Gabriel's Horn paradox (finite volume but infinite surface area)

Start learning today, click brilliant.org/... to check out Brillant.org. The first 200 people to sign up will get 20% off your annual premium subscription!
Finite Volume but Infinite Surface Area? Check out Gabriel's Horn paradox (aka painters paradox) here: brilliant.org/...
Review disc method, • Volume of the Solid of...
Review surface area, • Surface Area of Solid ...
☀️support bprp on Patreon: / blackpenredpen

[@osvaldomena1172]

So we can fill Gabriel's horn with paint, but we can't paint it 😅

[@btat16]

Osvaldo Mena Wouldn’t that, then, completely paint the inside of the horn? My head...

[@osvaldomena1172]

@@btat16 No because the internal surface of the horn is infinite too! 😅😫

[@btat16]

Osvaldo Mena but. But... if you fill it up wouldn’t it be touching everything? It hurts

[@morganthem]

@@parkershaw971 how we has so small no volume is in but area still there?

[@mithileshwadurkar8809]

We can paint it if the thickness of the layer coating the horn is infinitesimal

[@alejrandom6592]

I'm a simple man, I see Gabriel Iglesias and a suface of revolution in the same thumbnail, I click.

[@EchoHeo]

Or.....get a fractal bottle

[@1nd93dk3]

isn't it like a fractal?

[@maxhaibara8828]

Q1. Proove that there is an object that has finite volume but infinite surface area Ans : Notice that if there isn't, then BPRP won't make any video on it. Hence proved.

[@gdahs1473]

good trick :v

[@gamecoolguy619]

Ummn the area at x=1 is pi... So unless pi + {fraction of pi} = pi Which based on how your limit works is true, but logically you agree with this?

[@tubax926]

Proof by necessity

[@Nothing_serious]

It's referring to the angel Gabriel. Gabriel has a horn which he blows for seven days before the judgement day.

[@rayniac211]

Where judgement day refers to the point in time where Gabriel's neighbours finally lose patience and decide to do something about the noise.

[@JdeBP]

Actually, in the Bible it is seven angels with trumpets, and none of them are named; and Gabriel isn't described elsewhere as having a trumpet. "Gabriel's horn" is an affectation of 20th century recreational mathematics authors that didn't occur in the centuries before, or even before the 1980s that I can find. It isn't actually the proper name. Evangelista Torricelli named it the solidum hyperbolicum acutum, where more strictly this is a truncation of that solid, and the original theorem had a cylinder from x=0 to x=1 which the calculus proofs miss out.

[@shahjahonsaidmurodov]

And in the Quran, the angel blowing the horn is actually called Israfil

[@victoirevim9698]

Pi * r² * THICCness

[@seeseefok7659]

t h i c c

[@bengtbengt3850]

This is great, but I can’t believe you didnt mention the following: Since the area is infinite, there’s no way you can paint that horn with a finite amount of paint. However, since the volume is finite, you could just dump a finite amount of paint in it and the entire horn would be filled. If the entire horn is filled, then obviously the sides of it are painted as well. Therefore, you have painted the horn with a finite amount of paint after all, even though we just stated that that wasn’t possible with a finite amount of paint. I guess we have some sort of paradox, right?

[@Tumbolisu]

I think the problem comes from the fact that you are treating paint as both 2 and 3 dimensional.

[@dlevi67]

If you can make paint thinner than a molecule and find an infinitely tall ladder so you can pour the infinitely thinnable paint into the mouth of the horn, no paradox required. ;-) (No more paradoxical in any case than any infinite series converging)

[@stevethecatcouch6532]

@@dlevi67 You pour the paint into the bell of the horn. The mouth is inaccessible, so it can't be blown and the universe will never end.

[@stevethecatcouch6532]

No paradox. The paint, while in the paint can, has finite volume and finite surface area. As it is poured into the horn, the surface area increases. That's to be expected, because the surface area of paint always increases as you apply it to whatever surface you are painting. It just increases by a bit more than usual when it's poured into the horn.

[@Nothing_serious]

It was discussed in Numberphile I think.

