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.
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.
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.
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?
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)
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.
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!
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 ;-)
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.
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
I want to write my assignment on this theme, can you help me with the areas of math that will probably be included? The vids is cool as always, thanks! *.*
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.
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!
I remember finding surface area under this curve but only above x axis during my calc exam ... Although the limits were finite ... So nice to see that it could generate such an interesting shape when rotated around x axis .. 😀
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 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.
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.
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.
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?)
This is the horn Gabriel played that bit of Sledgehammer on. It sounds like a flute because you blow across a hole in it because it's infinitely long and you wouldn't be able to get to the end to blow into it.
Now to do a related rates problem: Place the horn vertically, and see how fast the fill line rises when you pour paint into it. Seems the filling rate at the limit of 0 volume poired in must be infinite.
The simpler version of this paradox would be the length of the curve from from 1 to infinity which is infinite and the area under the curve from 1 to infinity which is finite. It means if you walk upto infinity you can still paint all the surface under the curve with finite paint . If you take this problem seriously even apply infinite limit on an integral is paradox.
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.
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
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 ?
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
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?
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...
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?
That’s some fascinating paradox In fact, it works (the finite volume fills the horn with infinite surface area) only if we assume that it’s possible to fill the spaces that are smaller than atoms (and other stuff elementary stuff). In real life, the paint just wouldn’t go further that one point, where the A of the “disk” gets too small for the atoms to keep traveling inside the horn. From the other pont of view, we can paint the outside A with a finite volume of paint, all that matters is a thickness of the layer of the paint (here we go back to the problem with atoms).
It’s silly and nonsensical to be considering atoms in paint when you’re talking about a shape that itself could never be composed of atms, as it gets arbitrarily thin and has 0 thickness. Obviously we have to be working in a space where materials are infinitely divisible. Appealing to atoms is silly and misses the point.
I think the paradox could have been summarised better, like: To paint the outside of the horn, decide on the thickness of the coat of paint (greater than 0) first. Since the area is infinite, the volume of paint (of any non-zero coat thickness) has to be infinite too. Since the horn contains a finite volume of paint, it doesn't contain enough paint to paint itself even though the inner and outer areas are the same and the inside of the horn is in some sense painted since it's full of paint. As an exercise, figure out why it's not actually a paradox
It was probably Martin Gardner who mentioned this years ago in one of his columns. He teased his readers by putting the horn upright and imagined pouring pi liters of paint in it......
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? ;-)
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.
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.
In physics, Gabriel's Horn has been used as an analogy for certain physical systems that have infinite length or surface area but finite volume. One example is a long, thin tube that is closed at one end, such as a trumpet or a clarinet. If we imagine a trumpet as a Gabriel's Horn, the volume of air that can be contained in the horn is finite, but the length of the horn can be infinite. This means that the sound waves produced by the instrument can be thought of as propagating along an infinitely long path, even though the physical length of the instrument is finite. This property of Gabriel's Horn can help explain some of the unique characteristics of wind instruments, such as their ability to produce a wide range of pitches by changing the length of the air column inside the instrument. In another example, Gabriel's Horn has been used to describe the behavior of a particle falling into a black hole. In this case, the infinite surface area of the horn can be thought of as representing the event horizon of the black hole, which is the point of no return beyond which nothing can escape. The finite volume of the horn can be thought of as representing the mass of the black hole, which is concentrated at the singularity at the center of the event horizon. Overall, the use of Gabriel's Horn in physics is a useful tool for visualizing certain physical concepts and phenomena, and for highlighting the importance of carefully interpreting mathematical models and their implications.
