In this fast-paced world, we fail to observe many marvels of this expanding universe. What is this world without the queen of sciences? The place we reside in is full of wonderful, mysterious, and weird mathematical shapes and objects. Just because we don’t see something, doesn’t mean it isn’t there. Do you know that the symbol of infinity (∞) has a name? Yes, it does. It is called the lemniscates. What do you call an object that has the shape of a donut? Donut shaped? Try calling it torus next time. A dodecahedron is a platonic solid that has 12 faces, each of which is a pentagon itself. Similarly, we don’t know where other mystique are hidden and their properties. Our mythologies never fail to surprise us. They are full of fascinating objects with weird properties that may or may not exist in reality. Nevertheless, it keeps us captivated and motivates us to explore their properties. One such is the Gabriel’s horn.
What is a Gabriel’s horn?
Gabriel’s horn is from the Holy Bible. It is blown by the archangel Gabriel to announce the arrival of Judgment day associating the divine or the infinite with the finite. Infinity represents something that is boundless or unending. The word ‘Apeiron’ was used in the same sense before the word ‘infinity’ was widespread. The concept of infinity was explained by Aristotle back in the 4th century. Several mathematicians, philosophers, and physicists including Evangelista Torricelli, Kant, Poincare, Nete, Noel, Mancosii, Vailati, Pierri Gassendi, Cavalieri, Isaac Borrow have studied the concept of infinity and have made contributions on the same.
Relating the infinite with the finite, Gabriel’s horn is an infinitely long solid that takes the shape of a horn. Its properties were first studied by the Italian mathematician and physicist Evangelista Torricelli in the 17th century. Gabriel’s horn is a platonic object that is obtained by rotating a breathless and infinitely long curve. When a curve is rotated in 3-dimensions with respect to an asymptote, it forms a 3-d object. The asymptote here is the x-axis and the curve is y=1/x with the domain x≥1.
In the curve, as x tends to infinity, y approaches zero but never equals zero at any point. When rotated in three dimensions with respect to the x-axis, the radius of the discs gets smaller and smaller but never attains zero as well. (Think of the 3-d object as sections of discs or cylinders) Every object has surface area and volume, either finite or infinite and it could be obtained by using one of the several available methods.
Gabriel’s horn could be considered as constituted of discs or cylinders. To find the volume and surface area, they can be related with infinite series. While calculating the surface area, the radius of Gabriel’s horn can be broken down to the sum of the radius of discs
Volume of horn:
\int { \Pi \left( \frac { 1 }{ { x }^{ 2 } } \right) } dx=\quad \Pi
Surface area of horn:
{ { lim } }_{ a\rightarrow \infty }\int_{ 1 }^{ a }{ 2\Pi \left( \frac { 1 }{ x } \right) } \sqrt { 1+{ \left( -\frac { 1 }{ { x }^{ 2 } } \right) }^{ 2 } } dx=\quad \infty
Here, Gabriel’s horn is believed to have a finite volume i.e. π and infinite inner surface area. Though the volume is irrational, it is finite and bounded. Gabriel’s horn is an object which is not bounded on every side but still has finite volume. What is astonishing is not the object itself, but the Painter’s paradox.
What is a Paradox?
A paradox, in its use, is a statement that contradicts commonly held notions. Paradoxes have a long history in math, statistics, physics, and biology as well. Zeno’s paradox, Russel’s paradox, Abelson’s paradox, Friendship paradox, Potato paradox, Hilbert’s grand hotel paradox are few of the famous paradoxes in Mathematics. The time travel paradox is another paradox that is well known. A mathematical paradox is a statement that appears to be contradicting itself while simultaneously seeming to be completely logical.
Quine (1976) noted that “more than once in history the discovery of a paradox has been the occasion for major reconstruction at the foundations of thought”
Painter’s paradox:
As stated already, what makes Gabriel’s horn interesting is the painter’s paradox. Though it has never been mentioned in any classic textbooks as a paradox, in ‘Calculus- A complete course by Adams and Essex (2010)’, it’s been mentioned as a question. Find the volume of the infinitely long-horn that is generated by rotating the region bounded by y=1/x and y=0 and lying to the right of x=1 about the x-axis.
The original paradox is:
Is infinite amount of paint needed to paint the inner surface of the Gabriel’s horn?
Intuitively it is expected that an object generated by rotating a curve around the x-axis has either finite surface area and finite volume or infinite surface area and infinite volume. The painter’s paradox is realistic and requires decontextualization from the physical reality whereas Gabriel’s horn is contextual. The paradox is considered with the following assumptions: time is not a constraint and the paint can reach each and every part of the horn.
In reality, we have never dealt with an infinitely long object. This makes it hard to visualize the object, not to mention painting the same. Gabriel’s horn contradicts our expectation by having a surface area that is infinite and volume that is finite. People face difficulties while facing this paradox. One such difficulty is, people could not associate an object with an infinite surface area having a finite volume. They believe volume increases as surface area increases. Another one is, as the radius of the horn decreases, it gets infinitesimal at some point which is smaller than an atom. Painting at that point becomes impossible to them, as we associate painting the surface of Gabriel’s horn with painting a real object, which we can obviously see. Painting an ever-growing object with a finite amount of paint seems to be impossible for people. One problem they face while associating Gabriel’s horn with a real-life object is, in Mathematics, the surface has no thickness. But in reality, it does, at least infinitesimal.
Painting refers to applying a coat of paint on a surface. The amount of paint needed is in volume not in area. Paint required to coat an object with surface area A is same as painting a flat region of surface area A. To paint a flat surface of infinite area with a uniform thickness ‘c’ one needs an infinite amount of point as the amount needed to paint infinite area A is { lim }_{ A\rightarrow \infty }cA=\infty
If we consider painting the inner surface area with uniform thickness, the thickness of the paint needed is zero. Because, the radius of the discs keeps decreasing as we move towards the right tail and it nearly reaches zero. So, for the coat to be uniform, its thickness should be smaller than the smallest radius of discs (of Gabriel’s horn). At the same time, thickness could not be negative making zero thickness as our only possible solution i.e. cannot paint. Since the volume is finite we can fill it with paint.
Is Gabriel’s horn the only object with an infinite area and finite volume? No. There could be many such objects. For example, Koch’s curve or Koch’s snowflake, a fractal has infinite perimeter but a finite area. In Koch’s anti snowflake, area is zero and the perimeter is infinite. There are many such shapes with similar properties. In both cases, the measure with finite value is of higher dimension than the measure whose value is infinite. Volume’s dimension is higher than area and the area’s dimension is higher than that of the perimeter. There are many objects which have even weirder properties. This is just one example of the universe’s marvels.
Try our article on ‘The Intriguing divine ratio of Nature‘
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