[@WhattheHectogon]

What about the opposite? Is there any "object" with infinite volume, but finite surface area?

[@AnimeLawyers]

An inverted ball. The ball has a finite surface area. The volume of outside the ball extends to infinite time and space.

[@BinuJasim]

No. just check wikipedia en.wikipedia.org/wiki/Gabriel%27s_Horn It has a proof why the opposite can't happen

[@BinuJasim]

@@AnimeLawyers Why do you need an inverted ball? Simplify further. Let's say the universe and whatever beyond it. Infinite volume. Zero surface area.

[@stevethecatcouch6532]

@VeryEvilPettingZoo The surface area of A's boundary is nonexistent, not 0.

[@AnimeLawyers]

@@BinuJasim You....... took that far too seriously......

[@Dunkle0steus]

My first instinct is to imagine a cube with dimensions 1x1x1. Then add a rectangular prism with dimensions 1x1x0.5 on top of it. Then a prism 1x1x0.25 on top of that. You can add infinitely many prisms each half as thick as the previous. Each increases the surface area by at least 2, because it still has two faces with dimensions 1x1. The volume is 2 though.

[@MagnusSkiptonLLC]

That's pretty much a low-res version of Gabriel's Horn.

[@Dunkle0steus]

tbh I didn't watch the video, I just made this comment after the introduction

[@WhattheHectogon]

@@Dunkle0steus also vsauce

[@hedgechasing]

Dunkleosteus when the dev of TFC+ randomly pops up in the comments of an unrelated video... thanks for your work on that. :)

[@Dunkle0steus]

Whoa I never get recognized. Thanks!

[@NuptialFailures]

Interesting video! Gabriel's horn has an interesting property of having constant negative Gaussian curvature. Do you think you could do a video on the calculus of that in the future (as you can relate it to the surface area you calculated here)? Thanks!

[@blackpenredpen]

I can try!

[@xaxuser5033]

《Ofc the answer of this is yes cz otherwise how can i make this video》you're such a legend

[@l3igl2eaper]

A funny way to think about it, is that you can paint that horn of infinite surface area with a relatively small amount of paint.

[@ericherde1]

The angel Gabriel, from a scene in the fever dream know as the Book of Revelations.

[@XanderGouws]

Yeah, I think the mathematician that came up with it was called Torcellini, so sometimes it's reffered to as "Torcellini's trumpet/horn"

[@jessehammer123]

Xander Gouws Torricelli, not Torricellini. Close, though.

[@coleozaeta6344]

It was Squidward Tortellini.

[@WerewolfLord]

Revelation.... aka John on LSD.

[@blue_blue-1]

The revelation is serious.

[@mhm6421]

Blackpenredpen can't do perfect cuts- 5:22 timestamp: Are you sure about that?

[@zygaf6252]

Like fractals.

[@MrCigarro50]

But this is not surprising. The Gaussian integral has área sqrt(pi) and the length of the curve is infinite. Anyhow Gabriel´s horm is a fantastic example. Thanks Professor.

[@danielbenyair300]

1:17 what if we use zero and not 1? 1:22 how can we express it as 3D function? (X,Y,Z) 1:30 why not cut it horizontly? (V=2×pei×( area of 1/x) should work right?)

[@snfnsessizcocugu7884]

no, you can't evaluate the volume that way. try the same formula on a cone and you'll find a wrong value for its volume

[@shawnheneghan4110]

The paradox is simply resolved. Yes the volume is finite and so it can be filled. Unfortunately it would take an infinite amount of time to fill it. Therefore the surface cannot be painted.

[@ambrisabelle]

It doesn’t take an infinite time to fill, just pour in paint with a flow of pi volumetric units per second and it will only take a second.

[@rorythompson864]

omg u have solved years of scientific debating, i will call the noble prize people for you now !

[@siekensou77]

Pi is such a strange/mysterious and interesting number

[@blackpenredpen]

Definitely!

[@adaeptzulander2928]

This is a classic I learned WAY back in high school. You can fill it with water but you can't paint it. So what would happen if you filled it with paint? ! 🤯

[@WerewolfLord]

You get to a point where the diameter of the horn is smaller than the diameter of a molecule of paint.

