Building an Infinite Bridge | Infinite Series 

Tehom Yes, the image bothers me because it's wrong. I'm pretty sure that the mass with it's center of mass not touching a solid surface doesn't look down and say "I'm not supported but it's okay because there's this fancy little curve further down so I'll just pretend I'm supported by that instead."

Matthew Fennell But would it be stable? Where would its centre of mass be; would it even have a centre of mass?

TimGB Just place the center of mass even with the edge of the table. The infinitely long brick would stretch infinitely in both directions keeping the center... in the center. Trick 2: turn off gravity. I mean, while we're just doing _whatever_, might as well. Glue the bricks together so they don't float away.

Matthew Fennell an infinitly long brick wouldn't have a centre of mass or would at least never topple, because if the brick just slightly tilts, the points far away would have to travel faster than the speed of light, wich sadly isn't possible

You'll need an infinitely long table. I win.

Then you don't even have a bridge. You're disqualified.

So I'm not the only one who was irritated by that?

KinoftheFlames agreed

Well technically it's still possible to do it on a table with a curved edge except for one thing they were using the wrong balance point. The correct balance point would be where the base of support ends ie the moment that the surface shifts from being tangential to Earth's surface to curving towards the surface past that point it wasn't supporting the block anymore and the very first block would have rotated and flipped off the table as a result as they got the balance point wrong!

Proof that the CG/Animators did not listen to her at all

implying that a non curved edge exists outside a mathematical model.

Yeah I realize it doesn't make sense I just needed to make a -1/12 joke

And it's a good thing. If the maximum overhang went as the sum of natural numbers, and you had an infinite number of blocks, and you put the edge of the top block any further out than -1/12 - that means it's still entirely on the table by 1/12 of a block length - it would fall off.

-1/24, you need to divide it by two first.

You are right, I missed the factor of 1/2. So you can get the farthest edge of a block almost to the edge of the table before they all fall off.

-1/12 isn't the sum of all natural numbers. That's infinite. -1/12 is just the point on the Reimann Zeta function which would correspond to n= -1 if the function was actually defined there. Which it isn't.

I have watched 3 of her videos, this being the third, and have already subscribed and hit the notification bell.

but if it's infinite, how do you even find the edge to dive off of it?

CogniVision Good point. Maybe it's just an infinte balcony, then?

+Noel Goetowski Good point, but, uh, don't you need to get to the end of a diving board to make a dive?

You just need to go infinitely fast to get there.

They should just call it an infinite road and make it with yellow bricks.

I think she said, some infinte series converge to a finite value...

Maybe PBS Infinite series is actually finite and will not last forever.

This comment hasn't aged very well...

@@galladeguy123 :(

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Found the Physicist.

actually we found the engineer

I vote physicist. Physicists work with idealizations, and there's a lot of silly stuff you can tweak to idealize this, like... buoyant forces (put it in heavier air) static friction with the table (bricks are made of tape that sticks to the table) air flow (blow on the bottom of the bridge so it stays up) gravity (put another Earth-mass object on the other side of the bridge so it doesn't fall toward Earth) the weight of the duct tape (have it lighter than air to where it'll float upward with the same force that the tower would fall downward) Wingardium (Leviosa) thermal effects (heat the bridge so it expands until its density lowers enough to float) electromagnetism (think magnet trains) fluid effects (pressure within the bricks to make them extend like those long balloons do)

Free Diugh The engineer answer is "an infinitely long bridge is easy, just use an infinitely heavy counterweight"

char whick But that's only if the brick is infinitely strong. Otherwise, it'll bend downward under its own weight as it stretches further outward. Physicists can make those idealizations, while engineers can explore them to find a system's physical limits, but an engineer would lose the job pretty quickly if that was the kind of answer the engineer gave to an employer.

i am with you, they should really write fraction as simple fractures and not in the format like here it drives me nuts seeing that

That's pretty neat! Thanks!

What is that? It sounds cool.

That is a cool question. I will have to think about it.

Felix Windisch I have the same question, although I suspect there's some flaw in this construction that both of us don't realize.

Famfly It'd be physically impossible, but so would all the bridges that were discussed be

yeah I thought the same...

It would fall after some blocks, she even mentioned it in the video. She doesn't say upside down pyramid, but she says parabolic shape, which is exactly the same, unless you meant actual pyramid with base of triangle or square or other 2d shapes, in which case idk my guess is outcome would be the same.

Great question! In the article where they describe the parabolic stack, they explain why this doesn't work. It's linked in the description under "Overhang article"

share it on google disk or something

loljk

and then he died. lol

The situation assumes no welding- that is, the blocks have to be held together by gravity alone. If this were the case and it was infinitely strong, there's nothing stopping us from welding them end to end and placing it like one large first block extending infinitely far in both directions, which is kind of boring.