If we limit the length of Gabriel's Horn to be the same as the size of the observable universe, which is estimated to be around 93 billion light years in diameter, we can calculate an approximate value for the surface area of the horn. Recall that Gabriel's Horn is formed by revolving the curve y = 1/x around the x-axis between x = 1 and x = infinity. The surface area of the horn is given by the formula: S = 2π ∫(1 to ∞) y * sqrt(1 + (dy/dx)^2) dx Substituting y = 1/x, we get: S = 2π ∫(1 to ∞) (1/x) * sqrt(1 + (d/dx(1/x))^2) dx Simplifying the expression inside the square root, we get: S = 2π ∫(1 to ∞) (1/x) * sqrt(1 + (1/x^2)) dx Making the substitution u = 1/x^2, we get: S = 2π ∫(0 to 1) sqrt(1 + u) du This integral can be evaluated using the substitution u = tan^2θ, which gives: S = 2π ∫(0 to π/2) sqrt(1 + tan^2θ) * 2tanθ sec^2θ dθ Simplifying and integrating, we get: S = π^2 Therefore, the surface area of Gabriel's Horn, if its length is limited to be the same as the size of the observable universe, is approximately π^2 square units. This is an interesting result that shows how a surface with infinite length can have a finite surface area.
There are other natural phenomena that can be related to the Gabriel's Horn paradox, which involves a finite volume with an infinite surface area. One example is the shape of certain crystals and rocks, such as calcite crystals or stalactites and stalagmites in caves. These structures have a finite volume, but their surfaces can be extremely complex and have a high degree of fractal-like roughness, which can give rise to an effectively infinite surface area. This can have important implications for the chemical and physical properties of these materials, such as their ability to dissolve or absorb gases and liquids. Another example is the way that fluids behave in very small channels or capillaries. For example, the walls of small blood vessels or the narrow spaces between soil particles can have an effectively infinite surface area relative to their volume. This can give rise to unique fluid flow patterns and surface interactions that can influence many important physical and biological processes, such as nutrient transport, cell adhesion, and microbial growth. Overall, the Gabriel's Horn paradox is a fascinating and important concept in mathematics and science that can help us understand many different natural phenomena and the limitations of our mathematical models and physical intuitions.
The Gabriel's Horn paradox is a mathematical and physical concept that has important implications for our understanding of the universe and the limitations of our knowledge and models. While it may not have any direct benefit for the survival of humans or the universe, it can help us better understand the behavior of natural systems and phenomena, and improve our ability to make predictions and develop technologies based on this understanding. One way that the Gabriel's Horn paradox can be useful is in the development of models and simulations of complex natural systems, such as climate models, astrophysical simulations, and biomedical models. By taking into account the limitations and paradoxes that arise from the Gabriel's Horn paradox, we can develop more accurate and reliable models that better capture the behavior of these systems. In addition, the Gabriel's Horn paradox can help us appreciate the richness and complexity of the natural world, and inspire new ways of thinking and understanding the universe. By recognizing the limitations of our current models and theories, we can continue to push the boundaries of human knowledge and discovery, and develop new technologies and innovations that can benefit society and the world around us.
The paradox of Koch's snowflake is another fascinating example of a mathematical concept that challenges our intuitions and understanding of geometric shapes and their properties. Koch's snowflake is a fractal shape that can be generated by repeatedly replacing the middle third of each line segment in an equilateral triangle with two smaller line segments that form an equilateral triangle. This process is repeated infinitely many times, generating a shape that has an infinite perimeter but a finite area. The paradox arises when we consider the limit of this process, as the number of iterations goes to infinity. Intuitively, we might expect that the limit shape would be a smooth curve with a well-defined length and area. However, the resulting shape is a highly irregular and complex fractal, with an infinitely jagged perimeter that is larger than any finite number, and a finite area. This paradox challenges our intuitions about the relationship between the smoothness of a curve and its length, and raises important questions about the nature of infinity, limits, and continuity in mathematics and physics. The paradox of Koch's snowflake has many interesting implications for the study of fractals and their applications in fields such as computer graphics, image processing, and scientific visualization. It also has connections to other important concepts in mathematics and physics, such as chaos theory, dynamical systems, and non-linear dynamics.
I can show you the formulas for the area and perimeter of a Koch snowflake. Let "s" be the length of the initial equilateral triangle, and let "n" be the number of iterations in the Koch snowflake construction process. The formula for the perimeter of a Koch snowflake after n iterations is: P = s * (4/3) * (1 + (1/4)^n + (1/4)^(2n)) This formula gives the total length of the perimeter of the Koch snowflake after n iterations, in terms of the length of the original equilateral triangle. The formula for the area of a Koch snowflake after n iterations is: A = (sqrt(3)/4) * s^2 * (5/3)^n This formula gives the total area of the Koch snowflake after n iterations, in terms of the length of the original equilateral triangle. These formulas illustrate the paradox of the Koch snowflake, in which the perimeter of the shape increases without bound as the number of iterations goes to infinity, while the area remains finite.
