Topology Riddles | Infinite Series 

that only works for the mugs, although the thought of coffee in a shape that's homeomorphic to a donut is quite funny

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.

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!!

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.

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?

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.

You read my mind, seriously.

No because there is different passages that connect

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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!

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.

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.

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 ?

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.

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.

you mean besides everyone?

naturally xD

Anyone else think Matt is really attractive?

I'm seriously starting to ship these two.

It is a highly corrosive substance. Our bodies are slowly burning from the inside out. Why else would we be giving off so much heat?

As regards turning pants inside out, it'd be more practical to turn the knickers from time to time...

there is no donut.. was discussed here, get your thumbs and index fingers together and get your hands together. Are you a donut? no

+nooneofinterest I That one tricked me too at first, but then I noticed that the large hole in the middle isn't really a hole. The 2 ends of the loop aren't actually connected, so technically that "hole" is just a loop.

Greydon Iselmoe you are as long as the can is opened.

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look it up if you dont get it

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