[@stevethecatcouch6532]

@@WerewolfLord Mathematical paint molecules can be shrunk to any desired size. Besides, bringing up reality to explain why you can't paint a non-existent horn with impossible paint feels like cheating.

[@blackpenredpen]

: )

[@angelmendez-rivera351]

There is no paradox. The statement that you cannot paint it is simply false. Infinite surface area does not actually imply impossibility of painting. What implies impossibility of painting is the ratio of the volume to the amount of the surface area, and this is clearly finite.

[@sensei9767]

Physical paint can't fill areas, it can only fill the space on top of it, since it's an 3D object. So if you want to know how much paint you need, you have to calculate the volume of the paint. The paint is a small layer on top of the horn. The diameter of the paint is approching the paints thickness, since the slope is decreasing which means that the paints diameter isn't approching zero which means the paints volume is infinite. the diameter of the paint inside the horn is Always decreasing, so it will eventually be thinner than the regular thicknness of the paint.

[@anegativecoconut4940]

8:06 Hello this is Chef Steve from MathWishes.com

[@qillerdaemon9331]

You are after all the Major General for solving your integral!

[@fCauneau]

A similar case is the rectangle of area p*1/p : whatever the value for p, you can fill its area = 1, and the perimeter tends toward infinity whenever p goes to zero or infinity ;-)

[@spudhead169]

Gabriel got some serious braggin' rights.

[@U9191-e6s]

This a analog to any function with convergent surface area below but still a divergent arc length .. we want a video to discuss this issue 😁

[@Silver_G]

Please talk about the Sierpensky's Carpet or Manger's Sponge (too lazy to look it up for correct spelling but should be close)

[@dlevi67]

Sierpiński and Menger respectively, though the objects are generally indicated without the diacriticals ('Sierpinski')

[@sergioh5515]

Still treating infinity like a number...tsk tsk...sigh

[@TheBlueboyRuhan]

He didn't though; it was stated that it was limiting towards zero. And you can try deriving from first principles or look at cauchy's work - it works out

[@General12th]

I know, right? What a bother! Gauss is rolling in his grave right now. tsk tsk indeed. I wouldn't let anyone teach our children math like this.

[@mike4ty4]

It doesn't behave exactly like the real numbers but if you treat it according to its own laws, it won't bite you. It's an "extended real number", technically - or at least that's one way to make it rigorous. In particular, it is okay to: 1. Absorb any _nonzero_ number into oo multiplicatively: a*oo = sgn(a)*oo, so long as a != 0, and for -oo it's just the opposite 2. Absorb any number _greater than -oo_ into oo: a + oo = oo, so long as a > -oo, and the just opposite rules hold for -oo 3. Divide any number by oo _except +/-oo_ : a/(+/-oo) = 0. 3. Generally speaking, a function f(x) is evaluated at +/-oo by taking its limit as x -> +/-oo. If no limit exists, such evaluation is impossible - the function would have to be defined at those points separately if wanted. But you cannot: 1. Multiply oo (no matter the sign) by 0: 0*oo is a NO NO 2. Divide oo by oo (no matter the sign): oo/oo is NO NO Both of these are undefined operations. Moreover, unless you treat +oo = -oo, which forms a different kind of "extended real number line" called the "projective" extended real number line, you still cannot divide anything by 0, including oo itself. (Note that since subtraction and division correspond to adding negatives and multiplying reciprocals, their rules for oo follow from the above rules.) If you follow these rules, oo does indeed work consistently as a real mathematical object. Whether you want to call it a "number" or not is up to you.

[@blackflash9935]

mike4ty4 The author of the comment probably understood that before making his comment so I can safely do this: *breaths in* R/WOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOSH!!!!!!

[@blackflash9935]

Sir Rahmed This goes for you too: The author of the comment probably understood that before making his comment so I can safely do this: *breaths in* R/WOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOSH!!!!!!

[@mome6889]

imagine a simple cube you always cut in half. you can do it infinitely many times, it's surface area goes to infinity while it's volume stays the same. also you might consider having Gabriel's horn made up of some transparent material, fill it up with pi amounts of paint and boom, it will look painted from the outside

[@akasin8202]

Doreamon theme at beginning in background..