I was thinking the same thing but did notice 1 other thing. 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...
Although the surface area goes to infinity your set up of the calculation again only considers the area of the 1/x portion from 1 to infinity. However it ignores the area at the boundary condition of 1.
The area of the boundary at 1 is pi*r^2, so as r = 1/x this will be 1, so the area at the boundary of 1 is pi. At the other boundary condition, the area of the boundary would indeed converge upon zero.
My point is that based on the statement of the problem, you have to include the area of your boundary conditions, or you are answering a different problem.
@@deonmurphy6383 Sorry, but I'm not sure I understand your argument. The area of the infinitesimally thin _circumference_ AT 1 is zero (or pi dy^2 if you want to use non standard analysis, but that's another question), and it is included since the integration boundary is 1. What Steve (BPRP) is calculating is the surface area of the *horn* , i.e. the open-ended surface which has an infinitesimally thin opening at the mouth, and a finite opening at the bell (which as you rightly say has an area of pi), not the area of the hyperbolic *cone* (which as you rightly say should include the closing lid). The "lid" is not generated by the rotation of the hyperbola around the X axis and is not a part of the horn (or if you prefer, the segment representing the distance from the X axis to the hyperbola at 1 is not a part of the hyperbola). In either case you still have an infinite internal area (inf + pi = inf) and a finite volume, since the volume of the lid is zero (or pi * dx if you are using non-standard analysis, but it's still a finite quantity). The typical paradox mentioned in association with Gabriel's horn of a finite volume of paint covering an infinite internal area works either way, except that if there is a lid on the horn you can't pour paint into it without a tin opener. ;-) (edit: YT was screwing up formatting)
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
If the paint gets filled with finite volume then it would touch the infinite surface and indeed paint whole surface . Actually we don't need infinite paint to paint infinite surface we can paint infinite surface with finite paint mathematically because we can actually transform a finite volume object into infinite surface plane . So yes mathematically it gets painted . But if you look at it physically the horn gets so thinner the smallest molecule of paint can no more fall inside horn painting only the finite surface. The problem with this paradox is we approach the volume as mathematical in nature but treat surface as physical in nature
This produces the seemingly paradoxical result that you need a finite amount of paint to fill up the entire shape, but that paint that's filling it isn't enough to paint the entire surface.
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 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.
@@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.
@@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!
@@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.
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).
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 ...
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
So we can fill Gabriel's horn with paint, but we can't paint it 😅
Osvaldo Mena Wouldn’t that, then, completely paint the inside of the horn? My head...
@@btat16 No because the internal surface of the horn is infinite too! 😅😫
Osvaldo Mena but. But... if you fill it up wouldn’t it be touching everything? It hurts
@@parkershaw971 how we has so small no volume is in but area still there?
We can paint it if the thickness of the layer coating the horn is infinitesimal
It's referring to the angel Gabriel. Gabriel has a horn which he blows for seven days before the judgement day.
Where judgement day refers to the point in time where Gabriel's neighbours finally lose patience and decide to do something about the noise.
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.
And in the Quran, the angel blowing the horn is actually called Israfil
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.
good trick :v
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?
Proof by necessity
Pi * r² * THICCness
t h i c c
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.
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.
That's pretty much a low-res version of Gabriel's Horn.
tbh I didn't watch the video, I just made this comment after the introduction
@@Dunkle0steus also vsauce
Dunkleosteus when the dev of TFC+ randomly pops up in the comments of an unrelated video... thanks for your work on that. :)
Whoa I never get recognized. Thanks!
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?
I think the problem comes from the fact that you are treating paint as both 2 and 3 dimensional.
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)
@@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.
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.
It was discussed in Numberphile I think.
《Ofc the answer of this is yes cz otherwise how can i make this video》you're such a legend
What about the opposite? Is there any "object" with infinite volume, but finite surface area?
An inverted ball. The ball has a finite surface area. The volume of outside the ball extends to infinite time and space.
