting shirt
Do the ears go into the same cavity as the mouth?
You technically have the outer and middle ear separated by the ear drum, with the middle ear connected to your sinuses via the Eustachian tubes. Since these are dead ends they act like the "cup part" of a coffee cup and do not count as holes.
Yeah, that's what I was thinking.
I'm also pretty sure that the bladder is separate from the intestines. So that would leave only four holes going into the digestive system. Making a human a genus-3 surface, same as a t-shirt.
Cajer 1618 don't nostrils have dead end?
I spent a while at the beginning of the video. I paused the video as soon as I heard "if you were wearing really stretchy pants, could you remove them without lifting your feet."
My answer is no, but let's see if I'm wrong lol. I was stuck for a while, but soon realized that the pants can be reduced to essentially a rectangle with two holes, where the two holes come from the two pant legs. The waist part can be expanded and flattened to the floor in the shape of a rectangle. The rectangular part isn't so important, but it is important to note that the pants are homeomorphic to something with two "holes". So from there, I envisioned standing with my left foot in one of the holes, and my right foot within the other hole - While the rest of the rectangular part lay around my feet, on the floor. From here, I guess you could lift the other hole up and over to the other foot, but from there, with the two holes stacked on top of one another, there is no way to remove it from both legs. And I think another thing worth mentioning is that, while the pants have two holes essentially, our entire figure (with out legs, hips and floor making a sort of triangle) has just one hole. The arms and torso, etc., do not contribute to any additional holes.
Now if we take what we've learned from VSauce, we know that the body has a couple of "Through holes," such as the gastrointestinal tract. If we include this as a part of our solution space, MAYBE we can remove the pants!
Let's see if I'm wrong!
Edit: That wasn't even the right question. ;[
There seems to me to be an evolution across the PBS Digital series, where the subjects vary from the very deep to the very accessible, but doing so by being "accessibly deep" and "deeply accessible". That is, seeing the mystery in common everyday observations, and finding clarity in deep and obscure theory.
It's like a boxer working an opponent from low to high and back, but in this case seeking ways to impart knowledge rather than punches.
Thanks!
This is one of my favorite videos to come out of the Infinite Series.
The topic is enjoyable, and the comedy is perfect
Two shapes that are topologicaly equivalent?
A human digestive system. A donut.
You read my mind, seriously.
No because there is different passages that connect
Could you imagine a topological donut in real life? It'd be the ultimate fidget toy.
This happens with my pants every time I try to take them off when I'm drunk😂😂😂
10:57 "negative 1 plus positive 1 plus negative 1 never settles on a value" tell that to numberphile
Under the standard definition of convergence, the sum of (-1)^n diverges. But it converges under a broader definition of convergence called Cesaro convergence. en.wikipedia.org/wiki/Ces%C3%A0ro_summation
Dylan Rambow Indeed. They didn't say that in the video though, did they!
Nope. Usually with any youtube math video, you need to dig a bit deeper to get to the actual rigor of what's going on.
With the logic that 1+1-1+1-1+...=-1/12 you can also say that sqrt(-1)=infinity or -infinity, depending on how you execute the approximation algorithm.
I think you have your series mixed up. The series (-1)^n=1+1-1+1-1+1... converges to 1/2. The sum of all natural numbers, 1+2+3+4+5+... converges (through 'analytic continuation') to -1/12.
Sometimes people ask "what's the point of this?" on math videos.
I feel like the hanging picture puzzle is trolling such people.
I once managed to put a jumper on underneath my jacket while keeping the jacket still zipped up
+
Minecraftster148790 what is a jumper
Pi a sweater you American
You should do that every day to practice mental strength
Then something was probably wrong or out of place, that isn't possible!
"I left the proof in my other pants," has never been more true.
And despite all that topological pant action scenes, still not a single hair moved.
Quote of the day: "Let your pants be topological"
0:00 After careful consideration: yes, by pulling the waistband over one's head, closing it and treating the garment like a pantleg tube. Somewhat acrobatic.
