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Gabriel's Horn / Torricelli's Trumpet — finite volume, infinite area
100k ft — What is this?
Gabriel's Horn / Torricelli's Trumpet — finite volume, infinite area
50k ft — Why care?
@mathxmatrix explains Gabriel's Horn (Torricelli, 1643): rotate y=1/x around the x-axis to get a shape with finite volume but infinite surface area. Credit @jeedecodee.
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1k ft — Deep-dive / try it
What’s truly amazing about Gabriel’s Horn (also known as Torricelli’s Trumpet) is its mind-bending mathematical contradiction: it is a shape with a finite volume but an infinite surface area.
Discovered by Evangelista Torricelli in 1643, this shape is created by taking the curve y = 1/x(from x=1 to infinity) and rotating it around the x-axis.
Original content creation by @jeedecodee
raymondhedgecock
17w
This was figured out long ago. The reason is the integral of 1/x^p dx converges if p>1 and diverges otherwise. Thank Evangelista Torricelli for that. She's the one who figured it out and called it Acuta Infinita. Pretty on brand.
Transcript excerpt
This is the most dangerous shape in calculus. Gabriel's horn. We start with the simple curve y equals 1 over x, extending from 1 all the way to infinity. Now, let's use the power of integration. If we rotate this curve around the x-axis, we generate a 3D surface. Here is the paradox. If you calculate the volume, it converges to pi. That means this horn holds a finite amount of paint. You can fill it up, but if you calculate the surface area, the integral diverges to infinity. Mathematically, this means you can fill the horn with paint, but you can never paint the outside.