No. just check wikipedia en.wikipedia.org/wiki/Gabriel%27s_Horn It has a proof why the opposite can't happen
@@AnimeLawyers Why do you need an inverted ball? Simplify further. Let's say the universe and whatever beyond it. Infinite volume. Zero surface area.
@VeryEvilPettingZoo The surface area of A's boundary is nonexistent, not 0.
@@BinuJasim You....... took that far too seriously......
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!
I can try!
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 ;-)
i love how we don’t actually need to integrate this to prove it, fascinating!
Like fractals.
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.
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.
omg u have solved years of scientific debating, i will call the noble prize people for you now !
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
So, finally it's here.
Gabriel got some serious braggin' rights.
is there a solid of revolution with the opposite properties (infinite volume but finite area)?
Can you clear the concept of vectors multiplication and division is not defined in space.If you can, Please Please explain
I want to write my assignment on this theme, can you help me with the areas of math that will probably be included?
The vids is cool as always, thanks! *.*
What's the longest chain rule derivative you know of and could it make an education fun video?
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.
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!
I love the way you teach
I remember finding surface area under this curve but only above x axis during my calc exam ... Although the limits were finite ... So nice to see that it could generate such an interesting shape when rotated around x axis .. 😀
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? !
🤯
You get to a point where the diameter of the horn is smaller than the diameter of a molecule of paint.
@@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.
: )
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.
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.
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?)
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
Hey, why don't you record a video about how to derive the cartesian or polar formula of Gabriel's horn? The x, y and z of course
Pi is such a strange/mysterious and interesting number
Definitely!
what about a sierpinski tetrahedron?
This is the horn Gabriel played that bit of Sledgehammer on. It sounds like a flute because you blow across a hole in it because it's infinitely long and you wouldn't be able to get to the end to blow into it.
The angel Gabriel, from a scene in the fever dream know as the Book of Revelations.
Yeah, I think the mathematician that came up with it was called Torcellini, so sometimes it's reffered to as "Torcellini's trumpet/horn"
Xander Gouws Torricelli, not Torricellini. Close, though.
It was Squidward Tortellini.
Revelation.... aka John on LSD.
The revelation is serious.
In 2D space, the Koch Snowflake is another example of a shape with finite area and an infinite perimeter.
Excellent!
Are there any other functions with this beavior? Can i use e^x from negative infinity to, well, any real number?
Actually, e^x gets small enough fast enough it can have both finite surface area _and_ volume.
Please talk about the Sierpensky's Carpet or Manger's Sponge (too lazy to look it up for correct spelling but should be close)
Sierpiński and Menger respectively, though the objects are generally indicated without the diacriticals ('Sierpinski')
Didn t quite get it at the end why did you multiply 2πy by dl and not dx at the surface area integral
I'm a simple man, I see Gabriel Iglesias and a suface of revolution in the same thumbnail, I click.
It is good to know that the vuvuzela has only finite volume.
One liter paint can spread over the area
How much maximum area we covered with it if it's carbon molecules width required ????
This is good, relaxing video, strangely
Was relaxing to hear such a good presentation and I'm glad i didn't have to hear some slow version of indian presenter
Now to do a related rates problem:
Place the horn vertically, and see how fast the fill line rises when you pour paint into it.
Seems the filling rate at the limit of 0 volume poired in must be infinite.
The simpler version of this paradox would be the length of the curve from from 1 to infinity which is infinite and the area under the curve from 1 to infinity which is finite. It means if you walk upto infinity you can still paint all the surface under the curve with finite paint .
If you take this problem seriously even apply infinite limit on an integral is paradox.
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.
What about a square whose side lenght is √infinity?
Notice that the more excited he gets the faster he talks xD I love it
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
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 ?
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
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?
No cross section will be integral of f(x)dx from 0 to 1...and f(x) is 1/x...so infinite
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...
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?
That’s some fascinating paradox
In fact, it works (the finite volume fills the horn with infinite surface area) only if we assume that it’s possible to fill the spaces that are smaller than atoms (and other stuff elementary stuff).
In real life, the paint just wouldn’t go further that one point, where the A of the “disk” gets too small for the atoms to keep traveling inside the horn.