2:35 Seeing this part helped me visualize how the loops swap when just pulling the original depiction inside-out.
2:50 Oh, well there it is.
8:21 Well, that's more efficient.
12:34 Haha, awesome. Both of those challenge winners totally deserved it. Great submissions!
Amazing format for explaining a complex part of mathematics (and one of the most beautiful ones).
To the question posed at 11:15, if you add all of the terms defined by the sequence 1/n^p, the resulting series will converge as long as p>1, no matter how close you get, and it will diverge if p≤1. In a sense, 1/n is the series on the cusp of converging.
Jup, that's what I was thinking about as well. It confused me that 1/1+1/2+1/4+1/8+1/16 .... does not generalize to 1/n^2 but (1/2)^n. So I am not sure whether op means the geometric series or the p-series.
That series is on the verge of converging, but it's a wide verge. The series 1/n for n = 1, 3, 5, ... diverges more slowly so it's arguably "more convergent" than 1/n for n = 1, 2, 3 ... And you can always find a series that diverges even more slowly. So in addition to an infinite sequence of series that are "almost divergent", there is an infinite sequence of series that are "almost convergent".
It seems to me that the ideal answer would be a series for which convergence is undecidable. I have a memory, possibly a false one, of learning about such a series many, many years ago.
Convergence cannot be undecidable if you know the underlying sequence that you add together. Of course, you cannot decide whether
sum_{i=1}^{n}a_i
converges for n to infinity, without knowing what the sequence (a_i)_{i in IN} is. But as long as you know that, there are only two possible cases:
- There exists a real number R such that sum_{i=1}^{n}a_i comes arbitrarily close to zero for larger and larger n (and for any given tolerance epsilon, you can find a natural number n such that the difference is at most epsilon _for every natural number larger than n_).
- For every real number R there exists a given tolerance epsilon such that the distance of the sum up to n to the number R will stay bigger than epsilon _for infinitely many numbers n_.
The way the natural numbers work, you always have one of those two cases up there. The only way to not have infinitely many n with distance bigger than epsilon is to only have finitely many n (duh!), and finitely many n have a largest element. After that largest element, all others will have tolerance smaller than epsilon, or the largest element wouldn't in fact be the largest element for which the distance is bigger than epsilon.
+Franz Luggin I am not sure you know what undecidable means. The question whether an algorithm terminates or not has also only two possible answers but this problem is undecidable. In general, the question whether a series converges or not is indeed not decidable, simply for the reason that there are uncountable many series.
As I said, if you know what the sequence a_i is, i.e. if there is one sequence given, you have enough information to decide what the answer is.
There are convergence tests that work even though you do not know what the limit is.
By the way, genetics has ties to knot theory which is related to topology. Bacteria use special turing-like "algorithms" to untangle their own DNA when it gets knotted. They do so using a mathematically minimum number of DNA slices and how they do it exactly is not only not known by scientists but also requires solutions to problems in knot theory that we still don't know yet.
(correct me if I'm wrong - my information is not completely upto date.
Here's some fun: I thought of a good idea... find a set of 4 characters in the ascii extended chars that can represent the gene sequences A T G and C so that when negated, the image is it's opposite (A T) (GC).
Here's what I came up with:
A = • (black circle in a white box)
T = ◘ (white circle in a black box)
G = ○ (black ring in a white box)
C = ◙ (white ring in a black box)
If you express the Genes rather than atgtactgtca, but as it's RNA/inverse-RNA pairing is it would be in nature (nature's redundancy plan) then inverted blackwhite, equals it's flipped image...
cggatttagcgagtaattctacgagaatagcgactgtaagtacggacttggcaagtaatt
gcctaaatcgctcattaagatgctcttatcgctgacattcatgcctgaaccgttcattaa
◙○○•◘◘◘•○◙○•○◘••◘◘◙◘•◙○•○••◘•○◙○•◙◘○◘••○◘•◙○○•◙◘◘○○◙••○◘••◘◘
○◙◙◘•••◘◙○◙◘◙•◘◘••○•◘○◙◘◙◘◘•◘◙○◙◘○•◙•◘◘◙•◘○◙◙◘○••◙◙○◘◘◙•◘◘••
By the way, this may work better with a monospace font like Lucida Console.