[@blackpenredpen]

aka sin yup!!!!

[@cutieboo233]

I think Pi is already an infinite number. So the volume looks like a number, but this number is 3.141596......... to infinity

[@ayo6836]

Liyuan Boo to think about that, it’s an interesting detail ** the finite volume is not that finite after all)

[@RedefiningtheConcepts]

Can you clear the concept of vectors multiplication and division is not defined in space.If you can, Please Please explain

[@klausolekristiansen2960]

It is good to know that the vuvuzela has only finite volume.

[@3420undertaker]

Why do somethings converge and some don't? Sorry if it's really basic I just like the math and don't really know much

[@dlevi67]

There is no "because" other than "because it is that way", I'm afraid. You can work out definitions for convergence and divergence under different procedures (1+2+3+4+... "=" -1/12 anyone?), but ultimately there is no "reason" nor a way to predict in advance whether a _completely_ _arbitrary_ series or integral will converge. Why do you expect maths to make sense? ;-)

[@joluju2375]

Hi Mike, you don't need math. The key is to think backwards. First experiment : take an infinite rope, cut it in an infinity of pieces. Add them up, and you get infinity again : this sum diverges. Second experiment : take a 1 meter long rope, cut it in an infinity of pieces (yes you can, even if you're tired, pick one of the pieces, and divide it). Add them up, and you get your 1 meter long rope back again : this sum converges. There is no mystery, an infinite sum can be finite.

[@coolbionicle]

If we bring this design to the real world and made the horn with an hypothetical type of copper that is infinitely hard, since atoms are quantized particles the horn will eventually have an end where there are six atoms packed in the shape of an hexagon. If the starting radius were to be 1foot, how long would the resulting horn be?

[@dlevi67]

That rather depends on how much copper you started with in the first place, wouldn't it?

[@coolbionicle]

@@dlevi67 not really, the thickness of the surface wouldn't matter because of the nature of the function dictates that it would be infinitely long. So it would go as long as the atoms could maintain the cylindrical shape, that's why I stated that at the end it would only have six atoms in a hexagon packed together, that's a one atom thick copper circle as small as the atoms would allow. You could even infer that the trumpet is a solid and it should still give the same result. Another key detail that might indeed affect the solution is the temperature, make the trumpet room temperature to make it simple. And for the record, I don't know the answer. I just think it would make a very cool question based on this shape as food for thought.

[@dlevi67]

@@coolbionicle Sorry my friend, you cannot have it both ways: first you ask "how long would it be", then you say that "it would be infinitely long". No it wouldn't, if it were created by a finite quantity of material with a finite thickness (~2 * covalent atomic radius; for copper 264 pm). For example, assume you have 0.75 pg of copper - this is still about 7.2 billion atoms - and you would just about be able to get the first "ring" of 1 foot radius by stringing together individual copper atoms, so if you had 0.75 pg of copper your horn could only be a single atomic diameter long. A longer horn, even monoatomically thick, would require more copper. On the other hand if you have more copper than needed to get a single atom layer for _any_ given length, then you can choose: you keep beating the copper and it elongates at that "six atoms" size for however much copper you have, six atoms at a time, or you stop beating and you have a non-uniform thickness at some points in the horn. Which is why without specifying quite a few other conditions the answer depends on how much copper you have in the first place. If you assume single atomic thickness, and starting at an arbitrary point with a radius of 1 foot, the function 1/x will get to 8.66142* 10^-10 ft (covalent diameter of copper ~= 264 pm ~= "radius" of the horn at the narrowest point) at ~12 billion feet, or 3.8 million km. About 10 times the distance Earth-Moon.

[@coolbionicle]

@@dlevi67 I stated the curve would be infinite and the limiting factor would have been the "six atom" thing. But you got the gist of it. Damn, I though it would be long but damn, not that long! That's amazing! I will look into your calcs in more detail later but I think you got it correct!