John Plissken There is also the small problem of finding sufficiently many bricks.

Erdmännelchen The harmonic series is right on the border of being finite. For a series of elements 1/n^p, if p is greater then 1 then the series converges. If 0 < p

*Answer 1a*. 1/(n*log n) diverges and is between 1/n and 1/n^2. *Answer 1b*. 1/(n*log(n)*log(log(n))) diverges and is between 1/(n*log(n)) and 1/n^2. *Answer 1c*. ... continue in this fashion. *Answer 2*. Suppose you have a series of positive numbers x1+x2+x3+x4+... that diverges. Then you can find another series of positive numbers y1+y2+y3+... that diverges `infinitely slower'. By infinitely slower, I mean that the ratio (yn/xn) tends to zero as n tends to infinite. *One proof of Answer 2*: We know x1+x2+x3+x4+... diverges. This means that the sum eventually exceeds 1, and that it eventually exceeds 2, and eventually exceeds 3, and so on. For the sake of illustration and ease of notation, imagine it takes 100 terms to exceed 1, and the next 900 terms to exceed 2, and the next 9000 terms to exceed 3, and so on. Then we define the new series y like this: x_1,x_2,x_3,...x_100, (x_101)/2, (x_102)/2,...,(x_1000)/2, (x_1001)/3,(x_1002)/3,..., (x_10000)/3, (x_10001)/4,... Need to check two things: this new series diverges and that the ratio y_n/x_n tends to zero. *y_n/x_n tends to zero*: y_n/x_n starts off as 1, then after 100 terms the ratio becomes 1/2, then after 1000 terms the ratio becomes 1/3, and after 10000 terms becomes 1/4 and so on, getting closer and closer to zero. *y_n diverges*: By construction, y_1+...+y_100 >1. And y_1+...+y_1000 > 1 + 1/2. And y_1 + ... + y_10000 > 1 + 1/2 + 1/3. Continuing in this fashion, the right hand side is becoming the 1/n series, which we know diverges, and hence sum of y_n diverges. *Answer 3*. Naturally, your question actually still stands. Is there an edge? Sure, using Answer 1 or 2 repeatedly, we can create infinitely many (countably infinite) diverging series, each much slower than the previous one. But can you go beyond that? The answer is yes, using some kind of diagonal argument. You can find a series which diverges and which is slower than all the series constructed previously. And now we can keep on repeating. There is a technical sense in which this process has to end, which involves having to talk about ordinals! So it looks like you asked a deceptively simple yet highly interesting question - I am glad the next video in the series commented on your comment!

You can try to change value of exponentiential: www.wolframalpha.com/input/?i=sum+of+1%2Fn%5E1.5 and try to approach 1: www.wolframalpha.com/input/?i=sum+of+1%2Fn%5E1.01 it seems to work until you're at 1 ...

Christophe Andre. There are more things you can do than change the exponential. E.g. the sum of 1/n*log(n) diverges and is in between 1/n and 1/n^2.

You're actually asking about the value of the Riemann Zeta function : en.wikipedia.org/wiki/Riemann_zeta_function at the value s = 1 and at s = 2 This function is defined for all complex z such that Re(z) > 1 =, hence the series converges for all real numbers in ]1;+infinity[ . The function also has a pole at s= 1 since the series goes to infinity. So the value at the edge of being infinite would be something like 1+ds where ds is an infinetly small positive quantity.

heck, if we presume we can balance a brick on it's middle point, why wouldn't (1,2,3,4,5) work?

They also ignored forces between blocks and the fact that the force of gravity on each block would decrease as the bridge stretched farther from Earth. Frankly I doubt they've carried out an environmental impact assessment or even considered how to fund this boondoggle. This is why I love pure math!😃

+Karma Arachnid Indeed there are of course a whole bunch of issues preventing it being an engineering solution. Other factors would be the fact that eventually the weight of the blocks will be so great the bottom blocks would be crushed into dust even if you use infinitely strong blocks you also can't have loose blocks stacked to infinite height. Above 36,000km the top blocks would be at a height where the centripetal force needed to keep it rotating at 0.26 rad/hr would exceed the local value of g causing the blocks to drift upwards from inertia. Fun to think about but physics has to go spoil it heh.

What do you think this is??? A Physics channel??? ;)

Why so hostile?

Not hostile, just poking fun Edit: that's why there is a ";)" at the end but text isn't alway good at showing tone

Cool guy 49 I suspect that is a problem of geography.