Just bored and looking for things to entertain on the weekend. :-)
I'm in 10th grade and I prepared a project on Topology just a day before the Science Project Exhibition.
I explained it in such a way, so much mathematical way that even the research scientist failed to understand it.
For the interlinked rings and the double donut, you can also start by shrinking the big loop until it's just a short conduit between the rings; for convenience, you can rotate the rings to put the conduit at a point where the circles cross visually in the illustration. Then it's just a matter of straightening out the rings so they point in opposite directions.
For the pants, I first pulled the waist down to my ankles, then pulled the ankles up to my thighs; this turns the pants inside-out, but now they're upside-down. However, since they're topological pants, this is easy to fix! Your legs and the planet you're standing on form a ring shape like a donut; just rotate the pants around that ring till they're right side up! First the entire Earth goes through one leg, then your upper body goes through the other leg.
Topology is my new favorite math subject.
Here's how I'd hang the picture from two nails: Run the string clockwise over the left nail, then counterclockwise around the right nail, then counterclockwise around the left nail, then clockwise over the right nail. This is a 3-strand braid with two strands stretched straight and turned into nails. It's also symmetric.
Ah pleaso more on the fundamental group!
That question at the beginning is just how I greet people
I so love topology, it is just mindblowing on the simplest things :))))
Please do the rest of the series in those pants
Smooth.
I knew i wasn't the only one hounding
And turn them inside out at the end of every episode.
I want to know what black magic is keeping them on her hips.
Daniel Smith She's topologically equivalent to Shakira.
Ive only watched about 2 videos of yours and i am addicted to this!
Great session. Im terrible at all arithmetic disciplines,but I enjoy watching. Love the pants. Quite flattering. The colours well matched the form. I'll keep watching until I start to understand.
I have lived in a 3D space for my entire life but topology never fails to confuse me.
Why would anyone want to hang a picture using two nails so that if either breaks the picture falls?
Half the point of using two nails is so that if one fails there'll still be something holding the picture on the wall until it can be fixed.
Your hair-style perfectly explains topology.... Loved the vid :D
Why are you not breaking the rule about glueing an existing hole closed at 3:13?
I'm not really sure how to explain that hole disappearing. I was gonna say the hole was just an illusion, but it's not. Confusing. Anyhow, to get intuition about how this thing works, try this:
Press your thumb and middle finger together. On other hand, lock your other thumb and middle finger with first one by doing the same, so your thumbs and middle fingers now form 2 circles and without releasing your thumb and middle finger on either hand, your hands are stuck together. Now because these two rings are connected by your hands, arms and chest, you actually have topologically equivalent situation to the starting position there.(If you're re-reading this because you think you found an error, instead of thinking chest as the connecting piece, lock your elbows together so you get smaller loop)
What this morphing she described does, can be explained by simply this: Connect your elbows and wrists, pull your thumbs, sort-of trying to break free from their interlocked status. You now notice that you have managed to smoothly join your hands together and form the exact same shape as in the video.
Hey thanks for your reply! And yeah i see it now. I suppose you can straighten out the thicker curve and then shrink it so that the two rings are touching where the thicker curve connected them.
I don't see any hole being glued in that part of the video. It's the inflating the tube-shaped connector into a sphere shape.
It was not glued. It's still hollowed but the radius of the hole is small.
Why are they only 2 holes ? I see 3 holes in the arrangement... What exactly is defined as hole and what not ?
Another answer to the second question asked at the end: If a_n is a positive sequence whose series diverges to infinity, then there is always an asymptotically smaller sequence b_n which still diverges. By asymptotically smaller, i mean that b_n/a_n -> 0. The same is true for convergent series just in the other direction. This shows that there isn't really any edge for these things.