[@dlevi67]

​@@coolbionicle It's just a reciprocal. If I have screwed up somewhere it's in the conversion between metric and imperial/ACU. It happens to the best at NASA. BTW - by horn length I mean the length of the axis of the horn; if you want the arc length it's quite a bit messier: are the atoms "spherical" and just tangential to each other, or do the covalent radii overlap? Or do we assume no rippling? If we assume no rippling, then there is that nice integral of √(1+log²(x)) which has no elementary expression but that Wolframalpha tells me is ~80 million km.

[@koketsomohale8596]

I remember how aroused I was when I first came(haha) across this!

[@camerongray7767]

Hi, I would really appreciate if you could do a video on the formula of the surface area of a manga sponge. It might be a bit easy for a superhuman like you, but it is a similar problem to this in that a sponge where n is infinite has volume approaching 0 and infinite surface are. Please I have been trying to figure the formula out for weeks and I can’t get it!

[@t0mstone581]

what about a sierpinski tetrahedron?

[@thexoxob9448]

Mathematically there is no paradox, since you are comparing two entirely different quantities, which doesn't even make sense in the first place

[@IISH4RKZzII]

Wow, what is the level of math I need to be able understand all of this?

[@pietrotettamanti7239]

All you need to know is how an integral works and maybe a little bit of differential calculus. Nothing that highschool maths can't handle (and I'm not even good at maths so it's definitely doable).

[@TheMathias95]

B-A level maths, depending on your country. I believe the american equivalent is called pre-calculus?

[@leoallentoff]

@@TheMathias95 pre calculus doesn’t even get to derivatives

[@TheMathias95]

@@leoallentoff Yeah mb, I assumed 12th grade mathematics in the states could roughly be translated directly into someone who would be in 12th grade from my country. This is not the case.

[@RF-fi2pt]

see this other object, finite value at 2 dimensions paradox an infinite at 1 dimension: circle have finite area to one given radius R . One swirl line starting at center until that R have 1D Length infinite, as the line diameter is infinitesimal. Is the polar coordinates . The Integral from 0 to R of 2πr, gives exactly πR^2, but trying to see the integration process as that increasing swirl see the 1D line Length going to infinite. At Gabriel Horn is the same. 3D finite paradox a 2D infinite, one dimension below.

[@XZellTheBest]

Hi, I'm not a math student so maybe I'll say something stupid. This it happens also with a 3d version of a gaussian function? I don't know... Maybe...rotating e^(-x^2) along the y axes?

[@matengelo3812]

It is called the Gabriel's Horn because of the Archangel Gabriel and the horn he plays as the horn that announces the Judgement Day according to the Apocalypse chapter of the Bible

[@jall3ri]

So Doctor Who's TARDIS is a a reverse Gabriel's Horn finite surface area infinite volume

[@runneypo]

saw that edit at 5:49

[@toddtrimble2555]

How is this any more remarkable than a Flatlander painting the curve y = 1/x^2 over the infinite interval [1, infinity)?

[@rolandkarlsson7072]

This is no paradox and have never been a paradox. Yes, the volume is finite and the surface infinite. But, as the surface has 0 thickness, the surface has no volume.

[@leonardschreiter6871]

8:12 servers Area

[@Fran-fz7ud]

Gabriel horn makes sense. The volume will get so small it will aproach 0. The surface area will remain, it is greater than the one aproaching 0 so it will not aproach 0 and will be infinite

[@mychevysparkevdidntcatchfi1489]

If you take fractal snow (infinite circumference, finite area), extrude it to 3D "cylinder", you get infinite surface area, finite volume. In fact, most things in real world are like this (eg. coast line). This is why the world is discrete (quantized), not continuous. Continuous math is just approximation of the real world.

[@samueldavidrucker7514]

That's the 🔑

[@erik-ic3tp]

So the Planck length/volume is 1 pixel? :)

[@JeanSarfati]

But if you build the volume by inside successive russian-pupetts-horns, the sum of the infinite surfaces is... Pi ! Paradox or error mine ?

[@kuba9680]

The Gabriel is the Archangel Gabriel - he blows the horn to announce judgement day in Abrahamic traditions.

