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by Florin_Andrei
3035 days ago
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This was the first application of the Monte Carlo numerical method I've done in college. Take an N-dimensional cube with volume = 1. Inscribe an N-sphere in it. Calculate the volume of the N-sphere as a function of N. I was too lazy to do the strict proof, so I sprayed it with lots of Monte Carlo bullets. It's like a page of code in any language. It turns out, as the article says, the volume of the N-sphere keeps getting smaller and smaller as N increases. In higher dimensions, there's a lot more volume in the corners of the N-cube. It did seem like the N-sphere was shrinking down to nothing in spaces with lots of dimensions. Seems obvious after you read the article and look at the diagrams, but back then I had to think about it for a while. I thought my implementation was wrong somehow, but eventually I realized what was going on. Pretty amazing stuff. |
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Quote: "In his article “An Adventure in the Nth Dimension,” Brian Hayes explores how in high dimensions, balls have surprisingly little volume. As the dimension n increases, the volume of a ball of radius 1 increases until n = 5. Then for larger n the volume steadily decreases."
[1] https://www.johndcook.com/blog/2017/07/13/concentration_of_m...
[2] https://www.johndcook.com/blog/2017/07/19/corners-stick-out-...
[3] https://www.johndcook.com/blog/2012/10/23/dimension-5-isnt-s...