Without pushing together any prexisting holes 1:01 -> Now inflate the main loop bigger and bigger until it looks like a ball. 3:12
My solution to the pants problem was to turn the left.leg/Earth/right.leg loop into a torus. The pants became a toroidal sheath with a hole in it. I then stretched the hole all the way over both toroid and sheathe. Inside out pants!
12:47 "And they're being attacked by Pokemon". Thank you, Kelsey.
The simplest case I can think of is a sphere and a bowl being topologically identical. If you've ever handled a ball that is completely deflated so that it collapses in on itself, it forms a bowl shape. Add air, and it takes on a familiar sphere shape (at least approximately).
i can do this with regular pants/jeans, turn them inside out, but in the end they lose some of their purpose. you pull down the upper part, then pull the insides of your legs up to your waist. now you still "wear" your pants but they are inside out, without lifting a foot.
I was thinking to just nail one of the nails in deeper into the wall so the head of the nail is under the second one (because you put em close n shit), so when you take it out, the other one comes out without you doing anything :P
The pants thing was awesome.
the question about border between finite and infinite series, well, you can distort the sequence 1/n into 1/n², but the obvious one is to let the power change continuously. what I'm going at is the zeta function = 1 + 1/2^s + 1/3^s + ...
if this is what we choose, then actually s = 1 the harmonic series *is* the edge. because choosing any power slightly larger than 1 gives us a finite sum, and from 1 and below the series diverges to infinity. (for values less than 1 or 1+imaginary part, zeta is no longer defined as the series)
That reminds me of a part of topology called Non-Orientable Manifolds, and it specifically reminds me of möbius loops. You can connect two möbius loops (one left-handed and one right-handed), each one having only one edge, and the combination of them will produce a shape that requires four spatial dimensions to exist called a Klein Bottle. It has no edges and only one side. In three spatial dimensions it intersects with itself in a given location, but in a fourth spatial dimension this never happens.
I would totally use those pants. I would be constantly be flipping them just to mess with people
When iterating ƒ(x) = x² + c with certain real values of c, the number will bounce around infinitely between 2 and -2 never coming back to the same value. I just love that. ^_^ This is very strongly related to the Mandelbrot set.
Topology; studying surfaces in reference to holes
Bottomology; studying holes in reference to surfaces
There was a proof I saw somewhere that for any series that diverges to infinity, there is a sequence that diverges slower. There is similarly a proof that for any convergent series, there is a series that converges slower.
Thank you very much to you and youtube.
Genus Various: M. C. Escher's wood block prints included several in which the subject was two intertwined, but completely disconnected, worlds. Typically they would have different themes, such as light and dark or summer and winter. Anyway, both those worlds and the negative space surrounding them have so many holes, and (what is perhaps more interesting) they are quite constrained from any meaningful simplification without unacceptable crossing of boundaries. So that's my example, and it's weird like Escher.
I love this. Because I am highly interested in the topology of the universe, and the other fields that are around us must also have topologies and would be mind annihilating to keep in 3D, probably.
Hi! I loved this video. Back in the day, I couch surfed in the home of a UC Berkeley mathematician at the time when Grisha Perelman announced his proof of the Poincare Conjecture. It was all "yeah, we'll see" at the time. What ever became of it?
I remember a hastily called meeting at the Uni.
That framed picture is amazing
I remember a pictorial essay in Esquire mag about prisoners figuring out a way to take their pants off while wearing ankle shackles. They did pick up their feet!
1. yes but not in public-slide one pant over your body and bring the other pant back;
2. and do the hokey-pokey and turn yourself about-that's what topology is all about;
3. if you require using the waist opening, slide the pant over the whole Earth instead.
omg that fanart of Matt and Kelsey, tho lmaooooooooo
Loved the pants turning out. :)
The big ring with two connecting loops appears to have the big ring hole glued together into a big ball to make the 8 shape.
Fun stuff, I just finished BS in Math a few weeks ago and my last math course was an independent study in Algebraic Topology!
Are you familiar with the recent research in Homotopy Type Theory?