[@anilsharma-ev2my]

One liter paint can spread over the area How much maximum area we covered with it if it's carbon molecules width required ????

[@victorpaesplinio2865]

I guess that Gabriel is the Archangel in cristianismo. The Bible said that the angles will play horns to announce the doomsday. So I think that this is the Gabriel

[@evanpoole7829]

i love how we don’t actually need to integrate this to prove it, fascinating!

[@joaolisboa7775]

Gabriel is bc in Judeochristian tradition , Gabriel is the archangel wich have trumpets

[@aniltamang7506]

Was relaxing to hear such a good presentation and I'm glad i didn't have to hear some slow version of indian presenter

[@stevensiew4072]

Hey, Stop blowing Gabriel's horn

[@piratesofphysics4100]

I don't think that the inner surface area would be different than the outer surface area. So as we can fill this with pi amount of paint the paint will definitely paint the inner surface area. That means the paint can also paint the outer surface area. What the hell...

[@Rekko82]

So it isn't possible because the surface isn't finite. You can build only finite horns.

[@ian-ht1nf]

Infinite surface area but NOT INFINITE volume

[@zuccx99]

Yes Think of a cake Split if infinitely amount of times The surface area increases no matter how little you cut even if the volume is same You can eat the cake but cant ice it

[@angelmendez-rivera351]

ZRgaming You can ice it if your layer of icing is infinitesimally thin.

[@AndrewBarsky]

Check Vsauce if you are dumb like me. “Super tasks.”

[@soumyachandrakar9100]

So, finally it's here.

[@AP-wz6fb]

😂 the introduction music is Doraemon theme song. Childhood triggered

[@mspeir]

Wait... The volume = pi, an infinite sequence, but is finite?

[@bernardlemaitre4701]

and finite surface under an infinite length of the curve comprizing the surface ?

[@DavideCosmaro]

But pi IS infinite in its own way (just on the wrong side of the decimal dot)

[@anirudh7137]

A similar type of question came in my exam where we had to find the volume of the solid when rotated about its asymptote and I wrote infinite without even solving .Should have seen this video before. Now the teacher would award me 1/infinity for this question

[@meme_engineering4521]

Yes, it came in my paper also in dtu

[@anirudh7137]

@@meme_engineering4521 did you get it right?

[@meme_engineering4521]

@@anirudh7137 no..I am also in B14 batch roll number 52😂

[@anirudh7137]

@@meme_engineering4521 oh and I'm 53😂

[@mrhatman675]

Didn t quite get it at the end why did you multiply 2πy by dl and not dx at the surface area integral

[@MrCigarro50]

This is not rare. Any convergent integral in R2 from 0 to infinity has a finite area but the length of the line is infinite.

[@Trizzer89]

I suppose the surface area of the entire plane is infinite and the volume is 0

[@zakirreshi6737]

Same thing area under dirac delta function whose area is unit area and height is infinite

[@Lobyyyy]

The 3d graph like a horn🤣

[@isobar5857]

An example of where 'limits' fail...perhaps ?

[@jorgelenny47]

What about a square whose side lenght is √infinity?

[@gsittly]

also the area under y=1/x is infinite even it's inside the finite volume.

[@experimentbysaifanali7580]

I think you forget to include base area in the surface area

[@growtogether9332]

I thought fluffy become mathematician! Lol

[@bruzanHD]

Isn't it nonsensical to rotate regions that are missing a bound? I think so

[@gekkouga2828]

0:02 Is the background song the official theme of Doraemon?

[@AP-wz6fb]

I am 9 months late lol

[@shohamsen8986]

Isn't the Gabriel a reference to the angel Gabriel?

[@donovanb8555]

I think this the very first video I can understand

[@malhar073]

only video where both of my subscription are in it

[@dlevi67]

You keep your subscriptions into Gabriel's Horn? You'll run out of space at some point. Plus Gabriel may get annoyed.

[@I_like_pi_]

@@dlevi67 No, they're written on the surface.

[@dlevi67]

@@I_like_pi_ Well, Malhar said "in", not "on", but that might be a typo...