From the other pont of view, we can paint the outside A with a finite volume of paint, all that matters is a thickness of the layer of the paint (here we go back to the problem with atoms).
It’s silly and nonsensical to be considering atoms in paint when you’re talking about a shape that itself could never be composed of atms, as it gets arbitrarily thin and has 0 thickness. Obviously we have to be working in a space where materials are infinitely divisible. Appealing to atoms is silly and misses the point.
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 😁
I think the paradox could have been summarised better, like:
To paint the outside of the horn, decide on the thickness of the coat of paint (greater than 0) first.
Since the area is infinite, the volume of paint (of any non-zero coat thickness) has to be infinite too.
Since the horn contains a finite volume of paint, it doesn't contain enough paint to paint itself even though the inner and outer areas are the same and the inside of the horn is in some sense painted since it's full of paint.
As an exercise, figure out why it's not actually a paradox
It was probably Martin Gardner who mentioned this years ago in one of his columns. He teased his readers by putting the horn upright and imagined pouring pi liters of paint in it......
It's like the bell curve.
and finite surface under an infinite length of the curve comprizing the surface ?
Well if you think of the coastline paradox you can get shapes with finite area but infinite perimeter
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
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? ;-)
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.
Blackpenredpen can't do perfect cuts-
5:22 timestamp: Are you sure about that?
8:12 servers Area
8:06 Hello this is Chef Steve from MathWishes.com
You are after all the Major General for solving your integral!
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.
I think this the very first video I can understand
A 3D lookalike to the surfaces with finite area and infinite perimeter.
Same thing area under dirac delta function whose area is unit area and height is infinite
In physics, Gabriel's Horn has been used as an analogy for certain physical systems that have infinite length or surface area but finite volume. One example is a long, thin tube that is closed at one end, such as a trumpet or a clarinet.
If we imagine a trumpet as a Gabriel's Horn, the volume of air that can be contained in the horn is finite, but the length of the horn can be infinite. This means that the sound waves produced by the instrument can be thought of as propagating along an infinitely long path, even though the physical length of the instrument is finite. This property of Gabriel's Horn can help explain some of the unique characteristics of wind instruments, such as their ability to produce a wide range of pitches by changing the length of the air column inside the instrument.
In another example, Gabriel's Horn has been used to describe the behavior of a particle falling into a black hole. In this case, the infinite surface area of the horn can be thought of as representing the event horizon of the black hole, which is the point of no return beyond which nothing can escape. The finite volume of the horn can be thought of as representing the mass of the black hole, which is concentrated at the singularity at the center of the event horizon.
Overall, the use of Gabriel's Horn in physics is a useful tool for visualizing certain physical concepts and phenomena, and for highlighting the importance of carefully interpreting mathematical models and their implications.
If we limit the length of Gabriel's Horn to be the same as the size of the observable universe, which is estimated to be around 93 billion light years in diameter, we can calculate an approximate value for the surface area of the horn.
Recall that Gabriel's Horn is formed by revolving the curve y = 1/x around the x-axis between x = 1 and x = infinity. The surface area of the horn is given by the formula:
S = 2π ∫(1 to ∞) y * sqrt(1 + (dy/dx)^2) dx
Substituting y = 1/x, we get:
S = 2π ∫(1 to ∞) (1/x) * sqrt(1 + (d/dx(1/x))^2) dx
Simplifying the expression inside the square root, we get:
S = 2π ∫(1 to ∞) (1/x) * sqrt(1 + (1/x^2)) dx
Making the substitution u = 1/x^2, we get:
S = 2π ∫(0 to 1) sqrt(1 + u) du
This integral can be evaluated using the substitution u = tan^2θ, which gives:
S = 2π ∫(0 to π/2) sqrt(1 + tan^2θ) * 2tanθ sec^2θ dθ
Simplifying and integrating, we get:
S = π^2
Therefore, the surface area of Gabriel's Horn, if its length is limited to be the same as the size of the observable universe, is approximately π^2 square units. This is an interesting result that shows how a surface with infinite length can have a finite surface area.
There are other natural phenomena that can be related to the Gabriel's Horn paradox, which involves a finite volume with an infinite surface area.