Essentially, you apply the intuition of topology to the notions of term and type. Types are recognized as spaces in the same sense as a topological space, and terms with a certain type are regarded as points in those spaces. For example, the natural numbers are points in the space Nat. Under this scheme, equality of terms (a=b) is a path between terms (a path starting at a and ending at b).
Also regarding the question about the harmonic series, I'm reminded of the Kempner series: basically, if you remove terms from the harmonic series which contain any particular string of digits in the denominator in any particular base (originally, any term in base-10 containing a "9") the series converges. Given this and the incredibly slow rate at which the harmonic series diverges to infinity, it's always felt to me like the harmonic series itself exists on a sort of cusp between divergence and convergence, at least in the traditional (non p-adic) sense of convergence.
[05:22] how do you contract a circle to a point-without condensing its finite interior → 0 (you'd be able to do 'most-anything' by merging 0-width-points and expanding them out).
Theres no reason you shouldn't be allowed to do that
Well, it never actually gets to the point stage. It just approaches it
I don't know if this is helpful, but I'll try: The intuition behind those loops is that they start somewhere on the surface of the shape. It could be any point really, but let's mark that point with X. The idea is that, through point X, you feed loop, so that you hold both ends of the rope, then from your hands, each end of the rope connect to the point X, and then they go doing loop things on the surface of that shape.
And the intuition with that circle shrinking away is that for sphere, I can simply pull both ends of the string and take the rope back to me. But in case of Donut shape, it forms a loop, so I can't pull it back without letting go from one end or the other(which isn't allowed).
The rules are that, I can move the point X around freely(there are some rather mild restrictions to this in some edge cases, but for purposes of this video, it's completely free decision on your part where the point X is). I can also pull both ends of the rope, or give more rope, but I can never let go of either end of the rope. If given these rules, I manage to go from one loop to another, then these loops are considered the same. So in case of sphere, every loop can be made into "rope completely pulled back in", and every loop can likewise made from "rope pulled back in" position by releasing some rope, so every loop is the same. In donut case, you can see how there are different kinds of loops which we cannot make into one another. Like, no matter how much you pull or release the rope, two loops and one loop can't be made into one another.
Romaji, it does get to 0. Consider the map f(r, t) =(r*cos(t), r*sin(t)) for r between zero and one, and t measuring the angle. For r>0 this is a circle, but for r=0, f(0,t)=(0*cos(t), 0*sin(t))=(0,0) so is just a point, regardless of the angle t. This is is an example of a homotopy between two maps.
Why do you allow sewing holes in 2 dimensions like that but not in 3 dimensions? Inconsistent argument.
0:06 Yes! Don't wear the pants while turning them inide out!
When Kennedy said, "I'm a doughnut" he reminded that (wo)man's digestive system is like the hole in a doughnut, that we are topologically equivalent. Come to Berlin and run Axel Flinth's lecture!
As regards turning pants inside out, it'd be more practical to turn the knickers from time to time...
A pretzel and a bottomless cup with two handles.
i actually really like the picture in the picture frame.
1:00 you cant close the hole into a full ball
3:12 close the hole into a full ball
Beautiful and smart, nature should make lots of copies of u
I figured I could just drop my pants, and then pull both legs up through the waist until they're inside out.
You never said I had to be wearing the pants. Only that they had to end up inside out, and I couldn't move my feet.
You know that satisfying moment when you actually use the think break to your advantage and end up getting the right answer? (I got the one at 6:13)
YAY TOPOLOGY!!!!
For the first time, today I noticed that the picture hanging from two nails depicts Kelsey herself and Matt from SpaceTime!
Pfft, Mr Bean has you beat. He changed out his underwear for swim trunks without taking off his pants.
This made me think of math proofs. Where you can extend (bend) any of the current proofs to (shapes) to create a new figure (new proofs and branches of math), but they must all be topological equivalent, for else it is not the same. So no matter how much you extend math it is always the same, just we learned more ways to bend it.
That's broadly speaking one of the ideas behind homotopy type theory.
thanks to pbs and her for the extremely good videos.
