VIDEO: Napkin Ring Problem

If you were to core a sphere (remove a cylinder from it), you are left with a shape that looks like a napkin ring. This is a “bizarre” shape, as if you have two napkin rings with the same height, they will have the same volume regardless of the size of the initial spheres that they came from. How do you prove this?

Here’s a few hints to try and solve it yourself before watching the Vsauce video below which reveals the answer:

  • There are a few variables that need to be found: the height of the napkin ring, the radius of the starting sphere and the radius of the cylinder. Using these variables you can find a volume equation.
  • You don’t need to find the volume of the whole napkin ring in one go. This is because, as the two napkin rings have to be the same height, it’s enough to show that any slice of the napkin rings has to have the same area. If every pair of slices has the same area, then the napkin rings have the same volume.

Solution:

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VIDEO: Banach–Tarski Paradox

The Banach-Tarski Paradox is a theorem in geometry which states that:

“It is possible to decompose a ball into five pieces which can be reassembled by rigid motions to form two balls of the same size as the original.”

It was first stated in 1924, and is called a paradox as it contradicts basic geometric intuition.

An alternate version of this theorem tells us that:

“It is possible to take a solid ball the size of a pea, and by cutting it into a finite number of pieces, reassemble it to form a solid ball the size of the sun.”

Below is an awesome video explaining how this paradox works:

 

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