One example is the shape of certain crystals and rocks, such as calcite crystals or stalactites and stalagmites in caves. These structures have a finite volume, but their surfaces can be extremely complex and have a high degree of fractal-like roughness, which can give rise to an effectively infinite surface area. This can have important implications for the chemical and physical properties of these materials, such as their ability to dissolve or absorb gases and liquids.
Another example is the way that fluids behave in very small channels or capillaries. For example, the walls of small blood vessels or the narrow spaces between soil particles can have an effectively infinite surface area relative to their volume. This can give rise to unique fluid flow patterns and surface interactions that can influence many important physical and biological processes, such as nutrient transport, cell adhesion, and microbial growth.
Overall, the Gabriel's Horn paradox is a fascinating and important concept in mathematics and science that can help us understand many different natural phenomena and the limitations of our mathematical models and physical intuitions.
The Gabriel's Horn paradox is a mathematical and physical concept that has important implications for our understanding of the universe and the limitations of our knowledge and models. While it may not have any direct benefit for the survival of humans or the universe, it can help us better understand the behavior of natural systems and phenomena, and improve our ability to make predictions and develop technologies based on this understanding.
One way that the Gabriel's Horn paradox can be useful is in the development of models and simulations of complex natural systems, such as climate models, astrophysical simulations, and biomedical models. By taking into account the limitations and paradoxes that arise from the Gabriel's Horn paradox, we can develop more accurate and reliable models that better capture the behavior of these systems.
In addition, the Gabriel's Horn paradox can help us appreciate the richness and complexity of the natural world, and inspire new ways of thinking and understanding the universe. By recognizing the limitations of our current models and theories, we can continue to push the boundaries of human knowledge and discovery, and develop new technologies and innovations that can benefit society and the world around us.
The paradox of Koch's snowflake is another fascinating example of a mathematical concept that challenges our intuitions and understanding of geometric shapes and their properties.
Koch's snowflake is a fractal shape that can be generated by repeatedly replacing the middle third of each line segment in an equilateral triangle with two smaller line segments that form an equilateral triangle. This process is repeated infinitely many times, generating a shape that has an infinite perimeter but a finite area.
The paradox arises when we consider the limit of this process, as the number of iterations goes to infinity. Intuitively, we might expect that the limit shape would be a smooth curve with a well-defined length and area. However, the resulting shape is a highly irregular and complex fractal, with an infinitely jagged perimeter that is larger than any finite number, and a finite area.
This paradox challenges our intuitions about the relationship between the smoothness of a curve and its length, and raises important questions about the nature of infinity, limits, and continuity in mathematics and physics.
The paradox of Koch's snowflake has many interesting implications for the study of fractals and their applications in fields such as computer graphics, image processing, and scientific visualization. It also has connections to other important concepts in mathematics and physics, such as chaos theory, dynamical systems, and non-linear dynamics.
I can show you the formulas for the area and perimeter of a Koch snowflake.
Let "s" be the length of the initial equilateral triangle, and let "n" be the number of iterations in the Koch snowflake construction process.
The formula for the perimeter of a Koch snowflake after n iterations is:
P = s * (4/3) * (1 + (1/4)^n + (1/4)^(2n))
This formula gives the total length of the perimeter of the Koch snowflake after n iterations, in terms of the length of the original equilateral triangle.
The formula for the area of a Koch snowflake after n iterations is:
A = (sqrt(3)/4) * s^2 * (5/3)^n
This formula gives the total area of the Koch snowflake after n iterations, in terms of the length of the original equilateral triangle.
These formulas illustrate the paradox of the Koch snowflake, in which the perimeter of the shape increases without bound as the number of iterations goes to infinity, while the area remains finite.
But how about the other way around?
I'm so nooby at calculus, but why do you use the formula of the volume of a cylinder when it is more likely to be a cone?
You are insane man, no one ever replied me like that, thank you so much. I will read it tomorrow :)
Why can't we do surface area equals integration of 2(pi) (y) dx??
I was thinking the same thing but did notice 1 other thing.
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...
only video where both of my subscription are in it
You keep your subscriptions into Gabriel's Horn? You'll run out of space at some point. Plus Gabriel may get annoyed.