5:37 thats herself in the pic ? :)
About the question at the end of the video:
I think that in real life there are no infinite numbers, because infinity is a cycle with n-numbers and we think that it has no end and we do not know what is equal to n. The difference between them is that the first set has a bipolar separation: known and not known, and the second one is n-polar: the known, not known, and the order-of-cycle index, which in the example is shown as the power of the number.
Thanks for attention, sorry for my English.
Mathematics: Teaching us worse ways to hang a picture frame
As long as your head isn't against a surface (like your feet are), it can be done.
0:05 You don't need stretchy pants, you could do that with jeans. They're be on the floor rather than on you, but it's possible.
This is awesome!
A bit late, just saw your 11 May 2017 episode.
Here are two 4D (or perhaps 11D?) topological shapes with unknown topological complexity which I am pretty certain are topologically equivalent, at least on a coarse grain scale.
1. The entire universe right now.
2. The entire universe ... just now.
"Sketch Proof":
Use the laws of physics time cobordism to deform one into the other, assuming a scale at which smoothness assumptions of general relativity apply and that no new Black Hole singularity was created in the intervening seconds.
Another good answer to the question about the edge of infinity (regarding the sums, 1+1/2+... and 1+1/4 + 1/9+ ...) is the following:
The sum of 1/n diverges. But 1/n^2 converges.
The sum of 1/nlog n diverges. But the sum of 1/(n log^2 n) converges.
The sum of 1/ (n log n loglogn) diverges. But the sum of 1/ (n log n (loglog n)^2) converges, and so on.
So if you take the product n times log n times loglog n times logloglog n times ... loglog...n (k times). The sum of its reciprocals diverges. But if you square the last term, it converges. So the edge is sort of n log n loglogn logloglog n ....
Hey whadaya know! A math concept that comes naturally to me! Woo hoo!
Frequency merge gives topology. To remember topology know the frequency numbers. More than three is beyond scope of universe.
Superglue your shoes to the floor; While this step isn't strictly required, it helps for proving the authenticity of the solution. Saw circles into the floor around where your shoes are attached. Pop out the pieces of floor and pull the legs through past each respective wooden circle. Now that the pants are separated from your body, turn them inside out. Then slide your feet out of the shoes and put the pants back on. Now pay thousands of dollars to fix your floor.
+
So I tried taking off underwear without lifting my feet from the floor.
I suddenly need new underwear, and I have an increased appreciation for thought experiments...
😂
For the question posed at 11:30, I posed the same question on reddit's r/math. It's proven that there can be no such function for Sum(1/f(n)) such that any function asymptotically bigger converges and any function smaller diverges. However, there's a really interesting boundary where nlog(n)log(log(n)) diverges but nlog(n)log(log...log(n))^2 converges. However, that's not rigorous enough to be an answer.
Here's the post: www.reddit.com/r/math/comments/5h6ck0/smallest_function_whose_reciprocal_partial_sums/
The fact that two shapes can be continuously deformed into each other implies that they are topologically equivalent, but the reverse is not true. Two topological spaces are said to be topologically equivalent if there is a continuous bijection between them whose inverse is continuous. For instance, all knots are topologically equivalent [to a circle]. It would be good to be precise in a channel about math.
If the person wearing the pants were infinitely tall but the earth was still considered finite, you could still turn the pants inside out if you can pull it around the world.
You can turn a horshoe into a button! And vice versa!!! I solved a math problem good job brain! I want those pants!!!
If you take the 1/n in the harmonic series, and raise the n to any power greater than 1, even if it is only 1 plus an arbitrarily small epsilon, then the resulting series is convergent.
7:39 - Mind blown...
A human is topologically equivalent to a hand spinner with a hole in the center. Both has four holes.
best book for topology
I saw the linked chains unlinked and I STILL don't believe it.
when somebody asks you whats your favorite food and you dont want to make yourself look fat so you say "I like tori"
look it up if you dont get it
pun_resistance.exe has stopped working
Challenge problem:
A Hole in a Hole in a Hole is actually the same thing as a bowling bowl with its 3 holes piercing through.
But I only know that because I cheated and watched Numberphile.