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I stumbled upon a problem the result of which feels very counter intuitive.

Say I have a box in n-dimension each side of the box is length 1.

The volume is defined as 1^n = 1.

Now say I want to carve out a smaller box with each side x, and x < 1 such that the volume of the smaller box is 5% of the original. i.e. x^n = 5%.

This gives x = 5% ^ (1/n).

Now as I increase the dimension from n=1 to n=10 the value of x increases from 0.05 to 0.74.

This implies that to carve out only 5% of the volume I have to carve out bigger chunks from the original box as the dimension increases. This result seems to counter intuitive. Or did I run into a logical flaw ?

Any comments or help is much appreciated.

braX
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user1612986
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    This doesn't seem programming related. Why not post it on [mathematics.se] where it would be more on topic instead of on Stack Overflow (where it is off topic)? – John Coleman Jul 01 '17 at 18:20
  • In any event, it is related to this: https://en.wikipedia.org/wiki/Curse_of_dimensionality#Distance_functions . In higher dimensions most of the volume is far away from any of the corners. – John Coleman Jul 01 '17 at 18:31
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    I'm voting to close this question as off-topic because it is about mathematics rather than computer programming. – Rory Daulton Jul 01 '17 at 18:53

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Just consider the case of the square and the cube. Rather than 5% of the space consider 25%.

In the square, if you divide the side length by half you will get half the area. In the cube if you divided each side length by half you would get an eighth of the volume. Hence you would need to divide the side lengths by somewhat more than half to get a quarter of the volume.

As dimensions increase you get more and more space. In 4D there is so much space that you can do some quite odd things.

Salix alba
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