@@dlevi67 No, they're written on the surface.
@@I_like_pi_ Well, Malhar said "in", not "on", but that might be a typo...
Doreamon theme at beginning in background..
aka sin yup!!!!
Isn't the Gabriel a reference to the angel Gabriel?
But is it practically possible?
Nice explained this problem😊😊BPRP . any situation present in which object have infinite volume and finite surface area😶
Two things:
1. I think rotating it and solving it like a cone is better
2. I think it should have something to do with e or 1/x
Although the surface area goes to infinity your set up of the calculation again only considers the area of the 1/x portion from 1 to infinity. However it ignores the area at the boundary condition of 1.
Which is finite (in fact, zero), so I'm not sure what your point is?
The area of the boundary at 1 is pi*r^2, so as r = 1/x this will be 1, so the area at the boundary of 1 is pi. At the other boundary condition, the area of the boundary would indeed converge upon zero.
My point is that based on the statement of the problem, you have to include the area of your boundary conditions, or you are answering a different problem.
@@deonmurphy6383 Sorry, but I'm not sure I understand your argument. The area of the infinitesimally thin _circumference_ AT 1 is zero (or pi dy^2 if you want to use non standard analysis, but that's another question), and it is included since the integration boundary is 1.
What Steve (BPRP) is calculating is the surface area of the *horn* , i.e. the open-ended surface which has an infinitesimally thin opening at the mouth, and a finite opening at the bell (which as you rightly say has an area of pi), not the area of the hyperbolic *cone* (which as you rightly say should include the closing lid).
The "lid" is not generated by the rotation of the hyperbola around the X axis and is not a part of the horn (or if you prefer, the segment representing the distance from the X axis to the hyperbola at 1 is not a part of the hyperbola).
In either case you still have an infinite internal area (inf + pi = inf) and a finite volume, since the volume of the lid is zero (or pi * dx if you are using non-standard analysis, but it's still a finite quantity).
The typical paradox mentioned in association with Gabriel's horn of a finite volume of paint covering an infinite internal area works either way, except that if there is a lid on the horn you can't pour paint into it without a tin opener. ;-)
(edit: YT was screwing up formatting)
Understandable have a nice day
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
Yes, it came in my paper also in dtu
@@meme_engineering4521 did you get it right?
@@anirudh7137 no..I am also in B14 batch roll number 52😂
@@meme_engineering4521 oh and I'm 53😂
If the paint gets filled with finite volume then it would touch the infinite surface and indeed paint whole surface . Actually we don't need infinite paint to paint infinite surface we can paint infinite surface with finite paint mathematically because we can actually transform a finite volume object into infinite surface plane . So yes mathematically it gets painted . But if you look at it physically the horn gets so thinner the smallest molecule of paint can no more fall inside horn painting only the finite surface. The problem with this paradox is we approach the volume as mathematical in nature but treat surface as physical in nature
also the area under y=1/x is infinite even it's inside the finite volume.
saw that edit at 5:49
This produces the seemingly paradoxical result that you need a finite amount of paint to fill up the entire shape, but that paint that's filling it isn't enough to paint the entire surface.
WarpRulez There is no paradox. There is no reason to assume that it would not be sufficient to paint the entire surface.
How can you paint an infinite surface area with a finite volume of paint?
How is this any more remarkable than a Flatlander painting the curve y = 1/x^2 over the infinite interval [1, infinity)?
The 3d graph like a horn🤣
I wanted to make a video on this :O
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?
That rather depends on how much copper you started with in the first place, wouldn't it?
@@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.
@@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.
@@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!
@@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.
Mind blowing 😮😮😮😮😮
Manthan Mistry
I felt the same when I first saw it!
isn't it like a fractal?
I knew this from the mandelbulb set.
I got impaled by Gabriel's horn.
An example of where 'limits' fail...perhaps ?
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).
So Doctor Who's TARDIS is a a reverse Gabriel's Horn finite surface area infinite volume
0:02 Is the background song the official theme of Doraemon?
I am 9 months late lol
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 ...
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
Bro you look cool on this dress.
Isn't this equivalent to finite area and infinite length in 2D?