[@1495978707]

Gabriel’s horn is referring to the angel Gabriel

[@aditidas9978]

Nice explained this problem😊😊BPRP . any situation present in which object have infinite volume and finite surface area😶

[@akasin8202]

A question...that surface integral inequality used is same as cross section right? So what's confusing me is that why is cross section infinite when volume is finite?

[@akasin8202]

No cross section will be integral of f(x)dx from 0 to 1...and f(x) is 1/x...so infinite

[@gamecoolguy619]

If you integrate 2pi(1/x) it's far more simpler however unlike all other negative power of x the function we get after integrating doesn't share main properties of the function in this case 1/x diverges to 0 ln(x) does not so probably not accurate at infinite level but that's just my speculation...

[@rhombicuboctahedron7811]

invert the horn then fill it up the other way. HM

[@MiguelGarcia-zx1qj]

I didn’t know anything about this particular problem, nineteen years ago, when one of my students (the matter was Ordinary Differential Equations, by the way), posed me a paradoxical version: “as the area of the plane figure is infinite, you would need an infinite amount of paint, to paint it. But if you encase that area into the horn, as a kind of median partition, and fill the horn with paint, you only need a finite amount of paint. Where is the catch?” PS: I see that some people have posed this paradox. Sorry for the redundance. However, solving it is interesting ...

[@luisrosano3510]

I think is Gabriel the Arcangel. And there is his "Horn of the Apocalypse" on the Christian World.

[@manthanmistry1205]

Mind blowing 😮😮😮😮😮

[@blackpenredpen]

Manthan Mistry I felt the same when I first saw it!

[@MichaelJamesActually]

Extrapolate this to the 4th dimension. Finite 4d "volume", but infinite 3D volume on the surface?

[@MrChiddler]

You can do this with any cuboid If you half it’s thickness while doubling the area of its top and bottom surface it tends towards infinite surface area while volume does not change. Volume is not related to surface area.

[@mostafamhamed2606]

I have a challenge for you: Let g :R- -> R3 be a curve such that for every the vector g(t) is orthogonal to the tangent line to the curve at the point g(t). Prove that the curve lies on a sphere centered at (0,0,0).

[@seroujghazarian6343]

Finite surface but infinite perimeter?

[@philippelepilote7946]

As a Hilbert curve is infinite in length and covers the whole surface of the bounding square, if you rotate it in 3D around an axis parallel to one of the sides of the square, you get an infinite surface into a finite kind of a torus volume as well. No ?

[@vlibayramov8434]

The area under 1/x is also infinite

[@nithinsirimanne2924]

Understandable have a nice day

[@asameshimae6850]

Why can’t the surface integral be 2*pi*y*dx?

[@asameshimae6850]

And why volume integral isn’t pi*y^2*dL?

[@Kmrabhinav569]

For volume, we don't require the surface, we just need functions in terms of x,y

[@Kmrabhinav569]

But for surface, we basically use Pythagoras theorem, wherein we consider very small hypotenuse (here dL) and then find dL in function of x,y. Hope this helped

[@asameshimae6850]

Abhinav Kumar If the volume is a sum of volumes of cylinders with base area pi*y^2 and height dx, why can’t the surface be the sum of surfaces of these cylinders, namely 2*pi*y*dx?

[@kingbeauregard]

Think of it like this. If we're calculating volume, then the little diagonal bit on the edge is trivial and can be ignored. But if we're calculating surface area, we don't care about the volume, and the ONLY part we care about is the little diagonal bit. So we have to be precise about the length of the diagonal bit.

[@felixzz8598]

Take any convex quadrilateral ABCD split AB and CD into 3 equal parts each so 4 points, called Q1 Q2 P1 P2, are created. Proof that the area of Q1Q2P1P2 is the same as 1/3 * the area of ABCD

[@machobunny1]

You passed over a few steps faster than usual, and I did not exactly follow...but I see how you got there and you make very few mistakes of any kind, so I trust it, as far as it goes. Still, Infinity is a strange space. Perhaps our inability to deal with it relates to our inability to move beyond the limits of current quantum mechanics.

[@SidharthjainSingla]

But is it practically possible?